Performance Models and Risk Management in Communications Systems

PERFORMANCE MODELS AND RISK MANAGEMENT IN COMMUNICATIONS SYSTEMS Springer Optimization and Its Applications VOLUME 46...

Author: Nalân Gülpnar | Peter Harrison | Berç Rüstem

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PERFORMANCE MODELS AND RISK MANAGEMENT IN COMMUNICATIONS SYSTEMS

Springer Optimization and Its Applications VOLUME 46 Managing Editor Panos M. Pardalos (University of Florida) Editor–Combinatorial Optimization Ding-Zhu Du (University of Texas at Dallas) Advisory Board J. Birge (University of Chicago) C.A. Floudas (Princeton University) F. Giannessi (University of Pisa) H.D. Sherali (Virginia Polytechnic and State University) T. Terlaky (McMaster University) Y. Ye (Stanford University)

Aims and Scope Optimization has been expanding in all directions at an astonishing rate during the last few decades. New algorithmic and theoretical techniques have been developed, the diffusion into other disciplines has proceeded at a rapid pace, and our knowledge of all aspects of the field has grown even more profound. At the same time, one of the most striking trends in optimization is the constantly increasing emphasis on the interdisciplinary nature of the field. Optimization has been a basic tool in all areas of applied mathematics, engineering, medicine, economics and other sciences. The series Springer Optimization and Its Applications publishes undergraduate and graduate textbooks, monographs and state-of-the-art expository works that focus on algorithms for solving optimization problems and also study applications involving such problems. Some of the topics covered include nonlinear optimization (convex and nonconvex), network flow problems, stochastic optimization, optimal control, discrete optimization, multiobjective programming, description of software packages, approximation techniques and heuristic approaches.

For other titles published in this series, go to http://www.springer.com/series/7393

PERFORMANCE MODELS AND RISK MANAGEMENT IN COMMUNICATIONS SYSTEMS

By

ˆ GÜLPINAR NALAN Warwick Business School Coventry, UK PETER HARRISON Imperial College London, UK BERÇ RÜSTEM Imperial College London, UK

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Editors Nalân Gülpınar Warwick Business School The University of Warwick Coventry, CV4 7AL, UK [email protected]

Peter Harrison Department of Computing Imperial College London London, SW7 2BZ, UK [email protected]

Berç Rüstem Department of Computing Imperial College London London, SW7 2BZ, UK [email protected]

ISSN 1931-6828 ISBN 978-1-4419-0533-8 e-ISBN 978-1-4419-0534-5 DOI 10.1007/978-1-4419-0534-5 Springer New York Dordrecht Heidelberg London Mathematics Subject Classification (2010): 90B15, 90B18, 90C15, 90C90, 91A40, 93E03 Library of Congress Control Number: 2010937634 c Springer Science+Business Media, LLC 2011 All rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher (Springer Science+Business Media, LLC, 233 Spring Street, New York, NY 10013, USA), except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden. The use in this publication of trade names, trademarks, service marks, and similar terms, even if they are not identified as such, is not to be taken as an expression of opinion as to whether or not they are subject to proprietary rights. Printed on acid-free paper Springer is part of Springer Science+Business Media (www.springer.com)

Preface

Computer and telecommunication sectors are important for global dynamic economic development. The design and deployment of future networks are subject to uncertainties such as capacity prices, demand and supply for services, and shared infrastructure. Moreover, recent trends in those sectors have led to considerable increase in the level of uncertainty. Optimal policies of the operators and design of complex network functionalities require decision support methodologies that take into account uncertainty in order to improve performance, cost-effectiveness, risk and security and ensure robustness. Stochastic modeling, decision making, and game-theoretic techniques ensure the optimum end-to-end performance of general network systems. Optimal system design unifying performance modeling and decision making provides a generic approach. Real-time optimal decision making is inevitably intended to improve efficiency. Risk management injects robustness and ensures that effects of uncertainty are taken into account. The achievement of best performance may conflict with the minimization of the associated risk. Robustness in view of traffic variations, changes in network capacity, or topology, is important for both network operators and end users. A robust network allows operators to hedge against uncertainty and hence save costs and yield performance benefits to end users. Robustness can be achieved by introducing diversity at transport level and using flow or congestion control. This book considers recent developments in the design, operation, and management of telecommunication and computer network systems in performance engineering and addresses issues of uncertainty, robustness, and risk. The book consists of 10 chapters that provide a reference tool for scientists and engineers in telecommunication and computer networks. Moreover, it is intended to motivate a new wave of research in the interface of telecommunications and operations research. Coventry, UK London, UK London, UK

Nalân Gülpınar Peter Harrison Berç Rüstem

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Contents

Distributed and Robust Rate Control for Communication Networks . . . . . Tansu Alpcan

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Of Threats and Costs: A Game-Theoretic Approach to Security Risk Management . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 Patrick Maillé, Peter Reichl, and Bruno Tuffin Computationally Supported Quantitative Risk Management for Information Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 Denis Trˇcek Cardinality-Constrained Critical Node Detection Problem . . . . . . . . . . . . . . 79 Ashwin Arulselvan, Clayton W. Commander, Oleg Shylo, and Panos M. Pardalos Reliability-Based Routing Algorithms for Energy-Aware Communication in Wireless Sensor Networks . . . . . . . . . . . . . . . . . . . . . . . . . 93 Janos Levendovszky, Andras Olah, Gergely Treplan, and Long Tran-Thanh Opportunistic Scheduling with Deadline Constraints in Wireless Networks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127 David I Shuman and Mingyan Liu A Hybrid Polyhedral Uncertainty Model for the Robust Network Loading Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157 Ay¸segül Altın, Hande Yaman, and Mustafa Ç. Pınar Analytical Modelling of IEEE 802.11e Enhanced Distributed Channel Access Protocol in Wireless LANs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173 Jia Hu, Geyong Min, Mike E. Woodward, and Weijia Jia vii

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Contents

Dynamic Overlay Single-Domain Contracting for End-to-End Contract Switching . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191 Murat Yüksel, Aparna Gupta, and Koushik Kar Modelling a Grid Market Economy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225 Fernando Martínez Ortuño, Uli Harder, and Peter Harrison

Contributors

Tansu Alpcan Deutsche Telekom Laboratories, Technical University of Berlin, Ernst-Reuter-Platz 7, Berlin 10587 Germany, [email protected] Ay¸segül Altın Department of Industrial Engineering, TOBB University of Economics and Technology, Sö˘gütözü 06560 Ankara, Turkey, [email protected] Ashwin Arulselvan Center for Discrete Mathematics and Applications, Warwick Business School, University of Warwick, Coventry, UK, [email protected] Clayton W. Commander Air Force Research Laboratory, Munitions Directorate, and Department of Industrial and Systems Engineering, University of Florida, Gainesville, FL, USA, [email protected] Aparna Gupta Rensselaer Polytechnic Institute, Troy, NY 12180, USA, [email protected] Uli Harder Department of Computing, Imperial College London, Huxley Building, 180 Queens Gate, London SW7 2AZ, UK, [email protected] Peter Harrison Department of Computing, Imperial College London, Huxley Building, 180 Queens Gate, London SW7 2AZ, UK, [email protected] Jia Hu Department of Computing, School of Informatics, University of Bradford, Bradford, BD7 1DP, UK, [email protected] Weijia Jia Department of Computer Science, City University of Hong Kong, 83 Tat Chee Ave, Hong Kong, [email protected] Koushik Kar Rensselaer Polytechnic Institute, Troy, NY 12180, USA, [email protected] Janos Levendovszky Budapest University of Technology and Economics, Department of Telecommunications, H-1117 Magyar tud. krt. 2, Budapest, Hungary, [email protected] Mingyan Liu Electrical Engineering and Computer Science Department, University of Michigan, Ann Arbor, MI 48109, USA, [email protected] ix

x

Contributors

Patrick Maillé Institut Telecom; Telecom Bretagne, 2 rue de la Châtaigneraie CS 17607, 35576 Cesson-Sévigné Cedex, France, [email protected] Geyong Min Department of Computing, School of Informatics, University of Bradford, Bradford, BD7 1DP, UK, [email protected] Andras Olah Faculty of Information Technology, Peter Pazmany Catholic University, H-1083 Práter u. 50/A Budapest, Hungary, [email protected] Fernando Martínez Ortuño Department of Computing, Imperial College London, Huxley Building, 180 Queens Gate, London SW7 2AZ, UK, [email protected] Panos M. Pardalos Center for Applied Optimization, Department of Industrial and Systems Engineering, University of Florida, Gainesville, FL, USA, [email protected] Mustafa Ç. Pınar Department of Industrial Engineering, Bilkent University, 06800 Ankara, Turkey, [email protected] Peter Reichl Telecommunications Research Center Vienna (ftw.), Donau-City-Str. 1, 1220 Wien, Austria, [email protected] David I Shuman Electrical Engineering and Computer Science Department, University of Michigan, Ann Arbor, MI 48109, USA, [email protected] Oleg Shylo Center for Applied Optimization, Department of Industrial and Systems Engineering, University of Florida, Gainesville, FL, USA, [email protected] Long Tran-Thanh Budapest University of Technology and Economics, Department of Telecommunications, z, H-1117 Magyar tud. krt. 2, Budapest, Hungary, [email protected] Denis Trˇcek Faculty of Computer and Information Science, Laboratory of E-media , University of Ljubljana, Tržaška cesta 25, 1000 Ljubljana, Slovenia, [email protected] Gergely Treplan Faculty of Information Technology, Peter Pazmany Catholic University, H-1083 Práter u. 50/A Budapest, Hungary, [email protected] Bruno Tuffin INRIA Rennes – Bretagne Atlantique, Campus Universitaire de Beaulieu, 35042 Rennes Cedex, France, [email protected] Mike E. Woodward Department of Computing, School of Informatics, University of Bradford, Bradford, BD7 1DP, UK, [email protected] Hande Yaman Department of Industrial Engineering, Bilkent University, 06800 Ankara, Turkey, [email protected] Murat Yüksel University of Nevada - Reno, Reno, NV 89557, USA, [email protected]

Distributed and Robust Rate Control for Communication Networks Tansu Alpcan

1 Introduction Wired and wireless communication networks are an ubiquitous and indispensable part of the modern society. They serve a variety of purposes and applications for their end users. Hence, networks exhibit heterogeneous characteristics in terms of their access infrastructure (e.g., wired vs. wireless), protocols, and capacity. Moreover, contemporary networks such as the Internet are heavily decentralized in both their administration and resources. The end users of communication networks are also diverse and run a variety of applications ranging from multimedia (VoIP, video) to gaming and data communications. As a result of the networks’ distributed nature, users often have little information about the network topology and characteristics. Regardless, they can behave selfishly in their demands for bandwidth. Given the mentioned characteristics of the contemporary networks and their users, a fundamental research question is, how to ensure efficient, fair, and incentivecompatible allocation of network bandwidth among its users. Complicating the problem further, the mentioned objectives have to be achieved through distributed algorithms while ensuring robustness with respect to information delays and capacity changes. This research challenge can be quite open-ended due to the multifaceted nature of the underlying problems.

1.1 Summary and Contributions This chapter presents three control and game-theoretic approaches that address the described rate control problem from different perspectives. The objective here is to investigate the underlying mathematical principles of the problem and solution concepts rather than discussing possible implementation scenarios. However, it is Tansu Alpcan Deutsche Telekom Laboratories, Technical University of Berlin, Ernst-Reuter-Platz 7, Berlin 10587, Germany e-mail: [email protected]

N. Gülpınar et al. (eds.), Performance Models and Risk Management in Communications Systems, Springer Optimization and Its Applications 46, C Springer Science+Business Media, LLC 2011 DOI 10.1007/978-1-4419-0534-5_1,

1

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T. Alpcan

hoped that the rigorous mathematical analysis presented will be useful as a basis for engineering future rate control schemes. A noncooperative rate control game is presented in Section 3. Adopting a utilitybased approach, user preferences are captured by a fairly general class of cost functions [4]. Based on their own utility functions and external prices, the users (players) of this game use a standard gradient algorithm to update their flow rates iteratively over time, resulting in an end-to-end congestion control scheme. The game admits a unique Nash equilibrium under a sufficient condition, where no user has an incentive to deviate from it. Furthermore, a mild symmetricity assumption and a sufficient condition on maximum delay ensure its global stability with respect to the gradient algorithm for general network topologies and under fixed heterogeneous delays. The upper bound on communication delays given in the sufficient condition is inversely proportional to the square root of the number of users sharing a link multiplied by the cube of a gain constant. Section 4 studies a primal–dual rate control scheme to solve a global optimization problem, where each user’s cost function is composed of a pricing function proportional to the queueing delay experienced by the user, and a fairly general utility function which captures the user’s demand for bandwidth [5]. The global objective is to maximize the sum of user utilities under fixed capacity constraints. Using a network model based on fluid approximations and through a realistic modeling of queues, the existence of a unique equilibrium is established, at which the global optimum is achieved. The scheme is globally asymptotically stable for a general network topology. Furthermore, sufficient conditions for system stability are derived when there is a bottleneck link shared by multiple users experiencing non-negligible communication delays. A robust flow control framework is introduced in Section 5. It is based on an H∞ -optimal control formulation for allocating rates to devices on a network with heterogeneous time-varying characteristics [6]. H∞ methods are used in control theory to synthesize controllers achieving robust performance or stabilization. Here, H∞ analysis and design allow for the coupling between different devices to be relaxed by treating the dynamics for each device independently from others. Thus, the resulting distributed end-to-end rate control scheme relies on minimum information and achieves fair and robust rate allocation for the devices. In the fixed capacity case, it is shown that the equilibrium point of the system ensures full capacity usage by the users. The formulations presented in Sections 3, 4, and 5 are, on the one hand, closely related to each other. Each approach mainly shares the same common network model, which will be discussed in Section 2. Furthermore, they are totally distributed, end-to-end schemes with little information exchange overhead. All of the schemes are robust with respect to information delays in the system and their stability properties are analyzed rigorously. On the other hand, each approach brings the problem of rate allocation a different perspective. The rate control game of Section 3 focuses mainly on incentive compatibility and adopts Nash equilibrium as the preferred solution concept. The primal–dual scheme of Section 4 extends the basic fluid network model by taking into account the queue dynamics and is built upon available information to users for

Distributed and Robust Rate Control

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decision making. The robust rate control scheme of Section 5 emphasizes robustness with respect to capacity changes and delays. It also differs from the previous two, which share a utility-based approach, by focusing on fully efficient usage of the network capacity. The remainder of the chapter is organized as follows. A brief overview of existing relevant literature is discussed next. The network model and its underlying assumptions are presented in Section 2. Section 3 studies a rate control game along with an equilibrium and stability analysis under information delays. In Section 4, a primal– dual scheme is investigated. Section 5 presents a robust control framework and its analysis. The chapter concludes with remarks in Section 6.

1.2 Related Work The research community has intensively studied the challenging problem of rate and congestion control in the recent years. Consequently, a rich literature, which investigates the problem using a variety of approaches, has emerged. While it is far from a comprehensive survey, a small subset of the existing body of literature on the subject is summarized here as a reference. After the introduction of the congestion control algorithm for transfer control protocol (TCP) [18], research community has focused on modeling and analysis of rate control algorithms. Based on an earlier work by Kelly [20], Kelly et al. [21] have presented the first comprehensive mathematical model and posed the underlying resource allocation problem as one of constrained optimization. The primal and dual algorithms that they have introduced are based on user utility and link pricing (explicit congestion feedback) functions, where the sum of user utilities are maximized within the capacity (bandwidth) constraints of the links. They have also introduced the concept of proportional fairness, which is a relaxed version of min–max fairness [31], as a resource allocation criterion among users. Subsequent studies [14, 23, 24, 26, 29] have investigated variations and generalizations of the distributed congestion control framework of [20, 21]. Low and Lapsley [26] have analyzed the convergence of distributed synchronous and asynchronous discrete algorithms, which solve a similar optimization problem. Mo and Walrand [29] have generalized the proportional fairness and have proposed a fair end-to-end window-based congestion control scheme, which is similar to the primal algorithm. The main difference of this window-based algorithm from the primal algorithm is that it does not need explicit congestion feedback from the routers. Instead it makes use of measured queuing delay as implicit congestion feedback. La and Anantharam [24] have considered a system model similar to proposed in [29] with a window-based control scheme and static modeling of link buffers. They have investigated convergence properties of the proposed charge-sensitive congestion control scheme, which utilizes a static pricing scheme based on link queueing delays. In addition, they have established stability of the algorithm at a single bottleneck node. Kunniyur and Srikant [23] have examined the question of how to provide congestion feedback from the network to the user. They have proposed an explicit congestion notification (ECN) marking scheme combined with dynamic

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adaptive virtual queues and have shown using a timescale decomposition that the system is semi-globally stable in the no-delay case. In developing rate control mechanisms for the Internet, game theory provides a natural framework. Users on the network can be modeled as players in a rate control game where they choose their strategies or, in this case, flow rates. Players are selfish in terms of their demands for network resources and have no specific information on other users’ strategies. A user’s demand for bandwidth is captured in a utility function which may not be bounded. To compensate for this, one can devise a pricing function, proportional to the bandwidth usage of a user, as a disincentive to him to have excessive demand for bandwidth. This way, the network resources are preserved and an incentive is provided for the user to implement end-to-end congestion control [16]. A useful concept in such a noncooperative congestion control game is Nash equilibrium [10] where each player minimizes its own cost (or maximizes payoff) given all other players’ strategies. There is, consequently, a rich literature on game-theoretic analysis of flow control problems utilizing both cooperative [34] and noncooperative [1–3, 5, 7, 8, 30] frameworks. Robustness of distributed rate control algorithms with respect to delays in the network have been investigated by many studies [19, 27, 32]. Johari and Tan [19] have analyzed the local stability of a delayed system where the end user implements the primal algorithm. They have considered a single link accessed by a single user, as well as its multiple user extension under the assumption of symmetric delays. In both cases, they have provided sufficient conditions for local stability of the underlying system of equations. Massoulie [27] has extended these local stability results to general network topologies and heterogeneous delays. In another study, Vinnicombe [32] has also provided sufficient conditions for local stability of a user rate control scheme which is a generalization of the same algorithm. Elwalid [15] has considered stability of a linear class of algorithms where the source rate varies in proportion to the difference between the buffer content and the target value. Deb and Srikant [14], on the other hand, have focused on the case of single user and a single resource and investigated sufficient conditions for global stability of various nonlinear congestion control schemes under fixed information delays. Liu et al. [25] have extended the framework of Kelly and Coworkers [20, 21] by introducing a primal–dual algorithm which has dynamic adaptations at both ends (users and links) and have given a condition for its local stability under delay using the generalized Nyquist criterion. Wen and Arcak [33] have used a passivity framework to unify some of the stability results on primal and dual algorithms without delay, have introduced and analyzed a larger class of such algorithms for stability, and have shown robustness to variations due to delay.

2 Network Model A general network model is considered which is based on fluid approximations. Fluid models are widely used in addressing a variety of network control problems, such as congestion control [1, 7, 29], routing [7, 30], and pricing [11, 21, 34]. The


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topology of the network is characterized by a connected graph consisting of a set of nodes N = {1, . . . , N } and a set of links L = {1, . . . , L}. Each link l ∈ L has a fixed capacity Cl > 0 and is associated with a buffer of size bl ≥ 0. There are M active users sharing the network, M = {1, . . . , M}. For simplicity, each user is associated with a (unique) connection. Hence, the ith (i ∈ M) user corresponds to a unique connection between a source and a destination node, si , dei ∈ N . The ith user sends its nonnegative flow, xi ≥ 0, over its route (path) Ri , which is a subset of L. An upper bound, xi,max , is imposed on the ith users flow rate, which may be due to a user (device)-specific physical limitation. Define a routing matrix, A := [(al,i )] of ones and zeros, as in [21] which describes the relation between the set of routes R = {1, . . . , M} associated with the users (connections), i ∈ M, and links l ∈ L : 1, if source i uses link l Al,i = . (1) 0, if source i does not use link l It is assumed here, without any loss of generality, that no rows or columns in A are identically zero. Using this routing matrix A, the capacity constraints of the links are given by Ax ≤ C, where x is the (M ×1) flow rate vector of the users and C is the (L ×1) link capacity vector. The flow rate vector, x, is said to be feasible if it is nonnegative and satisfies this constraint. Let x−i be the flow rate vector of all users except the ith one. For a given fixed, feasible x−i , there exists a strict finite upper bound m i (x−i ) on flow rate of the ith user, x i , based on the capacity constraints of the links: m i (x−i ) = min(Cl − l∈Ri

Al, j x j ) ≥ 0 .

j=i

2.1 Model Assumptions Simplifying assumptions are necessary to develop a mathematically tractable model. The assumptions of this chapter are shared by the majority of the literature on the subject, including the works cited in Section 1. Furthermore, the analytical results obtained based on these assumptions are verified many times via realistic packetlevel simulations in the literature. The main assumptions on the network model are summarized as follows: 1. The network model is based on fluid approximations, where individual packets are replaced with flows. Fluid models are widely used in addressing a variety of

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2.

3. 4. 5.

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network control problems, such as congestion control [1, 7, 29], routing [7, 30], and pricing [11, 21, 34]. For simplicity, each user is associated with a unique connection and a corresponding fixed route (path). The routing matrix A is assumed to be of full row rank as non-bottleneck links have no effect on the equilibrium point due to zero queuing delay on those links. Bandwidth is focused on as the main network resource. Information delays are assumed to be fixed for tractability of analysis. The links are associated with first-in first-out (FIFO) finite queues (buffers) with droptail packet dropping policies.

Additional assumptions, being part of the specific optimization or game formulations, are explicitly introduced and discussed in their respective sections.

3 Rate Control Game A noncooperative rate control game is played among M users on the general network model, which is described in the previous section. The game is noncooperative as the users are assumed to be selfish in terms of their demand for bandwidth and have no means of communicating with each other about their preferences. Hence, each user tries to optimize his usage of the network independently by minimizing its own specific cost function Ji . This cost function is defined on the compact, continuous set of feasible rates of users, x := {x ⊂ R M : x ≥ 0, Ax ≤ C}. The cost function Ji not only models the user preferences but also includes a feedback term capturing the current network state. Thus, the ith user minimizes his cost, Ji , by adjusting his flow rate 0 ≤ xi ≤ m i (x−i ) given the fixed, feasible flow rates of all other users on its path, {x j : j ∈ (R j ∩ Ri )}. The cost function of the ith user, Ji , is defined as the difference between a user-specific pricing function, Pi , and a utility function, Ui . It is smooth, i.e., at least twice-continuously differentiable in all its arguments. The pricing function Pi depends on the current state of the network and can be interpreted as the price a user pays for using the network resources. The utility function Ui is defined to be increasing and concave in accordance with elastic traffic as well as with the economic principle, law of diminishing returns. The utility of each user depends only on its own flow rate. Thus, the cost function is defined as Ji (x; C, A) = Pi (x; C, A) − Ui (xi ).

(2)

Here, the pricing function Pi of user i does not necessarily depend on the flow rates of all other users; it can be structured to depend only on the flow rates of the users sharing the same links on the path of the ith user. The rate control game defined proposes “pricing” as a way to enforce a more favorable outcome for the system and users. If there is no pricing scheme, then the increasing and concave user utilities result in a solution where each user sends


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with the maximum possible rate. In practice, this would lead to a congestion collapse or “tragedy of commons.” To remedy this, the function P, proportional to the bandwidth usage of a user, is utilized as an incentive for the user to curb excessive demand. Thus, the network resources are preserved, and an incentive is provided for the user to implement end-to-end congestion control. On the other hand, the analysis in this section focuses on mathematical principles rather than architectural concerns or possible implementation of such pricing schemes.

3.1 Nash Equilibrium as a Unique Solution The defined rate control game may admit a (unique) Nash equilibrium (NE) as a solution. In this context, Nash equilibrium is defined as a set of flow rates, x∗ (and corresponding costs J ∗ ), with the property that no user can benefit by modifying its flow while the other players keep their flows fixed. Furthermore, if the Nash equilibrium, x∗ , meets the capacity constraints (e.g., Ax∗ ≤ C) as well as the positivity constraint (x∗ ≥ 0) with strict inequality, then it is an inner solution. Definition 1 (Nash Equilibrium) The user flow rate vector x∗ is in Nash equilibrium, when xi∗ of any ith user is the solution to the following optimization problem given that all users on its path have equilibrium flow rates, x∗−i : min

0≤xi ≤m i (x−i ∗ )

∗ Ji (xi , x−i , C, A) ,

(3)

where x−i denotes the collection {x j : j ∈ R j ∩ Ri } j=1,...,M . The assumptions on the user cost functions are next formalized: A1. Pi (x) is jointly continuous in all its arguments and twice continuously differentiable, non-decreasing, and convex in xi , i.e. ∂ Pi (x) ≥ 0, ∂ xi

∂ 2 Pi (x) ≥ 0. ∂ xi2

(4)

A2. U (xi ) is jointly continuous in all its arguments and twice continuously differentiable, non-decreasing, and strictly concave in xi , i.e. ∂Ui (xi ) ≥ 0, ∂ xi

∂ 2 Ui (xi ) ∂ xi2

0, ∀l. Hence, there exists at least one positive and feasible flow rate vector in the set X , which is an interior point. Thus, the set X has a nonempty interior. Let x1 , x2 ∈ X be two feasible flow rate vectors and 0 < λ < 1 be a real number. For any xλ := λx1 + (1 − λ)x2 , it follows that Axλ = A(λx1 + (1 − λ)x2 ) ≤ C. Furthermore, xλ ≥ 0 by definition. Hence, xλ is feasible and is in X for any 0 < λ < 1. Thus, the set X is convex. By a standard theorem of game theory (Theorem 4.4 p. 176 in [10]), the network game admits an NE. Next, uniqueness of the NE is shown. Differentiating (2) with respect to x i and using assumptions A1 and A2 results in f i (x) :=

∂ Ji (x) ∂ Pi (x) ∂Ui (xi ) = − . ∂ xi ∂ xi ∂ xi

(5)

As a simplification of notation, C and A are suppressed as arguments of the functions for the rest of this proof. Differentiating Ji (x) twice with respect to xi yields


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∂ fi (x) ∂ 2 Ji (x) ∂ 2 Pi (x) ∂ 2 Ui (xi ) = = − > 0. ∂ xi ∂ xi2 ∂ xi2 ∂ xi2 Hence, Ji is unimodal and has a unique minimum. Based on A3, f i (x) attains the zero value at m i (x−i ) > x i > 0 given a fixed feasible x−i . Thus, the optimization problem (3) admits a unique positive solution. 2 be denoted by Bi . Further introduce, for i, j ∈ To preserve notation, let ∂ J (x) 2 ∂ xi

M, j = i,

∂ 2 Ji (x) ∂ 2 Pi (x) = =: Ai, j , ∂ xi ∂ x j ∂ xi ∂ x j with both Bi and Ai, j defined on the space where x is nonnegative and bounded by the link capacities. Suppose that there are two Nash equilibria, represented by two flow vectors x0 and x1 , with elements xi0 and xi1 , respectively. Define the pseudogradient vector:

T g(x) := ∇x1 J1 (x)T · · · ∇x M JM (x)T .

(6)

As the Nash equilibrium is necessarily an inner solution, it follows from firstorder optimality condition that g(x0 ) = 0 and g(x1 ) = 0. Define the flow vector x(θ ) as a convex combination of the two equilibrium points x0 and x1 : x(θ ) = θ x0 + (1 − θ )x1 , where 0 < θ < 1. By differentiating x(θ ) with respect to θ , dg(x(θ )) d x(θ ) = G(x(θ )) = G(x(θ ))(x1 − x0 ) , dθ dθ

(7)

where G(x) is the Jacobian of g(x) with respect to x : ⎛

B1 A12 ⎜ G(x) := ⎝ ... A M1 A M2

⎞ · · · A1M . ⎟ .. . . .. ⎠ · · · B M M×M

Additionally note that, by assumption A4 l∈(Ri ∩R j )

∂ 2 Jl (x) = ∂ xi ∂ x j

l∈(Ri ∩R j )

∂ 2 Jl (x) ∂ xi ∂ x j

⇒ A(i, j) = A( j, i) i, j ∈ M .

(8)

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Hence, G(x) is symmetric. Integrating (7) over θ ,

0 = g(x ) − g(x ) = 1

0

1

G(x(θ ))dθ (x1 − x0 ) ,

(9)

0

1 where (x1 − x0 ) is a constant flow vector. Let Bi (x) = 0 Bi (x(θ ))dθ and 1 Ai j (x) = 0 Ai j (x(θ ))dθ . In view of A2 and A4, Bi (x) > Ai j (x) > 0 , ∀i, j. Thus, Bi (x) > Ai j (x) > 0, for any x(θ ). In order to simplify the notation, define 1 the matrix G(x1 , x0 ) := 0 G(x(θ ))dθ , which can be shown to be full rank for any fixed x. Rewriting (9) as, 0 = G · [x1 − x0 ], since G is full rank, it readily follows that x1 − x0 = 0. Therefore, the NE is unique. Under A3, the NE has to be an inner solution, as the following argument shows. First, x ≥ 0, with xi = 0 for at least one i, cannot be an equilibrium point since user i can decrease its cost by increasing its flow rate. Similarly, the boundary points {x ∈ R M : Ax ≤ C , x ≥ 0, with (Ax)l = Cl for at least one link l} cannot constitute NE, as users whose flows pass through the link can decrease their flow rates under A3. Thus, under A1–A4, the network game admits a unique inner NE.

3.2 Stability Analysis Consider a simple dynamic model of the defined rate control game where each user changes his flow rate in proportion with the gradient of his cost function with respect to his flow rate. Note that this corresponds to the well-known steepest descent algorithm in nonlinear programming [12]. Hence, the user update algorithm is d xi (t) dUi (xi ) ∂ Ji (x(t)) = − fl =− x˙i (t) = dt ∂ xi d xi l∈Ri

xj

:= θi (x),

(10)

j∈Ml

for all i = 1, . . . , M, where Ml (Ml ) is the set (number) of users whose flows pass through the link, l ∈ Ri ; t is the time variable, which we drop in the second line for a more compact notation; and fl is defined as fl (·) := ∂ Pl (·)/∂ xi . By assumption A4, the partial derivative of fl with respect to xi , ∂ fl (·)/∂ xi , is non-negative. Furthermore, since Pl (x) is convex and jointly continuous in xi for all i whose flows pass through the link l, on the compact set of feasible flow rate vectors, X := {x ∈ R M : Ax ≤ C , x ≥ 0}, the derivative ∂ fl (.)/∂ xi can be bounded above by a constant αl > 0. Hence, 0≤ where x¯l =

i∈Ml

xi .

∂ fl (x¯l ) ≤ αl , ∂ xi

(11)


11

It is now shown that the system defined by (10) is asymptotically stable on the set X , which is invariant by assumption A3 under the gradient update algorithm (10). In order to see the invariance of X , each boundary of X is investigated separately. If xi = 0 for some i ∈ M, then x˙i > 0 follows from (10) under assumption A3 due to the gradient descent algorithm of user i. Hence, the system trajectory moves toward inside of X . Likewise, in the case of x¯l = Cl for some l ∈ L, it follows from (10) and assumption A3 that x˙i < 0 ∀i ∈ Ml , and hence, the trajectory remains inside the set X . The unique inner NE, x∗ (see Theorem 1), of the rate control game constitutes the equilibrium state of the dynamical system (10) in X . Around this equilibrium, define a candidate Lyapunov function V : R M → R+ as 1 2 θi (x), 2 M

V (x) :=

i=1

which is in fact restricted to the domain X . Further let Θ := [θ1 , . . . , θ M ]. Taking the derivative of V with respect to t on the trajectories generated by (10), one obtains V˙ (x) =

M d 2 Ui (xi ) i=1

d xi2

θi2 (x) − Θ T (x)A T K AΘ(x),

where A is the routing matrix, and K is a diagonal matrix defined as ∂fM (x) ∂f1 (x) ∂f2 (x) , ,..., . K := diag ∂x ∂x ∂x

Since A T K A is non-negative definite and d 2 Ui /d xi2 is uniformly negative definite, V (x) is strictly decreasing, V˙ (x) < 0, on the trajectory of (10). Thus, the system is asymptotically stable on the invariant set X by Lyapunov’s stability theorem (see Theorem 3.1 in [22]). Theorem 2 Assume A1–A4 hold. Then, the unique inner Nash equilibrium of the network game is globally stable on the compact set of feasible flow rate vectors, X := {x ∈ R M : Ax ≤ C , x ≥ 0} under the gradient algorithm given by x˙i = −

∂ Ji (x) , ∂ xi

i = 1, . . . , M.

3.3 Stability under Information Delays Whether the user rate control (gradient) algorithm (10) is robust with respect to communication delays is an important question. This section investigates the rate

12

T. Alpcan

control scheme under bounded and heterogeneous communication delays. The distributed rate control algorithm under communication delays is defined as dUi (xi (t)) x˙i (t) = − fl d xi

x j (t − rli − rl j ) ,

(12)

j∈Ml

l∈Ri

where rli and rl j are fixed communication delays between the lth link and the ith and jth users, respectively. It is implicitly assumed here that queueing delays are negligible compared to the fixed propagation delays in the system. 3.3.1 Notation and Definitions Notice again that, fl is defined as fl (·) := ∂ Pl (·)/∂ xi and the pricing function of the ith user is defined in accordance with assumption A4 as Pi =

Pl (

x j ),

j∈Ml

l∈Ri

where Ri is the path (route) of user i and Pl is the pricing function at link l ∈ L. The notation is simplified by defining

x¯li (t − r ) :=

x j (t − rli − rl j ).

j∈Ml

In addition, let q be an upper bound on the maximum round-trip time (RTT) in the system q := 2 max i

rli − r(l−1)i ,

l∈Ri

where r0i = 0 ∀i. Finally, define xt := {x(t + s), −q ≤ s ≤ 0}, and by a slight abuse of notation let θi (xt ) denote the right-hand side of (12). Let φi ∈ C [−ri , 0], R be a feasible flow rate function (initial condition) for the ith user’s dynamics (12) at time t = 0, where C is the set of continuous functions. In addition, let x(φ)(t) be the solution of (12) through φ for t ≥ 0, and x˙ (φi )(t) be its derivative. In order to simplify the notation, x(φ) and x as well as θ (φ) and θ and their respective derivatives will be used interchangeably for the remainder of the chapter. Finally, a continuously differentiable and positive function V : C M → R+ is defined as 1 2 1 θi (xt (φ)) = Θ T (xt (φ))Θ(xt (φ)). 2 2 M

V (xt (φ)) :=

i=1


13

This constitutes the basis for the following candidate Lyapunov functional V : R+ × C M → R+ , V (t; φ) :=

sup

V (xs (φ)),

t−2q≤s≤t

where V (xs ) = 0, s ∈ [−2q, −q] without any loss of generality. 3.3.2 Stability Analysis Under Delays In order to establish global stability under delays, it is shown that the Lyapunov functional V (t; φ) is non-increasing. Furthermore, the stability theory for autonomous systems of [17] is utilized to generalize the scalar analysis of [35] and also of Chapter 5.4 of [17] to the multidimensional (multi-user) case. Let V˙ (t; φ) and V˙ (t; φ) be defined as the upper right-hand derivatives of V (t; φ) and V (t; φ) respectively along xt (φ). In order for V (t; φ) to be non-increasing, i.e., V˙ (t; φ) ≤ 0, the set Φ = {φ ∈ C : V (t; φ) = V (xt (φ)); V˙ (xt (φ)) > 0 ∀t ≥ 0}

(13)

has to be empty. This is established in the following lemma. Lemma 1 The set Φ, defined in (13), is empty if the following condition is satisfied 2 d¯ q≤√ , Mb3/2 where b := max i

d 2 Ui (xi ) − min + Ml αl , x i ∈X d xi2 l∈R i

and d 2 U (x ) i i d¯ := min min . i xi ∈X d xi2 Proof To see this consider the case when the set Φ is not empty. Then, by definition, there exists a time t and an h > 0 such that V˙ (xt+h (φ)) > V˙ (xt (φ)), and hence, V˙ (xt (φ)) cannot be non-increasing. It is now shown that the set Φ is indeed empty. Assume otherwise. Then, for any given t, there exists an ε > 0 such that

14

T. Alpcan

V (t; φ) = V (xt (φ)) =

M

θi2 (xt (φ)) = ε

(14)

i=1

and M

V (xs (φ)) =

θi2 (xs ) ≤ ε , s ∈ [t − 2q , t].

i=1

Thus, the following bound on θi , and thus on x˙i , follows immediately: |θi (xs )| = |x˙i (s)| ≤

√

ε , s ∈ [t − 2q , t].

(15)

Taking the derivative of x˙i (t) with respect to t, xï (t) =

∂ fl (x¯ i (t − r )) ∂ x˙i (t) d 2 Ui (xi ) l x ˙ (t)− x˙ j (t −ri −r j ). = θ˙i (xt ) = i ∂t d xi2 ∂ x¯li l∈R j∈M i

l

(16) d 2 Ui (xi ) Let di := − minxi ∈X > 0. Using (15) and (16), it is possible to bound d xi2 θ˙i (xs ) and xï (s) on s ∈ [t − q, t] with |θ˙i (xs )| = |xï (s)| ≤ di |x˙i (s)|+

∂ fl (x¯ i (s − r )) l

l∈Ri

∂ x¯li

|x¯li (s − r )| ≤ (di +

√ Ml αl ) ε.

l∈Ri

(17) To simplify the notation, define yi := di +

Ml αl .

l∈Ri

Hence, the following bound on θi (xs ), s ∈ [t − q, t] is obtained: √ √ θi (xt ) − qyi ε ≤ θi (xs ) ≤ θi (xt ) + qyi ε.

(18)

Subsequently, it is shown that V (xt (φ)) is non-increasing, and a contradiction is obtained to the initial hypothesis that the set Φ is not empty. Assume that ∂ fl (x¯li (t − j j r ))/∂ x¯li = ∂ fl (x¯l (t − r ))/∂ x¯l , ∀i, j ∈ Ml , ∀t for each link l. This assumption holds, for example, when fl is linear in its argument. Let B be defined in such a way that B T B := A T K A, where the positive diagonal matrix K is defined in Section 3.2. Also define the positive diagonal matrix D(x) := diag [|D1 (x1 )| , |D2 (x2 )| , . . . , |DM (xM )|] ,


15

where Di (x) := d 2 Ui (xi )/d xi2 . Then, using (18), V˙ (xt ) = −

M

Di (xi )θi2 (xt ) −

i=1

M

θi (xt )·

∂ fl (x¯ i (t − r )) l ∂ x¯li

l∈Ri

i=1

θ j (xt−rli −rl j )

j∈Ml

√ ≤ −Θ T DΘ − Θ T B T BΘ + q ε|Θ T B T By|, (19) where everything is evaluated at t. Now, for any fixed trajectory generated by (12), and for a frozen time t, a sufficient condition for V˙ (xt ) ≤ 0 is √ ||BΘ||2 + || DΘ||2 q ε≤ , ||BΘ|| ||By|| √

where || · || is the Euclidean norm. Let k := ||BΘ|| ||By|| > 0. Rewriting the sufficient condition one obtains √ 1 q ε ≤ k + μ, k √

DΘ|| where μ := || ||By|| > 0. The following worst-case bound on q can be derived by 2 a simple minimization: 2

√ √ q ε ≤ 2 μ.

(20)

√ Next a lower bound on μ is derived. From (14), it follows that || DΘ(xt )||2 ≥ 2 √ i) ¯ where d¯ := mini minxi ∈X d Ui (x dε, d xi2 , and D is the unique positive definite matrix whose square is D. Furthermore, ||By||2 ≤

M

yi

i=1

αl

l∈Ri

yj.

j∈Ml

Define also the following upper bound on yi : b := max di + Ml αl . i

l∈Ri

Since di > 0, one obtains ||By||2 ≤ Mb3 , and hence μ≥

¯ dε . Mb3

Thus, from (20) a sufficient condition for V (xt ) to be non-increasing is

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T. Alpcan

2 d¯ q≤ √ , Mb3/2

(21)

which now holds for all t ≥ 0.

˙ Based on Lemma 1, V (t; φ) is non-increasing, V (t; φ) ≤ 0. Then, using Definition 3.1 and Theorem 3.1 of [17] global asymptotic stability of system (12) is established. Let S := {φ ∈ C : V˙ (t; φ) = V˙ (xt (φ)) = 0}. From (12) and (19) it follows that S = {φ ∈ C : φ(τ ) = x∗ , −q ≤ τ ≤ 0} ⊂ S, as Θ(xτ ) = x˙ (τ ) = 0 ⇔ xτ = x∗ ⇒ V˙ (xτ ) = 0.

Hence, S is the largest invariant set in S, and for any trajectory of the system that belongs identically to S, we have xτ = x∗ . In other words, the only solution that can stay identically in S is the unique equilibrium of the system. This, then leads to the following theorem: Theorem 3 Assume that ∂ f l (x¯li (s − r )) ∂ x¯li

j

=

∂ fl (x¯l (s − r )) j

∂ x¯l

∀i, j ∈ Ml ∀t.

Then, the unique Nash equilibrium of the network game is globally asymptotically stable on the compact set of feasible flow rate vectors, X := {x ∈ R M : Ax ≤ C , x ≥ 0} under the gradient algorithm x˙i (t) =

dUi (xi (t)) − fl d xi l∈Ri

x j (t − rli − rl j ) ,

j∈Ml

in the presence of fixed heterogeneous delays, rli ≥ 0, for all users i = 1, . . . , M, and links l ∈ L, if the following condition is satisfied 2 d¯ , q≤√ Mb3/2 where b := max i

− min

x i ∈X

d 2 Ui (xi ) + M α l l , d xi2 l∈R i

and d 2 U (x ) i i d¯ := min min . i xi ∈X d xi2


17

If the user reaction function is scaled by a user-independent gain constant, λ, then the ith user’s response is given by x˙i = −λ

∂ Ji (x(t)) , ∂ xi

and the sufficient condition for global stability turns out to be q≤√

2 d¯ Mλ3/2 b3/2

.

Notice that, for any λ < 1, the upper bound on maximum RTT, q, is relaxed proportionally with λ3/2 . The upper bound on communication delays given in the sufficient condition of Theorem 3 is inversely proportional to the square root of the number of users multiplied by the cube of a gain constant. This structure is actually similar to those of local stability results reported in other studies [19, 27, 32]. The analysis above indicates a fundamental trade-off between the responsiveness of the users gradient rate control algorithm and the stability properties of the system under communication delays.

4 Primal–Dual Rate Control The distributed structure of the Internet makes it difficult, if not impossible, for users to obtain detailed real time information on the state of the network. Therefore, users are bound to use indirect aggregate metrics that are available to them, such as packet drop rate or variations in the average round trip time (RTT) of packets in order to infer the current situation in the network. Packet drops, for example, are currently used by most widely deployed versions of TCP as an indication of congestion. An approach similar to the one discussed in this section has been suggested in a version of TCP, known as TCP Vegas [13]. Although TCP Vegas is more efficient than a widely used version of TCP, TCP Reno [28], the suggested improvements are empirical and based on experimental studies. This section presents and analyzes a primal–dual rate control scheme based on variations in the RTT a user experiences based on [5]. Although users are associated with cost functions in a way similar to the game in Section 3, the formulation here is not a proper game as the users ignore their own effects on the outcome when making decisions. Consequently, the solution here is different from the concept of Nash equilibrium. The equilibrium solution discussed in this section maximizes the sum of user utilities under capacity constraints. The result immediately follows from a Lagrangian analysis and the concept of shadow prices. Furthermore, the solution becomes proportionally fair under logarithmic user utilities. A detailed analysis can be found in [20, 21].

18

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4.1 Extended Network Model An important indication of congestion for Internet-style networks is the variation in queueing delay, d, which is defined as the difference between the actual delay experienced by a packet, d a , and the fixed propagation delay of the connection, d p . If the incoming flow rate to a router l exceeds its capacity, packets are queued (generally on a first-come first-serve basis) in the existing buffer of the router of the link with bl,max being the maximum buffer size. Furthermore, if the buffer of the link is full, incoming packets have to be dropped. Let the total flow on link l be given by x¯l := i:l∈Ri xi . Thus, the buffer level at link l evolves in accordance with ⎧ ⎪[x¯l − Cl ]− ∂bl (t) ⎨ = x¯l − Cl ⎪ ∂t ⎩ [x¯l − Cl ]+

if bl (t) = bl,max if 0 < bl (t) < bl,max , if bl (t) = 0

(22)

where [.]+ represents the function max(. , 0) and [.]− represents the function min(. , 0). An increase in the buffers leads to an increase in the RTT of packets. Hence, RTT on a congested path is larger than the base RTT, which is defined as the sum of propagation and processing delays on the path of a packet. The queueing delay at the lth link, dl , is a nonlinear function of the excess flow on that link, given by ⎧ − 1 ⎪ ⎪ ⎪ (x¯l − Cl ) ⎪ ⎪ l ⎪ ⎨ C 1 d˙l (x, t) = (x¯l − Cl ) ⎪ C l ⎪ + ⎪ ⎪ 1 ⎪ ⎪ ⎩ (x¯l − Cl ) Cl

if dl (t) = dl,max if 0 < dl (t) < dl,max ,

(23)

if dl (t) = 0

in accordance with the buffer model described in (22), with dl,max := bl,max /Cl being the maximum possible queueing delay. Here, d˙l denotes (∂dl (t)/∂t). Thus, a user experiences is the sum of queueing delays on the total queueing delay, Di , its path, namely Di (x, t) = l∈Ri dl (x, t), i ∈ M, which we henceforth write as Di (t), i ∈ M. 4.1.1 Assumptions Additional assumptions of the extended model presented are as follows: 1. The effect of individual packet losses on the flow rates is ignored. This approximation is reasonable as one of the main goals of the developed rate control scheme is to minimize or totally eliminate packet losses. 2. The utility function Ui (xi ) of the ith user is assumed to be strictly increasing and concave in x i .


19

3. The effect of a user i on the delay, Di (t), she/he experiences is ignored. This assumption can be justified for networks with a large number of users, where the effect of each user is vanishingly small. Furthermore, from a practical point of view, it is extremely difficult, if not impossible, for a user to estimate its own effect on queueing delay.

4.2 Equilibrium Solution As in Section 3, define a cost function for each user as the difference between pricing and utility functions. However, here the pricing function of the ith user is linear in xi for each fixed total queueing delay Di of the user and is linear in Di with a fixed xi , i.e., it is a bi-linear function of xi and Di . The utility function Ui (xi ) is assumed to be strictly increasing, differentiable, and strictly concave in a similar way and it basically describes the user’s demand for bandwidth. Accordingly, variations in RTT are utilized as the basis for the rate control algorithm. The cost (objective) function for the ith user at time t is thus given by Ji (x, t) = αi Di (t) xi − Ui (xi ) ,

(24)

which she/he wishes to minimize. In accordance with this objective, again a gradient-based dynamic model is considered where each user changes its flow rate in proportion with the gradient of its cost function with respect to its flow rate, x˙i = −∂ Ji (x)/∂ xi . Taking into consideration also the boundary effects, the rate control algorithm for the ith user is ⎧ − dUi (xi ) ⎪ ⎪ ⎪ − αi Di (t) ⎪ ⎪ xi ⎪ ⎨ dUd(x i i) x˙i = − αi Di (t) ⎪ d xi ⎪ + ⎪ ⎪ dUi (xi ) ⎪ ⎪ ⎩ − αi Di (t) d xi

if xi = xi,max if 0 < xi < xi,max

(25)

if xi = 0.

Then, for a general network topology with multiple links, the generalized system is described by dUi (xi ) − αi Di (t) , i = 1, . . . , M , d xi x¯l − 1 , l = 1, . . . , L , d˙l (t) = Cl x˙i (t) =

(26)

with the boundary behavior given by (23) and (25). Define the feasible set Ω (as before) as

20

T. Alpcan

Ω = {(x, d) ∈ R M+L : 0 ≤ xi ≤ xi,max and 0 ≤ dl ≤ dl,max , ∀i , l}, where dl,max and xi,max are upper bounds on dl and xi , respectively. Define dmax := [d1,max , . . . , d L ,max ]. Existence and uniqueness of an inner equilibrium on the set Ω are now investigated under the assumption of xi,max > Cl , ∀l. Toward this end, assume that A is a full row rank matrix with M ≥ L, without any loss of generality. This is motivated by the fact that non-bottleneck links on the network have no effect on the equilibrium point and can safely be left out. Theorem 4 Let 0 ≤ αi,min ≤ αi ≤ αi,max , ∀i ∈ M where the elements of the vector αmax are arbitrarily large, and A be of full row rank. Given X , if αmin and dmax satisfy 0 < max d(αmin , x) < dmax , x∈X

where d(α, x) is defined in (30), then system (26) has a unique equilibrium point, (x∗ , d∗ ), which is in the interior of the set Ω. Proof Supposing that (26) admits an inner equilibrium and by setting x˙i (t) and d˙l (t) equal to zero for all l and i one obtains A x = C,

(27)

f(α, x) = A d ,

(28)

T

where d := [d1 , . . . , d L ]T is the delay vector at the links, C is the capacity vector introduced earlier, and the nonlinear vector function f is defined as

1 dU M 1 dU1 ,..., f(α, x) := α1 d x 1 αM d x M

T .

(29)

Define X := {x ∈ R M : Ax = C} as the set of flows, x, which satisfy (27). Multiplying (28) from left by A yields A f(α, x∗ ) = AAT d. Since A is of full row rank, the square matrix AAT is full rank, and hence invertible. Thus, for a given flow vector x and pricing vector α, d(α, x) = (AAT )−1 Af(α, x)

(30)

is unique. From the definition of f, d(α, x) is a linear combination of pi (xi )/αi and, hence, strictly decreasing in α. Since the set X is compact, the continuous function d(α, x) admits a maximum value on the set X for a given α. Therefore, for each > 0 one can choose the elements of αmax sufficiently large such that


21

0 < max d(αmax , x) < . x∈X

In addition, given X and dmax , one can find αmin such that 0 < max d(αmin , x) < dmax . x∈X

(31)

Hence, there is at least one inner equilibrium solution, (x∗ , d∗ ), on the set Ω, which satisfies (27) and (28). The uniqueness of the equilibrium is established next. Suppose that there are two different equilibrium points, (x∗1 , d∗1 ) and (x∗2 , d∗2 ). Then, from (27) it follows that A (x∗1 − x∗2 ) = 0 ⇔ (x∗1 − x∗2 )T AT = 0. Similarly, from (28) follows f(α, x∗1 ) − f(α, x∗2 ) = AT (d∗1 − d∗2 ) . Multiplying this with (x∗1 − x∗2 )T from left one obtains (x∗1 − x∗2 )T f(α, x∗1 ) − f(α, x∗2 ) = 0, which can be rewritten as M

(x∗1i − x∗2i )

i=1

1 αi

∗) ∗ ) dUi (x1i dUi (x2i = 0. − d xi d xi

Since Ui ’s are strictly concave, each term (say the ith one) in the summation is ∗ = x ∗ . Hence, the negative whenever x1∗ i = x 2∗ i with equality holding only if x1i 2i ∗ ∗ ∗ ∗ point x has to be unique, that is x = x1 = x2 . From this, and (26), it immediately follows that Di , i = 1, . . . , M, are unique. This does not, however, immediately imply that dl , l = 1, . . . , L, are also unique, which in fact may not be the case if A is not full row rank. The uniqueness of dl ’s, however, follow from (30), where a unique d∗ is obtained for a given equilibrium flow vector x∗ : d∗ = (AAT )−1 Af(α, x∗ ). Thus, (x∗ , d∗ ), following from (27) and (28), constitutes a unique inner equilibrium point on the set Ω.

4.3 Stability Analysis The rate control scheme and accompanying system described by (26) is first shown to be globally asymptotically stable under a general network topology in the ideal

22

T. Alpcan

case. Subsequently, the global stability of the system is investigated under arbitrary information delays, denoted by r , for a general network with a single bottleneck node and multiple users. The case of multiple users on a general network topology with multiple links is omitted since the problem in that case is quite intractable under arbitrary information delays. 4.3.1 Instantaneous Information Case The stability of the system below can easily be established under the assumption that users have instantaneous information about the network state. Alternatively, this case can be motivated by assuming that information delays are negligible in terms of their effects to the rate control algorithm. Defining the delays at links, dl , and user flow rates, xi , around the equilibrium as d˜l := dl − dl∗ and x˜i := xi − xi∗ , respectively, for all l and i, one obtains the following system inside the set Ω and around the equilibrium: x˜˙i (t) = gi (x˜i ) − αi D˜ i (t) , i = 1, . . . , M , 1 x˜i , l = 1, . . . , L , d˙˜l (t) = Cl

(32)

i:l∈Ri

where D˜ i =

l∈Ri

d˜l and gi (x˜i ) is defined as gi (x˜i ) :=

dUi (xi ) dUi (xi∗ ) − . d xi d xi

Define next a positive definite Lyapunov function ˜ = V (˜x, d)

M L 1 (x˜i )2 + Cl (d˜l )2 . αi i=1

(33)

l=1

˜ along the system trajectories is given by The time derivative of V (˜x, d) ˜ = V˙ (˜x, d)

M

(2/αi )gi (x˜i ) x˜i ≤ 0,

i=1

˜ is negative where the inequality follows because gi (x˜i ) x˜i ≤ 0, ∀i. Thus, V˙ (˜x, d) ˜ = 0}. It follows as before that ˜ ∈ R M+L : V˙ (˜x, d) semidefinite. Let S := {(˜x, d) ˜ ∈ R M+L : x˜ = 0}. Hence, for any trajectory of the system that belongs S = {(˜x, d) identically to the set S, we have x˜ = 0. It follows directly from (32) and the fact that gi (0) = 0 ∀i that x˜ = 0 ⇒ x˙˜ = 0 ⇒ D˜ i = 0 ∀i ⇒ d˜l = 0 ∀l,


23

where the last implication is due to the fact that D˜ = AT d˜ ∗ and the matrix A is of full row rank. Therefore, the only solution that can stay identically in S is the zero solution, which corresponds to the unique inner equilibrium of the original system. As a result, system (32) is globally stable under the assumption of instantaneous information. 4.3.2 Information Delay Case The preceding analysis is generalized to account for information delays in the system by introducing user-specific maximum propagation delays r = [r 1 , . . . , r M ] between a bottleneck link and the users. The system is assumed to have a unique inner equilibrium point (x∗ , d ∗ ) as characterized in Section 4.2. Modifying the system equations around this equilibrium point by introducing the associated maximum propagation delays, one obtains ˜ − ri ) , i = 1, . . . , M x˙˜i (t) = gi (x˜i (t)) − αi d(t M ˙˜ = 1 x˜ (t − r ). d(t) i i C

(34)

i=1

Then, the i th users rate control algorithm is ˜ + αi x˙˜i (t − ri ) = gi (x˜i (t − ri )) − αi d(t) C

0

M

−2ri j=1

x˜ j (t + s − r j )ds.

Define again a positive definite Lyapunov function ˜ = V (˜x, d)

M M 0 t 1 2 M ˜ (x˜i (t −ri ))2 +C(d(t)) + x˜i2 (u−ri )du ds. (35) αi C −2ri t+s i=1

i=1

Taking the derivative of V along the system trajectories yields M 2 ˜ = i=1 gi (x˜i (t − ri ))x(t ˜ − ri ) V˙ (˜x, d) αi 1 0 M M + 2x˜i (t − ri )x˜ j (t + s − r j )ds C −2ri i=1 j=1 M M 0 2 2 + i=1 −2ri [ x˜i (t − r ) − x˜ i (t + s − r )]ds. C This derivative V˙ is bounded from above by ˜ ≤ V˙ (˜x, d)

M 4Mri 2 2 gi (x˜i (t − ri ))x˜i (t − ri ) + x˜ (t − ri ). αi C i i=1

24

T. Alpcan

Hence, it can be made negative semi-definite by imposing a condition on the ˜ ∈ Ω˜ : maximum delay in the system, rmax := maxi ri . Let S := {(˜x, d) ˙ ˜ ˜ ˜ V (˜x, d) = 0}. It follows as before that S = {(˜x, d) ∈ Ω : x˜ = 0}. Therefore, for any trajectory of the system that belongs identically to the set S, x˜ = 0. It also follows directly from (34), and the fact that gi (0) = 0 ∀i, that x˜ = 0 ⇒ x˙˜ = 0 ⇒ d˜ = 0, where the fact that the matrix A is of full row rank is used. Consequently, the only solution that can stay identically in S is the zero solution, which corresponds to the unique equilibrium of the original system. As a result, system (34) is asymptotically stable by LaSalle’s invariance theorem [22] if the maximum delay in the system, rmax , satisfies the condition rmax C ⎠ = 1 − e−γ .

(11)

j=i−λi +1

The probability in (11) can be rewritten as ⎞ ⎛ ⎞ ⎛ K˜ N i 1 yi ⎝ g j + γi−λi ⎠ > C ⎠ P⎝ N k=1 i=1 j=i−λi +1 ⎛ ⎞ ⎞ ⎛ ˜ K N i 1 = ... P⎝ yi ⎝ g j +γi−λi ⎠ > C |λ1 =l1 , . . . , λ N =l N ⎠· N l1

lN

k=1

i=1

j=i−λi +1

· P (λ1 = l1 , . . . , λ N = l N ) = ⎛ ⎛ K˜ N i 1 = ... P⎝ yi ⎝ N l1

lN

k=1

i=1

⎞

⎞

g j + γi−li ⎠ > C ⎠

j=i−li +1

N "

P (λi = li ).

i=1

Expression ⎛

⎛ K˜ N i 1 ⎝ ⎝ P yi N k=1

i=1

j=i−li +1

⎞

⎞

g j + γi−li ⎠ > C |λ1 = l1 , . . . , λ N = l N ⎠

104

J. Levendovszky et al.

can be upper bounded by the Chernoff bound as ⎛ ⎛ N i K˜ 1 P⎝ yi ⎝ N k=1

⎞

⎞

g j + γi−li ⎠ > C |λ1 = l1 , . . . , λ N = l N ⎠ ≤

j=i−li +1

i=1

N

≤ ei=1

μi (s,Vi )− sNC ˜ K

,

where '

( ' ( μi (s, Vi ) := log E esyi Vi = log 1 − pi + pi esVi and N "

P (λi = li ) =

i=1

N "

⎛ ⎝ai−li

⎞ 1 − a j ⎠,

i " j=i−li +1

i=1

thus we obtain ⎛ ⎛ K˜ N 1 P⎝ yi ⎝ N k=1

...

l1

i=1

N

e

sNC i=1 μi (s,Vi )− K˜

lN

e

− sNC K˜

...

l1 − sNC K˜

N " l N i=1

N "

⎞

⎛

⎝ai−li ⎛

eμi (s,Vi ) ⎝ai−li

⎛

⎞

g j + γi−λi ⎠ > C ⎠ ≤

j=i−λi +1

N " i=1

e

i

⎛

⎝eμi (s,Vi ) ⎝ai−li

i " j=i−li +1 i "

⎞ 1 − aj ⎠ = ⎞ 1 − aj ⎠ =

j=i−li +1 i "

⎞⎞ 1 − a j ⎠⎠.

j=i−li +1

i=1 li

Introducing the extended logarithmic moment generation function as ⎛ ⎛ ⎝eμi (s,Vi ) ai−li βi (s, Vi ) := log ⎝ li

one can write

i " j=i−li +1

⎞⎞ 1 − a j ⎠⎠

(12)

Reliability-Based Routing Algorithms

105

⎛ ⎛ K˜ N 1 P⎝ yi ⎝ N k=1

i=1

≤e

− sNC ˜

⎞

i

⎞

g j + γi−λi ⎠ > C ⎠ ≤

j=i−λi +1

N "

K

eβi (s,Vi ) = e

N

sNC i=1 βi (s,Vi )− K˜

.

(13)

i=1

Comparing the bound with 1 − e−γ , we obtain N

i=1 βi

e where sˆ : arg min s

N

(sˆ,Vi )− sˆ NK˜C = 1 − e−γ ,

(s, Vi ) −

i=1 βi

sNC . K˜

(14)

The lifespan is the solution of the follow-

ing equation: K˜ :

N i=1

sˆ N C + log 1 − e−γ . βi sˆ , Vi = K˜

(15)

As it can be seen, the equation above determines the lifespan as a function of vector a; the components of which represent the probabilities of shortcut on a given node. This relationship is denoted by K˜ = Ψ (a). Using (12) the protocol optimization can take place by searching in the space of a-vectors to maximize Ψ (a). This can be done by gradient descent given as follows: # ai (n + 1) = ai (n) − Δsgn

$ Ψ (a (n)) − Ψ (a (n − 1)) , ai (n) − ai (n − 1)

i = 1, . . . , N , (16)

where a (n) is the probability vector at iteration n and Δ is the learning rate of the gradient descent. As a result, the protocol optimization is carried out as in Fig. 3. In the case of single-hop protocol we have ai = 1, i = 1, . . . , N . Here, we obtain ⎛

⎞ K˜ N 1 K˜ : P ⎝ yi (k) γi < C ⎠ = e−γ , N k=1

(17)

i=1

which yields the following lifespan: sˆ N C , −γ i=1 μi sˆ , γi + log 1 − e

K˜ = N

(18)

where μi (s, γi ) := log E esyi γi = log (1 − pi + pi esγi ) is the log moment generating function. The lifespan estimation (5) has been defined on the basis of the overall energy consumption of a packet. Taking into account that one may want to maximize the

106


Fig. 3 Algorithm to optimize the free parameters of the random shortcut protocol

minimum remaining energy after the packet transfer instead of minimizing the overall energy, this definition can be modified as follows: ⎧ ⎛ ⎫ ⎞ K˜ i ⎨ ⎬ K˜ : min P ⎝ ϑi (k) > c⎠ = e−γ , ⎭ i ⎩

(19)

k=1

where ϑi (k) denotes the average energy consumption on node i at time instant k and K˜ i denotes the lifespan of node i. The method discussed above can be extended to estimate the lifespan of any arbitrary protocol [14].

4.3 Performance Analysis and Numerical Results for Random Shortcut Protocols In this section, a detailed performance analysis of the chain, the shortcut and the single-hop protocols are given. The aim is to evaluate the lifespan of a sensor network containing N nodes placed in an equidistant manner. Figure 4 shows how the estimated lifespan changes as the function of the number of nodes (N ) in the case of the three methods described above. The distance between the base station and the farthest node was 20 m, the initial battery power was C = 10∧ 5, and the reliability parameter was set as (1 − e−γ ) = 0.95, while the probabilities of shortcut were uniform ai = a = 0.2. One can see that there is a maximum lifespan in the cases of chain and random shortcut protocols with respect to the number of nodes. Figure 4 shows that when the network is sparse then both methods result in relatively closer lifespan, while departing from the optimal number of nodes (either decreasing or increasing the number of nodes), the random shortcut model definitely gives much higher (it is more than


107

Fig. 4 Estimated lifespan as a function of the number of nodes

37% in the case of N = 7) lifespan. The figure also demonstrates that in the case of relatively low node density, the chain protocol yields higher longevity than the single-hop protocol. On the other hand, the random shortcut protocol always results in longer lifespan than any of the other two protocols. Figure 5 demonstrates the accuracy of lifespan estimation at different protocols. The settings were the same as earlier and the number of nodes was N = 7. From the figure one can see that the lifespan can be sharply estimated by the Chernoff bound (estimation error < 2%). Thus, based on this estimation the system parameters can be optimized accordingly in an off-line manner. The methods treated

Fig. 5 Lifespan and estimated lifespan values achieved by different protocols

108


in this section have provided increased lifespan but reliable packet delivery has not yet taken into account. This problem is going to be addressed in the forthcoming sections.

5 Reliability-Based Routing with Energy Balancing In this section, we develop new algorithms for optimal path selection over a random graph subject to the constraint that the probability of successful packet reception at the BS must exceed a given threshold. Packet forwarding to the BS is carried out over a path characterized by a set of indices = {i 1 , i 2 , . . . , i L }, where the indices identify the nodes which participate in the packet transfer (see Fig. 6).

Fig. 6 Packet forwarding over path from source node to the BS in WSN

In our notation, node i 1 is the sender node and i L+1 denotes the-BS. The transmis. sion energies used by the nodes contained by path are given as gi1 i2 , . . . , gi L B S . Thus, the overall energy required by getting the packet from sender node i 1 to the L gil il+1 . BS over the path = {i 1 , i 2 , . . . , i L } is given as l=1 If the probability of successfully forwarding the packet from node i 1 to node i L+1 reception of the is denoted by Pil il+1 = Ψ gil il+1 then the probability 'of successful ( !L packet at BS is P(correct reception at BS) = l=1 Ψ gil il+1 . Here, it is noteworthy to mention again that the transmission energies on the nodes can adaptively be changed to achieve reliable packet transfer at the cost of minimum energy. Hence reliable routing poses the following constrained optimization problems: 1. Minimizing the overall energy: In this case, our objective is to find the optimal path and the corresponding optimal transmission energies which minimize the overall energy needed for a packet to get to the BS subject


109

to the -reliability constraint. .More precisely, we seek the optimal path opt = i 1,opt , i 2,opt , . . 0 . , i L ,opt and the optimal transmission energies opt = / L opt opt opt gil il+1 subject to gi1 i2 , gi 2 i3 , . . . , gi L i L+1 for which opt , opt : min, l=1

P(correct reception at BS) =

L "

( ' opt Ψ gil il+1 ≥ 1 − ε.

(20)

l=1

The algorithm solving this problem will be referred to as Overall Energy Reliability Algorithm (OERA) [29]. 2. Maximizing the remaining energy of the bottleneck node: In this case, our objective is to find the optimal path and the corresponding optimal transmission energies which ensure that after the packet has been forwarded to the BS, the minimum remaining energy (the energy in the so-called bottleneck node) is maximized. More precisely, let ci (k), i ∈ V denote the energy state of node i at time instant k whereas c (k) is the energy state of the - network at time instant . k. Our objective is to find the optimal path / opt = i 1,opt , i 2,opt , . . . ,0i L ,opt and opt

opt

opt

the corresponding optimal energies opt = gi 1 i2 , gi2 i3 , . . . , gi L i L+1 for which max, minl cil (k + 1), where cil (k + 1) := cil (k) − gil il+1 and guarantee that P(correct reception at BS) =

L "

Ψ gil il+1 ≥ 1 − ε.

(21)

l=1

The algorithm solving this problem will be referred to as Bottleneck Energy Reliability Algorithm (BERA). We generally solve these problems in two phases. In the first phase, we assume that the path = {i1 , i 2 , . . . , i L } over which the packet is forwarded from node i 1 to 0 / opt opt opt the BS is given and only the transmission energies opt = gi 1 i2 , gi2 i3 , . . . , gi L i L+1 are to be optimized. In the second phase, we determine opt , the optimal packet forwarding path that guarantees the 1 − ε reliability and minimizes the overall consumption of the packet transfers. It can be proven that carrying out these two steps separately, one can find the optimum.

5.1 Optimization of the Overall Energy Consumption – The OERA Algorithm In this section, we will demonstrate that the optimal path solving problem (20) can be given in polynomial time which gives rise to the OERA algorithm [29]. As was mentioned before, let us first assume that the path is already given. In this case, our goal is to determine the optimal transmission energies opt =

110

/

opt

J. Levendovszky et al. opt

opt

gi1 i2 , gi2 i3 , . . . , gi L i L+1 (20). We state the following:

0 for which

L

l=1 gil il+1

is minimal subject to constraint

Theorem 1 Assuming that the packet transmission path = {i 1 , i 2 , . . . ., i L } from L gil il+1 can node i 1 to the BS is given, under the reliability parameter (1 − ε), l=1 only be minimal if √ √ √ √ gil il+1 = ( w1 + w2 + · · · + w L ) · wl ,

(22)

where wl :=

diαl il+1 Θσ Z2 − ln(1 − ε)

.

(23)

The proof of this theorem can be found in Appendix A. Thus, in the case of a given path = {i 1 , i 2 , . . . , i L } the optimal transmission energies which yield maximal lifespan are obtained from (22). Consequently, the overall energy consumption to get the packet to the BS along the path is given as L

√ √ √ gil ,il+1 = ( w1 + w2 + · · · + w L )2 .

(24)

i=1

Based on (24), the energy consumption of a packet transfer is E () =

L √

wi1 ,i2

)2 & L √ √ √ √ + wi2 ,i3 + · · · + wi L−1 ,i L wik ,ik+1 = wik ,ik+1 ,

k=1

k=1

(25) where wik ,ik+1 =

diαk ,ik+1 Θσ Z2 − ln(1 − ε)

.

(26)

We are seeking opt for which opt : min E().

(27)

√ √ As wik ,ik+1 is positive, minimizing (25) is equivalent to minimizing ( wi1 ,i2 + √ wi2 ,i3 + · · · + wi L−1 ,i L ). Hence problem (27) reduces to opt : min E () ∼ min

L √ l=1

wil ,il+1 .

(28)


111

√ It is, however, a shortest path problem if the value wu,v is assigned to the edge connecting the nodes (u, v), where wu,v is defined by (26). In this way, (28) can be solved by the Bellman Ford algorithm in polynomial time.

5.2 Reliable Packet Transfer by Maximizing the Remaining Energy of the Bottleneck Node . In this case, we seek opt = i 1,opt , i 2,opt , . . . , i L ,opt and the corresponding optimal 0 / opt opt opt energies opt = gi1 i2 , gi 2 i3 , . . . , gi L i for which max, minl cil (k + 1) subject L+1 to the condition that the packets arrive at the BS with a given reliability, as indicated below P(correct reception at BS) =

L "

Ψ gil il+1 ≥ 1 − ε.

(29)

l=1

We will demonstrate that this problem can also be solved in polynomial time by using the BERA algorithm. Let us again first assume that the packet forwarding path is already given. Then we state the following: Theorem 2 Assuming that the packet transmission path = {i 1 , i 2 , . . . , i L } from node i 1 to the BS is given, under the reliability parameter (1 − ε), then minil cil (k) − gil il+1 can only be maximal if the residual energy of each node is the same, expressed as cil (k) − gil il+1 = A, and A satisfies the following equation: "

Ψ cil − A = 1 − ε.

(30)

il ∈

The proof of this theorem can be found in Appendix B1. It is easy to note that the left-hand side of (30) is monotone decreasing with respect to parameter A. Thus (30) will have a unique solution over the interval 0, mini j ci j . If there is no solution then there is no such energy set opt = / 0 opt opt opt gi1 i2 , gi2 i3 , . . . , gi L i which could fulfill the reliability constraint. Due to its L+1 monotonicity, one can develop fast methods to solve (30), like the Newton–Raphson algorithm. Having A at hand, we can search for the most reliable the maximiza path when tion of reliability is equivalent to the minimization of − il ∈ log Ψ cil (k) − A . This formula reduces the search for the most reliable pathinto a shortest path optimization problem where the weight − log Ψ cil (k) − A is assigned to each link. The task opt : min − log Ψ dil il+1 , cil − A (31)

il ∈

112


can be solved in polynomial time by the Bellman–Ford algorithm. (Note that − log Ψ dil il+1 , cil − A ≥ 0). By applying the Rayleigh model (as described by (1)) and with setting A = 0, one obtains opt : min

il ∈

%) & α −dil il+1 Θσ Z2 ∼ − log Ψ cil ∼ min − log exp cil il ∈

∼ min

diαl il+1 Θσ Z2 il ∈

cil

. (32)

In this special case, expression (32) is equivalent to the optimization problem solved by the PEDAP-PA algorithm [28]. On the other hand, it is easy to see that the solution of (31) depends on the value of A. Furthermore, the optimal value of A depends on the path itself. Therefore, let us solve (31) and (30) recursively, one after another. This implies that we search for the most reliable path and then for the path found we make sure that the reliability constraint holds obtaining the proper value of A belonging to the given reliability parameter. This algorithm will have a fix point and will stop when there are no changes in the obtained paths any longer. The convergence to the optimal solution is stated by the following theorem: Theorem 3 Let A(k) indicates the series obtained by recursively solving (31) and (30) one after another. A(k) is monotonically increasing and will converge to the fix point of (31) and (30). Furthermore (31) and (30) have a unique fix point. Hence the algorithm described above and depicted by Algorithm 1 converges to the global optimum. The proof of this theorem can be found in Appendix B2. Algorithm 1 This algorithm calculates variables [A, path], where OptResEnergies is an equation solver which solves equation (30). DIJKSTRA is the well-known minimal path selection algorithm which solves the optimization task indicated with (31). The initial path is a one-hop path between source and BS. Require: ci > 0, ∀i Ensure: [A, path] A←0 path ← [S OU RC E, B S] while path = path old do path old ← path El j ← − log Ψ dl j , cl (k) − A , ∀l j path ← DIJKSTRA(E) A ← O pt Res Energies( path) end while

Furthermore, it can be shown that the speed of convergence is O M N 2 , where M is the upper bound on number of time the decision operation is performed, while O N 2 is the complexity of the Dijkstra algorithm. Note that the M is independent of the network size. Hence the convergence speed is still O N 2 .


113

5.3 Performance Analysis and Numerical Results In this section, the performance of the two new reliability-based routing algorithms (OERA and BERA) are analyzed and compared with the standard WSN routing algorithms. However, we also run simulations to analyze the network behavior when some nodes had already died and in this case the lifespan is defined as the time being to the latest death. We actually define earliest death as the time for the first node going dead and we define latest death as the time till the last node goes flat. In each time instant a new packet has been generated randomly by one of the nodes still operational. The fading model was the well-known Rayleigh fading with the exponent of fading attenuation α = 3, sensitivity threshold Θ = 50, and energy of noise σz2 = 10−4 . The methods were tested on multiple networks with size N =5, 10, 20, 50, and 100 nodes, respectively. The nodes had been distributed randomly or deterministically according to a uniform distribution over an area of 100 m2 . The BS was placed in a corner. The test topologies of the network is indicated in Fig. 7.

Fig. 7 Topologies: (a) Random topology with 100 sensor nodes and (b) grid topology with 100 sensor nodes

Among the traditional methods the first one tested is the single-hop protocol in which every node transmits directly to the BS; thus, the reliability can be easily ensured. In the case of LEACH, the number of hops is 2; hence, the reliability criteria can also be ensured. On the other hand, for reversed path forwarding algorithms (e.g., PEGASIS, PEDAP-PA, DD) reliability cannot be ensured directly. Therefore, the traditional protocols had to be modified in order to guarantee the reliability of delivery. Hence, if the length of the selected path was M, in order to ensure (1 − ε) level of reliability the nodes participating in the transmission must transmit with, such that (1 − ε1 ) M = (1 − ε).

(33)

In Fig. 8a, one can see that the newly proposed BERA and OERA algorithms outperform the traditional protocols, yielding longer lifespan (the longevity is twice or

114


Fig. 8 Comparing the lifespan of the BERA and OERA algorithms with lifespan achieved by traditional protocols: (a) Time to the earliest death and (b) time to the latest death

three times longer compared to the single-hop protocol). On the other hand, single hop is better in the sense of last node dying, since it is the nearest node to the BS (Fig. 8b). The probability of successful packet transfer to BS has also been evaluated in the case of all protocols, where parameter ε was set as ε = 0.05 . The results depicted in Fig. 8 are made from multiple test, where 20 different random networks were tested and averaged. In the next simulations the lifespan was defined as the longevity of the longest lasting node. Figure 9 indicates the percentage of the operational node as a function of time.

Fig. 9 The percentage of the operational node as a function of time

One can see that in the case of BERA algorithm the nodes go flat more or less at the same time. BERA is the best in the load balancing effect, as using energy awareness. OERA makes a good compromise between the earliest and latest death of the network, i.e., the latest death is occurring much later than in the case of BERA; however, the first death occurs earlier. Here first and last death are consid-


115

ered timewise. In the case of BERA the longevity of the first node going flat has been significantly improved compared with the traditional methods. Figure 10 depicts the average number of packets transferred to the BS with respect to the size of the network when all protocols started to run with the same initial energy. The packets were generated subject to uniform distribution in the case of each protocol. The optimal strategy can be selected easily if we know the network density. If this density is low PEGASIS is good choice, on the other hand BERA guarantees the highest throughput (the maximum number of delivered packets) if the density is high. From Fig. 10, it can be seen that the performance of some protocols depends on the number of nodes. Furthermore, DD underperforms the rest of the protocols due to the fact that they minimize the total energy consumption instead of maximizing the longevity of node which goes flat earliest. One may also observe that the new BERA protocol performs rather well, implying that more packets can be transferred with a given level of initial energy than in the case of the classical protocols. The simulation results are made from a multiple test (running the simulations several times and averaging the results).

Fig. 10 Average number of delivered packets using a given energy level

Figure 11 depicts the lifespan with respect to the reliability parameter. Analyzing the different protocols, one can see that the new BERA protocol can achieve higher lifespan in the case of all ε. One can also observe that decreasing ε will increase the lifespan exponentially. In this example, algorithms perform very differently, but PEDAP-PA has proven to be a good strategy for this specific topology. Figure 12 demonstrates the load-balancing capabilities of the different protocols in the case of a 20-node network. The results were obtained from a single test, as we have analyzed the battery reduction of all nodes as a function of time. As can be seen, the single hop and DD do not enforce load balancing at all, while LEACH, PEDAP-PA, and the new protocols achieve good load balancing. One can see that the new BERA protocol provides the best load balancing. If longevity is defined as the death of the last node, BERA still performs well, as the nodes will die more or less at the same time. The PEGASIS protocol does not perform well because

116


Fig. 11 Lifespan of the network depends on the required service

satisfying the reliability criterion in the case of a large number of nodes in a chain topology needs a large amount of energy consumption. So far, we have adopted the Rayleigh model for performance analysis, which is valid if the node antennas do not “see” each other. This typically occurs in indoor applications. If the antennas can see each other then the Rice fading [1, 32] describes the radio channel better. Figure 13 indicates, how the lifespan increases (in the case of BERA) when the amplitude of the dominant wave grows and it also demonstrates the effect of errors in the fading parameter estimation on the lifespan. The Rician factor F means the ratio of the LOS signal energy and the non-LOS signal energy [23, 28]. As can be seen, by underestimating the fading parameters the lifespan will increase but it puts the reliability in jeopardy. In the case of overestimation the reliability criterion is satisfied but with excessive energy consumption (too large transmission energies are selected) and the lifespan is decreased. The expected value and the standard deviation of the delay in the case of the different protocols are depicted in Fig. 14b. From the figure it can be inferred that with increasing lifespan the latency is also going to be increased. In Fig. 14a one can see the dependency of the size of the network and delay. In the case of PEGASIS, the packet is forwarded along a very long chain, which may result in extremely long delays. The long chain is also disadvantageous with respect to the transmission energies as the reliability can only be maintained by high transmission energies in the case of several hops. In Tables 3, 4, and 5, respectively the investigated protocols are compared with each other. The values are always normalized by the lifespan provided by the best protocol, e.g., 79% for PEDAP-PA with N = 50 implies that the lifespan of PEDAP-PA is 79% of the lifespan of BERA for the multiple test. From Table 3 one


117

Fig. 12 Illustration of the load balancing performance of different protocols

Fig. 13 The impact of fading parameter estimation on the lifespan and reliability (the reliability parameter was set as ε = 0.05, while the exponent of fading attenuation was set as α = 3)

118


Fig. 14 Average delay of the different protocols: (a) delay of the analyzed protocols and (b) delay in the function of network size

can see that the single-hop protocol performs rather poorly, due to the fact that the node closest to BS carries a high load and dies prematurely. However, in Tables 4 and 5 the single-hop protocol exhibits increasing performance as κ increases (for the definition of κ, see Section 2.3 on page 8). In the case of a low node number the PEGASIS algorithm proves to be the best. However, when the number of nodes is increasing then the lifespan of PEGASIS deteriorates fast. LEACH can only be used in large networks as it requires the clustering of nodes. Analyzing Tables 4 and 5 one can see that OERA performs the best in the case of choosing κ = 0.2 or 0.4 because it minimizes the overall energy. One can clearly see that the traditional protocols are less efficient than the new ones. Furthermore, the traditional protocols do not guarantee reliability, either.

N 100 50 20 10 5

N 100 50 20 10 5

PEDAP-PA (%) 67 79 82 91 88

PEDAP-PA (%) 60 65 83 72 71

Table 3 Comparison of lifespans till the first node dies Single PEGASIS LEACH DD hop (%) (%) (%) (%) 8 16 24 42 68

10 3 6 38 108

19 10 10 19 28

20 17 20 26 55

Table 4 Comparison of lifespans with parameter κ = 0.2 Single PEGASIS LEACH DD hop (%) (%) (%) (%) 25 28 32 52 42

9 10 25 39 63

20 15 23 53 24

44 45 53 47 72

OERA (%)

BERA (%)

42 47 48 67 82

100 100 100 100 100

OERA (%)

BERA (%)

100 100 100 100 76

94 89 92 86 100


N 100 50 20 10 5

PEDAP-PA (%) 58 64 67 70 60

119

Table 5 Comparison of lifespans with parameter κ = 0.4 Single PEGASIS LEACH DD hop (%) (%) (%) (%) 42 55 62 57 70

10 21 35 70 100

22 30 34 45 35

41 47 51 55 60

OERA (%)

BERA (%)

100 100 100 100 75

74 73 69 75 53

Based on the numerical results, as a summary one can conclude that the new BERA protocol can outperform the traditional ones and it can be applied in any application when longevity and reliability are of major concerns. Moreover, one can see that increasing κ the OERA algorithm will also perform well as it minimizes the overall energy without balancing.

6 Protocol Implementation In this section we summarize the implementation of the novel algorithms. At first we describe the special protocol stack which must be implemented to use the new routing protocols. Second, we demonstrate how to develop distributed implementations.

6.1 Protocol Stack Assumptions The proposed novel reliability-based routing algorithms (OERA and BERA) can directly be implemented as routing protocols if the protocol layers are operating as follows: • Application layer: The BS collects the measurements made by the nodes and determines the reliability and other QoS parameters which is then handed over to the network layer. • Network layer: When requesting data from node i the routing protocol determines the optimal path and transmission energies based on OERA or BERA algorithm taking into account the network topology, energy levels, and QoS parameters. As the information will be conveyed to node i via the nodes contained by the optimal path, each node in the path is informed about its role and the transmission power can be set accordingly. • Data link layer: It implements a time division multiple access (TDMA) protocol with synchronous access, where each node uses a different time slot for access but most of the time the nodes are switched off (in this way the resources are not misused and collision is avoided). • Physical layer: It can operate on any platform (e.g., Mica2, BTmote, Skymote,TI ez430 [9, 21, 31]).

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The two requirements for the operation of the WSN indicated above are (i) the knowledge of the current state of the network as far as the available node powers and fading state are concerned and (ii) time synchronization for TDMA. These conditions can be ensured by running a “maintenance protocol” periodically, which estimates the fading parameters and ensures the clock synchrony. As the data traffic is known on the BS the available node energies can be evaluated there.

6.2 Distributed Implementations for Novel Packet Forwarding and Routing Algorithms In this section distributed versions of the aforementioned algorithms are developed based on the ad hoc on-demand distance vector (AODV) protocol [19]. AODV is a proactive table-driven routing protocol for mobile ad hoc networks (MANET), implying that every node maintains a routing table: • Random shortcut protocol: It is easy to see that the novel random shortcut packet forwarding method can be combined with an AODV protocol implementation. Decision algorithm must only be modified by using a Bernoulli random number generator, which can overwrite the next destination address. • Overall energy routing algorithm: In the case of fading-aware routing with minimal overall energy consumption, the metric of the links is given by expression (26). Then, we can run the algorithm described in Section 5.1, by using AODV for each of the steps. • Bottleneck energy routing algorithm: In the case of BERA, let the metric now be defined as given in (31). In this way, we select the route which has maximum reliability. Unfortunately the nodes are not aware of Aopt which is ci (k + 1) if the solution is optimal. Theorem 3 states that finding the most reliable route in a case where the residual energy is optimal, we achieved the global optimum. As we cannot run the AODV protocol recursively due to its huge signaling overhead, one may run the algorithm only once with an estimated Aopt . Let this estimation ˆ We can be sure that be denoted by A. min ci < Aopt < max ci − gmax

(34)

holds. This algorithm runs as follows: according to (34) we can choose Aˆ = max ci − gmax which is propagated in the network and each node calculates the corresponding transmission energy according to (30). Then we run AODV with the calculated link measures (if some of the nodes do not have the necessary ˆ then the connecting edges will be set amount of residual energy dictated by A, zeros; hence, they will not be part of the path). Aˆ is very close to A opt because gmax is much smaller than ci . If we change Aˆ all the time then the routing tables are updated continuously, which would present an intolerable overhead. Hence it is advised to update the routing tables only with a given frequency, which may


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ease the routing complexity and decrease the amount of signaling. This frequency parameter can also be optimized with respect to the dynamics and the size of the network; however, this problem is not the subject of the present discussion.

7 Conclusion In this chapter, optimal packet forwarding algorithms and reliability-based energyaware routing for WSN have been studied. First, the statistical analysis of the socalled random shortcut protocol was given by large deviation theory and the gain in the achieved lifespan was demonstrated. Then we have proposed two novel routing algorithms (OERA and BERA), which are capable of providing reliability-based routing in polynomial time. With the help of the new protocols the probability of packet loss can be kept under a predefined threshold, while the transmission energies are minimized along the paths. It has also been shown that these novel algorithms outperform the traditional routing algorithms with respect to both longevity and reliability. Furthermore, the new protocols can be implemented in a distributed fashion. As a result, they can be applied to applications where lifespan and reliable communication to the BS are of major concerns.

Appendix A – Proof of Theorem 1 As the reliability of packet transfer is L ! l=1

⇒

Ψ gil il+1 = exp L l=1

−diα i Θσ Z2 l l+1 gil il+1

L

−diα i

l=1

Θσ Z2

l l+1 gil il+1

≥ (1 − ε) (35)

≥ ln(1 − ε).

Using the definition of wil ,il+1 in (23), we can reformulate (35) as L −wil il+1 l=1

gil il+1

≥ −1, wil il+1 > 0, gil il+1 > 0.

Hence, we have the following constraint optimization (CO) problem: Let

f (G) =

L l=1

and

gil il+1

(36)

122


g(G) =

L −ail il+1 l=1

gil il+1

+ 1,

-

. where G = gi0 ,i1 , gi1 ,i 2 , . . . , gi L ,i L+1 . The CO is G opt : min f G s.t. g G ≥ 0.

(37)

G

Let L(G, λ) = f (G)−λg(G) be the Lagrangian function of the problem. Therefore, its Lagrange dual problem can be written as the following: max L(G, λ)s.t. λ ≥ 0; G,λ

∂L ∂G

= 0.

After solving (38), we have the following solution: gil il+1 = L √ k=1 wi k ,i k+1 . Thus the optimal solution is gil il+1

(38) λwil ,il+1 and

λopt =

) & L √ √ = wik ,ik+1 · wil ,il+1 . k=1

Appendix B1 – Proof of Theorem 2 First, we show that the solution of optimization task defined in Section 5 must fulfill "

Ψ gil il+1 = 1 − ε.

(39)

il ∈

Let us assume that (39) does not hold, then because of (29) the next expression holds "

Ψ gil il+1 > 1 − ε.

(40)

il ∈

1 for which gîl i < gil i , il = arg min(ci j − gi j i ) In this case, there exists a G l+1 l+1 j+1 ij

and (39) is satisfied. This will yield a better solution; thus, the path in (40) is not needed. Theorem 2 states that if G is a solution of (21) then cil (k + 1) = A, for ∀il . Let us assume that it does not hold, meaning that cil (k + 1) = Ail is a better solution implying A < A il Then the values can be arranged as follows:

∀il .

(41)


123

Ai1 < Ai2 ≤ · · · ≤ Ai L

(42)

as the remaining energies are different. However, if (41) and (42) are true then "

Ψ dil il+1 , gil il+1 < 1 − ε

(43)

il ∈

as Ψ (.) is monotone decreasing with respect to the remaining energy and A is the solution of (30) which contradicts (29).

Appendix B2 – Proof of Theorem 3 First, we show that Ak is monotone increasing. Let us denote the solution of (30) as a function F (.) and similarly the solution of (31) is represented by a function G (.), namely A = F (), = G (A), respectively. Furthermore, let us introduce H (, A) =

"

Ψ dil il+1 , cil − A ,

(44)

il ∈

where dil il+1 argument represents the distance between the nodes. Then Ak = F (k )

(45)

gives us the path k with an optimal Ak selection which satisfies the (1−ε) criterion. Then we select a new path as follows: k+1 = G ( Ak )

(46)

is more reliable, since function G (.) seeks the most reliable path based on A, thus k+1 will be more reliable than k : "

Ψ dil il+1 , cil − A > 1 − ε.

(47)

il ∈k+1

If there is no more reliable route, then we get stuck in a fix point. The monotonicity of Ψ (.) and (47) implies that Ak+1 = F (k+1 )

(48)

will give a solution where Ak+1 > Ak . Second, it will be shown that if A = F (G ( A))

(49)

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is a fix point then ∃A∗ > A, A∗ = F G A∗ .

(50)

In other words there exists only one fix point of Algorithm 1. It is trivial that H (G ( A) , A) = 1 − ε,

(51)

because of (49). Since G (.) selects the most reliable path it holds that ∀ = G (A) ,

H (, A) ≤ 1 − ε.

(52)

Let us assume indirectly that there exists A∗ > A fix point. In this case, ∀ = G ( A) ,

H , A∗ < 1 − ε.

(53)

But A∗ cannot be a fix point because (30) always ensures that the chosen strategy fulfills the QoS requirement.

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Opportunistic Scheduling with Deadline Constraints in Wireless Networks David I Shuman and Mingyan Liu

1 Introduction We consider a single source transmitting data to one or more users over a wireless channel. It is highly desirable to operate such a wireless system in an energyefficient manner. When the sender is a mobile device relying on battery power, this is obvious. However, even when the sender is a base station that is not power constrained, it is still desirable to conserve energy in order to limit potential interference to other base stations and their associated mobiles. Due to random fading, wireless channel conditions vary with time and from user to user. The key realization from a transmission scheduling perspective is that these channel variations are not a drawback, but rather a feature to be beneficially exploited. Namely, transmitting more data when the channel between the sender and receiver is in a “good” state and less data when the channel is in a “bad” state increases system throughput and reduces total energy consumption. Doing so is commonly referred to as opportunistic scheduling. In this chapter, we are particularly interested in delay-sensitive applications, such as multimedia streaming, voice over Internet protocol (VoIP), and data upload or transfer with a time restriction. In such applications, packets are often subject to hard deadline constraints, after which time their transmission provides limited or no benefit. Our objective is to review energy-efficient opportunistic transmission scheduling policies that also comply with deadline constraints. Specifically, we want to provide an intuitive understanding of how the deadline constraints affect the scheduler’s optimal behavior. David I Shuman Electrical Engineering and Computer Science Department, University of Michigan, Ann Arbor, MI 48109, USA e-mail: [email protected] Mingyan Liu Electrical Engineering and Computer Science Department, University of Michigan, Ann Arbor, MI 48109, USA e-mail: [email protected]


127

128

D.I Shuman and M. Liu

The remainder of the chapter is organized as follows. In the next section, we introduce the concept of opportunistic scheduling and provide some examples. In Section 3, we review some common modeling issues in opportunistic scheduling problems in wireless networks. We formulate three related opportunistic scheduling problems with deadline constraints, and then elucidate the role of the deadline constraints in Section 4. In Section 5, we relate the models of Section 4 to models from inventory theory. Section 6 concludes the chapter.

2 Opportunistic Scheduling We motivate opportunistic scheduling with a few simple example problems. Example 1 Consider a channel that can be in one of M channel conditions, with probabilities p1 , p 2 , . . . , p M , respectively. Associated with each channel condition is a known convex, increasing, differentiable power-rate function, f 1 (z), f 2 (z), . . . , f M (z), respectively, describing the power required to transmit at data rate z (or equivalently the energy required to transmit z packets in a discrete time slot or time unit). The objective is to minimize the average power consumed ¯ This probover an infinite horizon, subject to a minimum average rate constraint, R. lem reduces to the following convex optimization problem: min

M i=1

pi · f i z i

(z 1 ,z 2 ,...,z M )∈IR+M M i i ¯ s.t. i=1 p · z ≥ R,

(1)

where z i represents the number of packets transmitted per time slot when the channel is in condition i. The solution to (1) is found by reducing (in the same manner as [11, Example 5.2, p. 245]) the Karush–Kuhn–Tucker (KKT) conditions to ' ( ∗ ∗ ∗ ∗ z i ≥ 0, z i · pi · ν ∗ + f i (z i ) = 0, and ν ∗ + f i (z i ) ≥ 0 ∀i ∈ {1, 2, . . . , M}, and pT z∗ = R¯ , where ν ∗ is the Lagrange multiplier associated with the rate constraint. Graphically, the so-called inverse water-filling solution is found by fixing the slope of a tangent line and setting the number of packets to be transmitted under condition i to be a z i such that f i (z i ) is equal to the slope, or zero if f i (z i ) is greater than the slope for continuously repeated as the slope of the tangent line is all z i ≥ 0. This process is M ¯ The resulting optimal solution z∗ has pi · z i = R. gradually increased until i=1 ∗ the property that for every channel condition i, the optimum number of packets z i is either equal to zero or satisfies fi (z i ) = −ν ∗ , where −ν ∗ is the slope of the final tangent line. See Fig. 1 for a diagram of this solution. Example 2 Next, we consider the same infinite horizon average cost problem as (1), with the additional stipulations that (i) the power-rate function in each channel

Opportunistic Scheduling in Wireless Networks

129

Fig. 1 Pictorial representation of the solution to (1). The vector z∗ of the optimal number of packets ∗ to transmit under each channel condition has the property that f i (z i ) is the same for all channel ∗ i conditions i such that z > 0

condition is linear, with slope φ i , and (ii) there is a power constraint P in each slot. In other words, ⎧ P i i i ⎨ φ · z , if z ≤ φ i ' ( ⎪ i fi z = . ⎪ ⎩ ∞, if z i > P i φ We assume without loss of generality that φ 1 ≤ φ 2 ≤ · · · ≤ φ M (i.e., φ 1 is the slope of the power-rate function under the best channel condition and φ M is the slope under the worst condition). With these assumptions, the problem becomes (

min

z 1 ,z 2 ,...,z M

M )

M ∈IR+

s.t.

i=1

M i=1

zi ≤

and

pi · φ i · zi p i · z i ≥ R¯

P φi

,

(2)

∀i ∈ {1, 2, . . . , M}

where z i represents the number of packets transmitted per time slot when the channel is in condition i. The solution to (2) is found by defining ⎧ ⎫ j ⎨ ⎬ P j ∗ := min j ∈ {1, 2, . . . , M} : p m · m ≥ R¯ . ⎩ ⎭ φ m=1

Then the optimal amount of data to send under each channel condition is given by

∗

z m :=

⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩

P φm ,

j ∗ −1 m P ¯ R− m=1 p · φ m pj

∗

0,

if m < j ∗ , if m = j ∗ .

(3)

if m > j ∗

See Fig. 2 for a diagram of this solution. Examples 1 and 2 illustrate the main idea of exploiting the temporal variation of the channel via opportunistic scheduling; namely, we can reduce energy con-

130


P

P

z1

0 *

z1

P

P

z2

0 z2

*

z3

0 *

z3

zM

0 zM

*

Fig. 2 Pictorial representation of the solution to (2). Each plot represents the power-rate curve under a different channel condition. The full power available is used for transmission when the channel is in its best condition(s), and no packets are transmitted when the channel is in its worst condition(s)

sumption by sending more data when the channel is in a “good” state, and less data when the channel is in a “bad” state. Much of the challenge for the scheduler lies in determining how good or bad a channel condition is, and how much data to send accordingly. In Examples 1 and 2, the scheduler was scheduling packets for a single receiver, but it is often the case in wireless communication networks that a single source sends data to multiple users over a shared channel. Such a downlink system model is shown in Fig. 3. In this situation, the scheduler can exploit both the temporal variation and the spatial variation of the channel by sending data to the receivers with the best conditions in each slot. The benefit from doing so is commonly referred to as the multiuser diversity gain [77]. It was introduced in the context of the analogous uplink problem where multiple sources transmit to a single destination (e.g., the base station) [45].

Fig. 3 Multiuser downlink system model. A single source transmits data to multiple users over a shared wireless channel

3 Modeling Issues and Literature Review There is a wide range of literature on opportunistic scheduling problems in wireless communications. This section is by no means intended to be an exhaustive survey of problems that have been examined, but rather an introduction to some of the most common modeling issues. For more complete surveys of opportunistic scheduling studies in wireless networks, see [50, 52].


131

3.1 Wireless Channel Modeling the wireless channel deserves an entire book in its own right. For a good introduction to the topic, see [74]. Here, we restrict attention to modeling the wireless channel at the simplest level required for opportunistic scheduling problems, without considering any specific modulation or coding schemes. In this context, the condition of the time-varying wireless channel is usually modeled as either (i) independently and identically distributed (IID) over time; or (ii) a discrete-time Markov process. In the case of multiple receivers, as shown in Fig. 3, the channels between the sender and each receiver may or may not be correlated in each time slot. For a detailed introduction to modeling fading channels as Markov processes, see [63]. In general, the transmitter can reliably send data across the channel at a higher rate by increasing transmission power. For each possible channel condition, there is a corresponding power-rate curve that describes how much power is required to transmit at a given rate. In the low signal-to-noise ratio (SNR) regime, this powerrate curve is commonly taken to be linear and strictly increasing. In the high SNR regime, the power-rate curve is commonly taken to be convex and strictly increasing [74, Section 5.2]. For a justification of the convex assumption, see [76]. Specific convex power-rate curves that have been considered in the literature include (i) z c(z, s) = α2 1−1 (s) (see, e.g., [47]), motivated by the capacity of a discrete-time additive μ

white Gaussian noise (AWGN) channel, and (ii) c(z, s) = α2z (s) (see, e.g., [48]), where in both cases, c(z, s) is the power required to transmit at rate z under channel condition s, μ is a fixed parameter, and the αi ’s are parameters that may depend on the channel condition.

3.2 Channel State Information In this chapter, we assume that, through a feedback channel, the transmission scheduler learns perfectly (and for free) the state of the channel between the sender and each receiver at the beginning of every time slot. Thus, its scheduling decisions are based on all past and current states of the channel(s), but none of the future channel realizations. This set of assumptions is commonly referred to as causal or full channel state information. Some papers such as [17, 75] also refer to problems resulting from this assumption on the scheduler’s information as online scheduling problems, to differentiate them from offline scheduling problems, where the scheduler learns all future channel realizations at the beginning of the time horizon. For a recent survey of research on systems with limited feedback, which may cause the channel state information to be outdated or suffering from errors, see [53]. References [26, 29, 64, 70, 82] also discuss ways to deal with restrictions on the timing and amount of feedback in an opportunistic scheduling context. A second relaxation of the perfect channel state information assumption is to force the scheduler to decide whether or not to attain channel state information at some cost, which represents the time and energy consumed in learning the channel

132


state. The process of learning the channel is often referred to as probing, and [12– 14, 32, 33, 37, 57, 62] are examples of studies that examine the best joint strategies for probing and transmission in different contexts.

3.3 Data The simplest and often most tractable way to model the data is that the sender has an infinite backlog of data to send to each receiver. Analysis under this assumption gives a bound on the maximum achievable performance of a system in terms of throughput. Alternatively, one can assume that data arrive to the sender’s buffer over time and explicitly model the arrival process. The arrival process may be deterministic (often the case in offline scheduling problems, where the scheduler is assumed to learn the times of all future arrivals at the beginning of the horizon), an IID sequence of random variables (as in [18]), a Poisson process (as in [3]), a discretetime Markov process (as in [2]), or just about any other stochastic process appropriate for a given application. With an arriving packet model, the scheduler’s control policies often depend on both the current queue length of packets backlogged at the sender and the statistics of future arrivals. It may also be the case, as in [3], that the sender’s buffer to store arriving packets is of finite length. If so, the scheduler must take care to avoid having to drop packets due to buffer overflow. Finally, the opportunistic scheduling literature is divided on the treatment of a “packet.” Some studies take the data to be some integer number of packets that cannot be split, while others consider a fluid packet model that allows packets to be split, with the receiver reassembling fractional packets.

3.4 Performance Objectives Broadly speaking, opportunistic scheduling problems in wireless networks focus on the trade-offs between energy efficiency, throughput, and delay. With some exceptions (e.g., [6]), delay is usually modeled as a QoS constraint (a maximum acceptable level of delay), rather than a quantity to be directly minimized. In many opportunistic scheduling problems, delay is not even considered, with the justification that some applications are not delay sensitive. We discuss delay further in the next section. Thus, the two most basic setups are (i) to maximize throughput, subject to a constraint on the maximum average or total energy expended, and (ii) to minimize energy consumption, subject to a constraint on the minimum average or total throughput (as in Examples 1 and 2). These two problems are dual to each other, and many similar techniques can therefore be used to solve both problems. Examples of studies that solve both problems for a similar setup and relate their solutions are [25, 48].


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3.5 Resource and Quality-of-Service Constraints In this section, we provide a brief introduction to some common resource and QoS constraints. 3.5.1 Transmission Power Due to hardware and/or regulatory constraints, a limit is often placed on the sender’s transmission power in each slot. Some models allow the sender to transmit to multiple users in a slot, with the total transmission power not exceeding a limit, while others only allow the sender to transmit data to a single user in each slot. This power constraint is often left out of problems where the power-rate curve is strictly convex, as the increasing marginal power required to increase the transmission rate prevents the scheduler from wanting to increase transmission power too much. However, the absence of a power constraint in a problem with a linear power-rate curve would often result in the scheduler wanting to increase transmission power well beyond a reasonable limit in order to send a large amount of data when the channel condition is very good (see, e.g., [25]). 3.5.2 Delay Delay is an important QoS constraint in many applications. Different notions of delay have been incorporated into opportunistic scheduling problems. One proxy for delay is the stability of all of the sender’s queues for arriving packets awaiting transmission. The motivation for this criterion is that if none of these queues blows up, then the delay is not “too bad.” With stability as an objective, it is common to restrict attention to throughput optimal policies, which are scheduling policies that ensure the sender’s queues are stable, as long as this is possible for the given arrival process and channel model. References [2, 58, 65, 72] present such throughput optimal scheduling algorithms and examine conditions guaranteeing stabilizability in different settings. When an arriving packet model is used for the data, then one can also define end-to-end delay as the time between a packet’s arrival at the sender’s buffer and its decoding by the receiver. A number of opportunistic scheduling studies have considered the average end-to-end delay of all packets over a long horizon. For instance, [1, 6, 7, 9, 18, 20, 30, 31, 43, 44, 61, 78] all consider average delay, either as a constraint or by incorporating it directly into the objective function to be minimized. However, the average delay criterion allows for the possibility of long delays (albeit with small probability); thus, for many delay-sensitive applications, strict end-to-end delay is often a more appropriate consideration for studies with arriving packet models. References [12, 16, 54, 61] consider strict constraints on the end-to-end delay of each packet. A strict constraint on the end-to-end delay of each packet is one particular form of a deadline constraint, as each arriving packet has a deadline by which it must be transmitted (which happens to be a fixed number of slots after its arrival). This

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notion can be generalized to impose individual deadlines on each packet, whether or not the packets are arriving over time or are all in the sender’s buffer from the beginning, as with the case of infinite backlog. Studies that impose such individual packet deadlines include [17, 68]. In [24, 25, 46–49, 71, 75], the individual deadlines coincide, so that all the packets must be received by a common deadline (usually the end of the time horizon under consideration). We further examine the role of these deadline constraints in Section 4. 3.5.3 Fairness If, in the multiuser setting shown in Fig. 3, the scheduler only considers total throughput and energy consumption across all users, it may often be the case that it ends up transmitting to a single user or to the same small group of users in every slot. This can happen, for instance, if a base station requires less power to send data to a nearby receiver, even when the nearby receiver’s channel is in its worst possible condition and a farther away receiver’s channel is in its best possible condition. Thus, fairness constraints are often imposed to ensure that the transmitter sends packets to all receivers. A number of different fairness conditions have been examined in the literature. For example, [5, 51] consider temporal fairness, where the scheduler must transmit to each receiver for some minimum fraction of the time over the long run. Under the proportional fairness considered by [2, 35, 77], the scheduler considers the current channel conditions relative to the average channel condition of each receiver. Reference [51] considers a more general utilitarian fairness, where the focus is on system performance from the receiver’s perspective, rather than on resources consumed by each user. The authors of [10] incorporate fairness directly into the objective function by setting relative throughput target values for each receiver and maximizing the minimum relative long-run average throughput.

4 The Role of Deadline Constraints In this section, we elucidate the role of deadline constraints by examining a series of three related problems. The overarching goal in all three problems is to do energyefficient transmission scheduling, subject to deadline constraints.

4.1 Problem Formulations In all three problems, we consider a single source transmitting data to a single user/receiver over a wireless channel. Time evolution is modeled in discrete steps, indexed backwards by n = N , N −1, . . . , 1, with n representing the number of slots remaining in the time horizon. N is the length of the time horizon, and slot n refers to the time interval [n, n − 1).


135

The wireless channel condition is time varying. Adopting a block fading model, we assume that the slot duration is within the channel coherence time such that the channel condition within a single slot is constant. The user’s channel condition in slot n is modeled as a random variable, Sn . We take the channel condition to be independent and identically distributed (IID) from slot to slot, and denote its sample space by S. At the beginning of each time slot, the transmitter or scheduler learns the channel’s state through a feedback channel. It then allocates some amount of power (possibly zero) for transmission. If the channel condition is in state s, then the transmission of z data packets incurs an energy cost of c(z, s). Note that we allow data packets to be split, so the number of packets sent in a given slot can be any nonnegative real number. We also assume the channel condition evolution is independent of any of the transmitter’s scheduling decisions. Our primary objective in deriving a good transmission policy is to minimize energy consumption. However, in doing so, we must also meet the deadline constraint(s), and possibly a power constraint in each slot. Thus, all three problems we discuss in this section can be formulated as Markov decision processes (MDPs) with the following common form: min IE π

π ∈

# N

$ c(Z n , Sn ) | F N

n=1

s.t. Per-Slot Power Constraints and Deadline Constraint(s),

(4)

where F N denotes all information available at the beginning of the time horizon, and Z n = πn (Z N , Z N −1 , . . . , Z n+1 , S N , S N −1 , . . . , Sn ) is the number of packets the scheduler decides to transmit in slot n. The sequence π = (π N , π N −1 , . . . , π1 ) is called a control law, control policy, or scheduling policy, and denotes the set of all randomized and deterministic control laws (see, e.g., [34] definition 2.2.3, p. 15). Next, we specify the precise variant of (4) for each of the three problems. 4.1.1 Single Deadline Constraint, Linear Power-Rate Curves, and a Power Constraint in Each Slot The first problem we consider features linear power-rate curves, a power constraint in each slot, and a single deadline constraint. For each possible channel condition s, there exists a constant cs such that c(z, s) = cs · z. The maximum transmission power in any given slot is denoted by P, and the total number of packets that need to that be transmitted by the end of the horizon is denoted by dtotal . In order to ensure ' ( it is always possible to satisfy the deadline constraint, we assume that N · cs P ≥ worst

·P , where csworst is the energy cost per packet dtotal , or, equivalently, csworst ≤ dNtotal transmitted under the worst possible channel condition. Thus, even if the channel is in the worst possible condition for the entire duration of the time horizon, it is

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still possible to send dtotal packets by transmitting at full power in every slot. The general formulation (4) becomes

min IE

π∈

π

# N

$ c Sn · Z n | F N

n=1

. s.t. c Sn · Z n ≤ P, w. p.1 ∀n ∈ 1, 2, . . . , N and

N

Z n ≥ dtotal , w. p.1 .

n=1

We refer to this problem as problem (P1). It was introduced and analyzed by Fu et al. [24, 25, Section III-D].

4.1.2 Strict Underflow Constraints, Linear Power-Rate Curves, and a Power Constraint in Each Slot The second problem we consider features exactly the same per-slot power constraint and linear power-rate curves as problem (P1); however, the deadline constraints come in the form of strict underflow constraints. Namely, the single receiver maintains a buffer to store received packets, as shown in Fig. 3 (here, M = 1). Following transmission in every slot, d packets are removed from the receiver buffer. Strict underflow constraints are imposed so that the transmitter must send enough packets in every slot to guarantee that the receiver buffer contains at least d packets following transmission. The primary motivating application for this problem is wireless media streaming, where the packets removed from the receiver buffer in each slot are decoded and used for playout. Underflow is undesirable as it may lead to jitter and disruptions to the user playout. These strict underflow constraints can also be interpreted as multiple deadline constraints: the source must transmit at least d packets by the end of the first slot, 2d packets by the end of the second slot, and so forth. As with problem (P1), we need to make an assumption to ensure that it is always possible to satisfy the deadline constraints; namely, we assume that cs P ≥ d, so worst that if the receiver buffer is empty at the beginning of the slot, the source can still send the required d packets, even if the channel is in the worst possible condition. For this problem, the general formulation (4) becomes

min IE

π ∈

π

# N

$ c Sn · Z n | F N

n=1

s.t. c Sn · Z n ≤ P, w. p.1 and

N n=k

. ∀n ∈ 1, 2, . . . , N

Z n ≥ (N − k + 1) · d, w. p.1

. ∀k ∈ 1, 2, . . . , N .


137

We refer to this problem as problem (P2). It was introduced and analyzed by Shuman et al. [66–68]. 4.1.3 Single Deadline Constraint and Convex Monomial Power-Rate Curves The third problem we consider features the same single deadline constraint as problem (P1); however, there is no per-slot power constraint imposed, and the energy cost from transmission is taken to be a convex monomial function of the number of packets sent. Namely, for every channel condition s, there exists a constant ks such μ that c(z, s) = zks , where μ > 1 is the fixed monomial order of the cost function. As mentioned in Section 3.1, such a power-rate curve may be more appropriate in the high SNR regime. The general formulation (4) becomes

min IE

π∈

s.t.

N

π

# N (Z n )μ n=1

k Sn

$ | FN

Z n ≥ dtotal , w. p.1 .

n=1

We refer to this problem as problem (P3). It was introduced and analyzed by Lee and Jindal [48].

4.2 Structures of the Optimal Policies In this section, we present the structures of the optimal policies for each of the three problems as straightforwardly as possible, without changing drastically the original presentations. All three problems can be solved using standard dynamic programming (see, e.g., [8, 34]), and the structures of the optimal policies follow from properties of the value functions or expected costs-to-go. For problem (P1), Fu et al. [25] take the information state at time n to be the pair (Q n , Sn ), where Q n represents the number of packets remaining to be transmitted at time n, and Sn denotes the channel condition in slot n. The dynamics of packets remaining to be transmitted are Q n−1 = Q n − Z n , as Z n packets are transmitted during slot n. The dynamic programming equations for this problem are given by Vn (q, s) =

min/

0≤z≤min q, cPs

= cs · q + # V0 (q, s) =

0

. cs · z + IE Vn−1 (q − z, Sn−1 )

/ min 0 max 0,q− cPs ≤u≤q

-

. −cs · u + IE Vn−1 (u, Sn−1 ) , n = N , N − 1, . . . , 1

0, if q = 0 ∞, if q > 0

∀s ∈ S.

(5) (6)

138


Here, the transition from (5) to (6) is done by a change of variable in the action space from Z n to Un , where Un = Q n − Z n . The controlled random variable Un represents the number of packets remaining to be transmitted after 0 takes place in / transmission the nth slot. The restrictions on the action space, max 0, q − cPs ≤ u ≤ q, ensure the following: (i) a nonnegative number of packets is transmitted; (ii) no more than dtotal packets are transmitted over the course of the horizon; and (iii) the power constraint is satisfied. For problem (P2), Shuman et al. [68] take the information state at time n to be the pair (X n , Sn ), where X n represents the number of packets in the receiver buffer at time n, and Sn denotes the channel condition in slot n. The simple dynamics of the receiver buffer are X n−1 = X n + Z n − d, as Z n packets are transmitted during slot n, and d packets are removed from the buffer after transmission. The dynamic programming equations for this problem are given by Vn (x, s) =

-

min

max{0,d−x}≤z≤ cPs

= −cs · x +

. cs · z + IE Vn−1 (x + z − d, Sn−1 )

min

max{x,d}≤y≤x+ cPs

-

(7)

. cs · y + IE Vn−1 (y − d, Sn−1 ) , (8) n = N , N − 1, . . . , 1

V0 (x, s) = 0

∀x ∈ IR+ , ∀s ∈ S.

Here, the transition from (7) to (8) is done by a change of variable in the action space from Z n to Yn , where Yn = X n + Z n . The controlled random variable Yn represents the queue length of the receiver buffer after transmission takes place in the nth slot, but before playout takes place (i.e., before d packets are removed from the buffer). The restrictions on the action space, max {x, d} ≤ y ≤ x + cPs , ensure the following: (i) a nonnegative number of packets is transmitted; (ii) there are at least d packets in the receiver buffer following transmission, in order to satisfy the underflow constraint; and (iii) the power constraint is satisfied. Note that the dynamic programming equations (6) and (8) have the following common form: Vn (x, s) = f (x, s) +

min

w1 (x,s)≤a≤w2 (x,s)

-

. h 1 (a) + IE Vn−1 (h 2 (a), s) ,

(9)

n = N , N − 1, . . . , 1 , where (x, s) is the current state and a represents the action. The key realizations for both problems are (i) h 1 (a) is convex in a and h 2 (a) is an affine function of a; and (ii) for any fixed s, f (x, s), w1 (x, s), and Vn−1 (x, s) are all convex in x, and w2 (x, s) is concave in x. These functional properties can be shown inductively using the following key lemma, which is due to Karush [40], and presented in [60, pp. 237–238].


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Lemma 1 (Karush [40]) Suppose that f : IR → IR and that f is convex on IR. For v ≤ w, define g(v, w) := min f (z). Then it follows that z∈[v,w]

(a) g can be expressed as g(v, w) = F(v) + G(w), where F is convex nondecreasing and G is convex nonincreasing on IR. (b) Suppose that S is a minimizer of f over IR. Then g can be expressed as ⎧ ⎨ f (v), if S ≤ v g(v, w) = f (S), if v ≤ S ≤ w. ⎩ f (w), if w ≤ S Using Lemma 1, we can write Vn (x, s) = f (x, s) + F(w1 (x, s)) + G(w2 (x, s)), which is convex in x for a fixed s, because F(w1 (x, s)) is the composition of a convex nondecreasing function with a convex function and G(w2 (x, s)) is the composition of a convex nonincreasing function with a concave function (see, e.g., [11, Section 3.2] for the relevant results on convexity-preserving operations). Furthermore, by Lemma 1, if, for a fixed s, βn (s) is a global minimizer of h 1 (a) + IE Vn−1 (h 2 (a), s) over all a, then the optimal action has the form ⎧ ⎨ w1 (x, s), if βn (s) < w1 (x, s) an∗ (x, s) := βn (s), if w1 (x, s) ≤ βn (s) ≤ w2 (x, s) . ⎩ w2 (x, s), if w2 (x, s) < βn (s)

(10)

The optimal transmission policy in (10) is referred to as a modified base-stock policy. Applying this line of analysis to the dynamic program (6) for problem (P1), we see that when the channel condition is in state s at time n, and there are q packets remaining to be transmitted by the deadline, the optimal action is given by ⎧ / 0 0 / P P ⎪ max 0, q − , if β (s) < max 0, q − ⎪ n ⎨ cs cs 0 / P , (11) u ∗n (q, s) := βn (s), ≤ β if max 0, q − n (s) ≤ q ⎪ cs ⎪ ⎩ q, if q < βn (s) for some sequence of critical numbers {βn (s)}s∈S . Changing variables back to the original action variable Z n and noting that βn (s) ≥ 0 for all n and s, (11) is equivalent to ⎧P if βn (s) + cPs < q ⎨ cs , ∗ (12) z n (q, s) := q − βn (s), if βn (s) ≤ q ≤ βn (s) + cP . s ⎩ 0, if q < βn (s) See Fig. 4 for diagrams of this optimal policy. When the number of possible channel conditions is finite (i.e., the sample space S of each random variable Sn is finite) and for every s ∈ S,

140


Fig. 4 Optimal action for problem (P1) in slot n when the state is (q, s), the number of packets remaining to be transmitted before transmission and the current channel condition. (a) z ∗ , the optimal transmission quantity. (b) u ∗ , the optimal number of packets remaining to be transmitted after transmission in slot n

cs =

csworst for some lˆ ∈ IN , lˆ

(13)

then the critical numbers {βn (s)}s∈S can be calculated recursively. For further details on the calculation of these critical numbers, see [25]. A similar line of analysis for problem (P2) leads to the following structure of the optimal control action with n slots remaining: ⎧ if x ≥ bn (s) ⎨ x, P yn∗ (x, s) := bn (s), if bn (s) − cs ≤ x < bn (s) . ⎩ P x + cs , if x < bn (s) − cPs

(14)

Furthermore, for a fixed s, bn (s) is nondecreasing in n, and for a fixed n, bn (s) is nonincreasing in cs ; i.e., for arbitrary s 1 , s 2 ∈ S with cs 1 ≤ cs 2 , bn (s 1 ) ≥ bn (s 2 ). At time n, for each possible channel condition realization s, the critical number bn (s) describes the ideal number of packets to have in the user’s buffer after transmission in the nth slot. If that number of packets is already in the buffer, then it is optimal to not transmit any packets; if there are fewer than ideal and the available power is enough to transmit the difference, then it is optimal to do so; and if there are fewer than ideal and the available power is not enough to transmit the difference, then the sender should use the maximum power to transmit. See Fig. 5 for diagrams of this optimal policy. When the number of possible channel conditions is finite and for every s ∈ S, P = l · d for some l ∈ IN , cs

(15)

then the critical numbers {βn (s)}s∈S can be calculated recursively. Condition (15) says that the maximum number of packets that can be transmitted in any slot covers


141

Fig. 5 Optimal action for problem (P2) in slot n when the state is (x, s). (a) the optimal transmission quantity. (b) the resulting number of packets available for playout in slot n

exactly the playout requirements of some integer number of slots. For further details on the calculation of these critical numbers, see [68]. Like problem (P1), Lee and Jindal [48] take the information state for problem (P3) to be the pair (Q n , Sn ), where Q n represents the number of packets remaining to be transmitted at time n, and Sn denotes the channel condition in slot n. The dynamics of packets remaining to be transmitted are once again Q n−1 = Q n − Z n . The dynamic programming equations for problem (P3) are given by #

$ zμ Vn (q, s) = min + IE Vn−1 (q − z, Sn−1 ) , z≥0 ks n = N , N − 1, . . . , 1 # 0, if q = 0 V0 (q, s) = . ∞, if q > 0

(16)

The key idea of Lee and Jindal is to show inductively that IE Vn−1 (q − z, Sn−1 ) = ξn−1,μ · (q − z)μ for some constant ξn−1,μ that depends on the time n − 1 and the known monomial order μ. Therefore, z n∗ (q, s) = argmin z≥0

#

zμ + ξn−1,μ · (q − z)μ ks

$ .

(17)

Differentiating the inner term of the right-hand side of (17) with respect to z and setting it equal to zero yields z n∗ (q, s) = λn,μ (s) · q, for some λn,μ (s) ∈ [0, 1]. Thus, with n slots remaining in the time horizon, the optimal control action is to send a fraction, λn,μ (s), of the remaining packets to be sent. Here, the fraction to send depends on the time remaining in the horizon, n; the current condition of the channel, s; and the parameter representing the monomial order of the cost function, μ. The fractions λn,μ (s) can be computed recursively. Note that plugging the optimal z n∗ (q, s) back into (16) yields

142


μ (λn,μ (S) · q)μ = ξn,μ · q μ IE Vn (q, S) = IE + ξn−1,μ · q − λn,μ (S) · q kS forsome constant ξn,μ , completing the induction step on the form of IE Vn (q, S) . Finally, Lee and Jindal also show that for each fixed channel state s, the fraction λn,μ (s) is decreasing in n. In other words, the scheduler is more selective or opportunistic when the deadline is far away, as it sends a lower fraction of the remaining packets than it would under the same state closer to the deadline. This makes intuitive sense as it has more opportunities to wait for a very good channel realization when the deadline is farther away.

4.3 Comparison of the Problems In this section, we provide further intuition behind the role of deadlines by comparing the above problems. First, we show that problems (P1) and (P2) are equivalent when a certain technical condition holds. Next, we examine how the extra deadline constraints in problem (P2) affect the optimal scheduling policy, as compared with problem (P1). We finish with some conclusions on the role of deadline constraints. 4.3.1 A Sufficient Condition for the Equivalence of Problems (P1) and (P2) In this section, we transform the dynamic programs (6) and (8) to find a condition under which problems (P1) and (P2) are equivalent. In problem (P1), there is just a single deadline constraint; however, because the terminal cost is set to ∞ if all the data are not transmitted by the deadline, the scheduler must transmit enough data in each slot so that it can still complete the job if the channel is in the worst possible condition in all subsequent slots. Thus, the scheduler can leave no more than cs P packets for the final slot, no more than worst

2 · cs P packets for the last two slots, and so forth. So there are in fact implicit worst constraints on how much data can remain to be transmitted at the end of each slot. If we make these implicit constraints explicit, then the dynamic program (6) becomes Vn (q, s) = cs · q +

min/ 0 / max 0,q− cPs ≤u≤min q,(n−1)· cs

P worst

0

. −cs · u + IE Vn−1 (u, Sn−1 ) , n = N , N − 1, . . . , 1

V0 (q, s) = 0

∀q ∈ [0, dtotal ], ∀s ∈ S.

Next, we change the state space from total packets remaining to be transmitted to total packets transmitted since the beginning of the horizon (Tn = dtotal − Q n ), and we change the action space from total packets remaining after transmission in the


143

nth slot to total packets sent after transmission in the nth slot ( An = dtotal − Un ). The resulting dynamic program is Vn (t, s) = −cs · t + / max

min0

t,dtotal −(n−1)· cs P worst

-

/ 0 cs ≤a≤min t+ cPs ,dtotal

. · a + IE Vn−1 (t, Sn−1 ) ,

n = N , N − 1, . . . , 1 V0 (t, s) = 0

∀t ∈ [0, dtotal ], ∀s ∈ S.

(18)

In problem (P2), it is never optimal to fill the buffer beyond n · d at time n. This is easily shown through a simple interchange argument, and we can therefore impose this as an explicit constraint. We can also do similar changes of variables as above to change the state space from packets in the receiver’s buffer to total packets transmitted since the beginning of the horizon (Tn = X n + (N − n) · d), and to change the action space from packets in the receiver’s buffer following transmission to total packets sent after transmission in the nth slot (An = Yn + (N − n) · d). With these changes of variables, the dynamic program (8) becomes Vn (t, s) = −cs · t +

min

/ 0 max{t,(N −n+1)·d}≤a≤min t+ cPs ,N ·d

-

. cs · a + IE Vn−1 (t, Sn−1 ) , n = N , N − 1, . . . , 1

V0 (t, s) = 0 ∀t ∈ [0, N · d], ∀s ∈ S.

(19)

The dynamic programs (18) and (19) associated with problems (P1) and (P2), respectively, become identical when the following two conditions are satisfied: (C1) dtotal = N · d (i.e., the total number of packets to send over the horizon of N slots is the same for both problems). (C2) cs P = d (i.e., the maximum number of packets that can be transmitted under worst the worst channel condition is equal to the number of packets removed from the receiver’s buffer at the end of each slot in problem (P2)). Furthermore, if condition (C2) is not satisfied, then

P csworst

> d, because we require

≥ d for problem (P2) to be well defined. Thus, when condition (C1) is that cs worst satisfied, but condition (C2) is not satisfied, the action space at time n and state (t, s) in (18) contains the action space at the same time and state in (19). This is because the explicit deadline constraints resulting from the strict underflow constraints in problem (P2) are more restrictive than the implicit deadline constraints in problem (P1). P

144


4.3.2 Inverse Water-Filling Interpretations In this section, we interpret problems (P1) and (P2) within the context of the inverse water-filling procedure introduced in Section 2. The aim is to show how the extra deadline constraints in problem (P2) affect the optimal scheduling policy. We again start with problem (P1), which features a single deadline constraint. If, at the beginning of the horizon, the scheduler happens to know the realizations of all future channel conditions, s N , s N −1 , . . . , s1 , then problem (P1) reduces to the following convex optimization problem: (

min

N )

N z N ,z N −1 ,...,z 1 ∈IR+

s.t. and

n=1 csn

N

n=1 z n z n ≤ cPs n

· zn (20)

≥ dtotal ∀n ∈ {1, 2, . . . , N } .

It should be clear that (20) is essentially the same problem as (2), and the solution can be found by scheduling data transmission during the slot with the best condition until all the data are sent or the power limit is reached, and then scheduling data transmission during the slot with the second best condition until all the data are sent or the power limit is reached, and so forth. See Fig. 6 for a diagram of this solution. P

P

z6

0 0

2d

4d

P

z5

0 0

2d

4d

c s5

P

z4

0 0

2d

4d

c s4

P

z3

0 0

2d

4d

c s3

P

z2

0 0

2d

4d

cs2

z1

0 0

2d

4d

c s1

Fig. 6 Pictorial representation of the solution to problem (P1) in the somewhat unrealistic case that all future channel conditions are known at the beginning of the horizon. Packets are scheduled in slots in ascending order of csn , until all the data are transmitted or the power constraint for the slot is reached. In the example shown, the time horizon to send the data is N = 6, the total number of data packets to be sent is 6d, and the power constraint in each slot is P = 4d. One optimal policy is to transmit 4d packets in slot 2, which has the best channel condition, and the remaining 2d packets in slot 5, which has the second best channel condition. This policy results in a total cost of 2P

If we are focused on finding the optimal amount to transmit in the current slot, we can also aggregate the power-rate functions of all future slots, by reordering them according to the strength of the channel, as shown in Fig. 7. The aggregate power-rate curve shown is defined by c˜ N −1 (˜z , s N −1 , s N −2 , . . . , s1 ) :=

N −1

n=1 csn · z n (z N −1 ,...,z1 )∈IR+N N −1 s.t. n=1 z n = z˜ and z n ≤ cPs ∀n ∈ {1, 2, . . . , N − 1} , n (21)

min


145

5P

4P

3P

2P

P

P

z6

0 0

2d

~ z

0 0

4d

2d

4d

6d

8d

10d

cs6 Fig. 7 The aggregate power-rate function for future slots. Aggregating the power-rate functions of the final five slots from Fig. 6 allows us to determine the optimal number of packets to transmit in the current slot by comparing the current slope to the slopes of the aggregate curve. In this case, the slope of the current curve is greater than the slope of the aggregate curve at all points up to dtotal = 6d, so it is optimal to not transmit any packets in the current slot

where z˜ is the aggregate number of packets to be transmitted in slots N − 1, N − 2, . . . , 1. The optimal number of packets/to transmit in the current slot 0is then determined as follows. Define γ N := min z˜ 0 : ψ˜ N −1 (˜z ) ≥ cs N , ∀˜z > z˜ 0 , where ψ˜ N −1 (·) is the slope from above of the aggregate power-rate curve, c˜ N −1 (·, s N −1 , s N −2 , . . . , s1 ), shown in Fig. 7. Then the optimal number of packets to transmit in slot N is given by z ∗N

#

$ P = min , max {dtotal − γ N , 0} . cs N

(22)

This policy says that if the current per packet energy cost from transmission is greater than the slope of the aggregate curve at all points up to dtotal , then it is optimal to not transmit any packets in the current slot. Otherwise, the optimal number of packets to transmit in the current slot N is the minimum of the maximum number of

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packets that can be transmitted under the current channel condition and the number of packets that would otherwise be transmitted in worse channel conditions in future slots. Now, as Fu et al. explain in [24, 25, Section III-D], in the more realistic case that the channel condition in slot n is not learned until the beginning of the nth slot, a very similar aggregate method can be used as long as the number of possible channel conditions is finite and condition (13) is satisfied. In this situation, however, the slopes of the piecewise-linear aggregate power-rate function for future slots are not defined in terms of the actual channel conditions of future slots (which are not available), but rather by a series of thresholds that only depend on the statistics of future channel conditions. Condition (13) ensures that the slopes of this aggregate expected power-rate curve only change at integer multiples of cs P . The form of the worst optimal policy at time N is the same as (22), with dtotal being the number of packets remaining to transmit at time N . Because the slopes of the aggregate expected power-rate curve only change at integer multiples of cs P , we have worst

# γ N ∈ 0,

P csworst

,2 ·

P csworst

, . . . , (N − 1) ·

P csworst

$ .

We now return to the wireless streaming model considered in problem (P2), with d packets removed from the receiver’s buffer at the end of every slot. Let us once again begin by considering the unrealistic case that the scheduler knows all future channel conditions at the beginning of the horizon. The optimal solution can be found by using the same basic inverse water-filling type principle of transmitting as much as possible in the slot with the best channel condition, and then the second best, and so forth; however, due to the additional underflow constraints, one needs to solve N sequential problems of this form. The first problem is the trivial problem of sending d packets in the first slot, [N , N − 1). The second problem is to send 2d packets in the first two slots. If the power limit in the first slot has not been reached after allocating the initial d packets there, then the scheduler may choose to send the second batch of d packets in either the first or second slot, according to their respective channel conditions. For each sequential problem, whatever packets have been allocated in the previous problem must be “carried over” to the subsequent problem, where there is one additional time slot available and the next d packets are allocated. The solution to the Nth problem represents the optimal allocation. See Fig. 8 for a diagram of this solution. Comparing Fig. 8 to Fig. 6, we see that when N · d = dtotal and the known sequence of channel conditions is the same for both problems, the additional underflow constraints cause more data to be scheduled in earlier time slots with worse channel conditions. When all future channel conditions are known ahead of time, as in Fig. 8, we can also use the same aggregation technique from above to represent problems 2 to N as comparisons between the current channel condition and the aggregate of the future channel conditions. Furthermore, when the future channel conditions are not known ahead of time and condition (15) is satisfied, we can once again define the aggregate expected power-rate function for future slots in terms of a series of


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Fig. 8 Pictorial representation of the solution to problem (P2) in the somewhat unrealistic case that all future channel conditions are known at the beginning of the horizon. In the example shown, the time horizon is N = 6, d packets are removed from the receiver’s buffer at the end of every slot, and the power constraint in each slot is P = 4d. To satisfy the underflow constraints, six sequential problems are considered, with an additional d packets allocated in each problem. Packets allocated in one problem are “carried over” to all subsequent problems and shown in solid black filling. The optimal policy, given by the solution to problem 6, is to transmit d packets in slots 6 and 3, and 2d packets in slots 5 and 2. This policy results in a total cost of 3P

thresholds that only depend on the statistics of future channel conditions. Due to the underflow constraints, however, these thresholds are computed differently than those in problem (P1). The net result for this more realistic case is the same as the case when all future channel conditions are known – the additional underflow constraints make it optimal to send more data in earlier time slots with worse channel conditions.

4.4 Extensions and Other Energy-Minimizing Transmission Scheduling Studies Featuring Strict Deadline Constraints We have presented these three deadline problems in their most basic form in order to enable comparisons, but they can also be extended in a number of different ways. For instance, the structures of the optimal policies for problems (P1) and (P2), presented in Section 4.2, also hold for a Markovian channel. For problem (P2), a

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modified base-stock policy is also optimal if the sequence of packet removals from the receiver buffer is nonstationary or if the optimization criterion is the infinite horizon discounted expected cost. When the linear power-rate curves of problem (P2) are generalized to piecewise-linear convex power-rate curves, a finite generalized base-stock policy, which is discussed further in Section 5, is optimal. Perhaps most interesting from a wireless networking standpoint and most difficult from a mathematical standpoint is the extension to the case of a single source transmitting data to multiple receivers over a shared channel. In [71], Tarello et al. extend problem (P1) (without the power constraint) to the case of multiple identical receivers and assume that the source can only transmit to one user in each slot. The extension of problem (P2) to the case of multiple receivers is discussed in [66, 68]. In addition to the three problems discussed above and their extensions, there have been a few other studies of energy-minimizing transmission scheduling that feature a time-varying wireless channel and strict deadline constraints. In [46, 47, 49], Lee and Jindal consider the same setup as problem (P3), except that the convex powerz z or e α−1 , which are based on the Gausrate curves are of the form c(z, s) = 2 α−1 s s sian noise channel capacity. The earlier models of Zafer and Modiano in [79, 80] also include essentially the same setup as problem (P3), with the exception that the underlying timescale is continuous rather than discrete. Using continuous-time stochastic control theory, they also reach the key conclusion that the optimal number of packets to transmit under convex monomial power-rate curves is the product of the number of packets remaining to be sent and an “urgency” fraction that depends on the current channel condition and the time remaining until the end of the horizon. Chen et al. [16, 17] and Uysal-Biyikoglu and El Gamal [75] consider packets arriving at different times, analyze offline scheduling problems, and use the properties of the optimal offline scheduler to develop heuristics for online (or causal) scheduling problems. An overview of the models considered in each of these studies is provided in Table 1. Additionally, Luna et al. [54] consider an energy minimization problem subject to end-to-end delay constraints, where the scheduler must select various source coding parameters in addition to the transmission powers. Finally, there is a sizeable literature on energy-efficient transmission scheduling studies such as [76] that feature a time-invariant or static channel and strict deadline constraints; however, we do not discuss these studies further, so as to keep the focus on the opportunistic scheduling behavior resulting from the time-varying wireless channel.

4.5 Summary Takeaways on the Role of Deadline Constraints As mentioned earlier, the main idea of opportunistic scheduling is to reduce energy consumption by sending more data when the channel is in a “good” state and less data when the channel is in a “bad” state. However, deadline constraints may force the sender to transmit data when the channel is in a relatively poor state. One strategy when faced with such deadline constraints would be to deal with them as they come, by always sending just enough packets when the channel is “bad” to ensure the deadline can be met, and holding out for the best channel conditions to send


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Table 1 Overview of models for energy-minimizing transmission scheduling that feature a time-varying wireless channel and strict deadline constraints Study

Time

Num. of receivers

Fu et al. [24, 25]

Discrete

Shuman et al. [66–68]

Discrete

Lee and Discrete Jindal [48] Lee and Discrete Jindal [46, 47, 49]

Data

Deadline constraints

Scheduler’s information

Power-rate curves

1

Infinite backlog

Single deadline

Non-causal; causal

Multiple (focus on 1, 2)

Infinite backlog

Multiple Causal deadlines (underflow constraints)

1

Infinite backlog Infinite backlog

Single deadline Single deadline

Causal

Infinite backlog; random packet arrivals Packet arrivals

Causal

Convex; linear with power constraint Linear and piecewiselinear convex with power constraint Convex monomial Convex (Gaussian noise channel capacity) Convex; convex monomial

1

Zafer and Modiano [79, 80]

Continuous 1

Chen et al. [16, 17]

Discrete

UysalDiscrete Biyikoglu and El Gamal [75] Tarello et al. Discrete [71]

Multiple (focus on 2)

Packet arrivals

Single deadline; multiple variable deadlines Individual packet deadlines Single deadline

Multiple

Infinite backlog

Single deadline

1

Causal

Non-causal; causal

Convex

Non-causal; causal

Convex

Non-causal; causal

Linear; convex

We use the term “infinite backlog” to include problems where there are a finite number of packets to be sent, all of which are queued at the beginning of the time horizon. “Non-causal” refers to the offline scheduling situation where the transmission scheduler has knowledge of future channel states and packet arrival times

a lot of data. Yet, a key conclusion from the analysis of the three problems we presented in this section is that it is better to anticipate the need to comply with these constraints in future slots by sending more packets (than one would without the deadlines) under “medium” channel conditions in earlier slots. In some sense, doing so is a way to manage the risk of being stuck sending a large amount of data over a poor channel to meet an imminent deadline constraint. We also saw that the extent to which the scheduler should plan for the deadline by sending data under such “medium” channel conditions depends on the time remaining until the

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deadline(s), and on how many deadlines it must meet; namely, the closer the deadlines and the more deadlines it faces, the less opportunistic the scheduler can afford to be. So perhaps the essence of opportunistic scheduling with deadline constraints is that the scheduler should be opportunistic, but not too opportunistic.

5 Relation to Work in Inventory Theory The models outlined in Section 4.1 correspond closely to models used in inventory theory. Borrowing that field’s terminology, our abstractions are multi-period, singleechelon, single-item, discrete-time inventory models with random ordering costs, a budget constraint, and deterministic demands. The item corresponds to the stream of data packets, the random ordering costs to the random channel conditions, the budget constraint to the power available in each time slot (there is no such budget constraint for problem (P3)), and the deterministic demands to the packet deadline constraints. In problems (P1) and (P2), the random ordering costs are linear, and in problem (P3), they are convex monomial. In problem (P2), the deterministic demands are stationary (d packets are removed from the inventory at the end of every time slot); in problems (P1) and (P3), the deterministic demand sequence is nonstationary, and equal to {0, 0, . . . , 0, dtotal }. To the best of our knowledge, the particular problems introduced in Section 4.1 have not been studied in the context of inventory theory, but similar problems have been examined. References [22, 27, 28, 38, 41, 42, 55, 56] all consider single-item inventory models with random ordering prices (linear ordering costs). The key result for the case of deterministic demand of a single item with no resource constraint is that the optimal policy is a base-stock policy with different target stock levels for each price. Specifically, at each time, for each possible ordering price (translates into channel condition in our context), there exists a critical number such that the optimal policy is to fill the inventory (receiver buffer) up to that critical number if the current level is lower than the critical number, and not to order (transmit) anything if the current level is above the critical number. Of the prior work, Kingsman [41, 42] is the only author to consider a resource constraint, and he imposes a maximum on the number of items that may be ordered in each slot. The resource constraint in problems (P1) and (P2) is of a different nature in that we limit the amount of power available in each slot. This is equivalent to a limit on the per-slot budget (regardless of the stochastic price realization), rather than a limit on the number of items that can be ordered. Of the related work on inventory models with deterministic ordering prices and stochastic demand, [23, 73] are the most relevant; in those studies, however, the resource constraint also amounts to a limit on the number of items that can be ordered in each slot and is constant over time. References [4, 69, 81] consider singleitem inventory models with deterministic piecewise-linear convex ordering costs and stochastic demand. The key result in this setup is that the optimal inventory level after ordering is a piecewise-linear nondecreasing function of the current inventory


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level (i.e., there are a finite number of target stock levels), and the optimal ordering quantity is a piecewise-linear nonincreasing function of the current inventory level. Porteus [59] refers to policies of this form as finite generalized base-stock policies, to distinguish them from the superclass of generalized base-stock policies, which are optimal when the deterministic ordering costs are convex (but not necessarily piecewise-linear), as first studied in [39]. Under a generalized base-stock policy, the optimal inventory level after ordering is a nondecreasing function of the current inventory level, and the optimal ordering quantity is a nonincreasing function of the current inventory level. Finally [15, 19, 21, 36] consider multi-item discretetime inventory systems under deterministic ordering costs, stochastic demand, and resource constraints. These studies are more relevant for the multiple receiver extensions discussed in Section 4.4. We have found some of the techniques used to solve these related inventory models to be quite informative in examining the opportunistic scheduling problems from wireless communications. For a more in-depth comparison of problem (P2) to the related inventory theory literature, see [70]. For more background on common models and techniques used in inventory theory, see [60, 83].

6 Conclusion In this chapter, we introduced opportunistic scheduling problems in wireless networks and specifically focused on the role of deadline constraints in this class of problems. We presented three opportunistic scheduling problems with deadline constraints, along with their solutions and outlines of the techniques used to analyze the problems. The roots of some of these techniques lie in inventory theory studies developed three to four decades earlier. By comparing the problems to each other and interpreting their solutions with water-filling-type principles, we were able to better understand the effect of the deadline constraints on the optimal scheduling policies. In particular, we concluded that the scheduler must anticipate the impending deadlines and adjust its behavior in earlier time slots to manage the risk of being stuck sending a large amount of data over a poor channel just before the deadline.

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A Hybrid Polyhedral Uncertainty Model for the Robust Network Loading Problem Ay¸segül Altın, Hande Yaman, and Mustafa Ç. Pınar

1 Introduction For a given undirected graph G, the network loading problem (NLP) deals with the design of a least cost network by allocating discrete units of capacitated facilities on the links of G so as to support expected pairwise demands between some endpoints of G. For a telephone company, the problem would be to lease digital facilities for the exclusive use of a customer where there is a set of alternative technologies with different transmission capacities. For example, DS0 is the basis for digital multiplex transmission with a signalling rate of 64 bits per second. Then, DS1 and DS3 correspond to 24 and 672 DS0s in terms of transmission capacities, respectively. The cost of this private service is the total leasing cost of these facilities, which is determined in a way to offer significant economies of scale. In other words, the least costly combination of these facilities would be devoted to a single customer to ensure communication between its sites. Then the customer would pay just a fixed amount for leasing these facilities and would not make any additional payment in proportion to the amount of traffic its sites exchange with each other. The structure of the leasing cost, which offers economies of scale, complicates the problem [31]. Although the traditional approach is to assume that the customer would be able to provide accurate estimates for point-to-point demands, this is not very likely to happen in real life. Hence, we relax this assumption and study the robust NLP to obtain designs flexible enough to accommodate foreseen fluctuations in demand.

Ay¸segül Altın Department of Industrial Engineering, TOBB University of Economics and Technology, Söˇgütözü 06560 Ankara, Turkey e-mail: [email protected] Hande Yaman Department of Industrial Engineering, Bilkent University, 06800 Ankara, Turkey e-mail: [email protected] Mustafa Ç. Pınar Department of Industrial Engineering, Bilkent University, 06800 Ankara, Turkey e-mail: [email protected]


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Our aim is to design least cost networks which remain operational for any feasible realization in a prescribed demand polyhedron. Efforts for incorporating uncertainty in data can be divided into two main categories. The first one is stochastic optimization(SO), where data are represented as random variables with a known distribution and decisions are taken based on expectations. However, it is computationally quite challenging to quantify these expectations. Moreover, given that limited or no historical data are available most of the time, it is not realistic to assume that exact distributions can be obtained reliably. Besides, SO yields decisions that might become infeasible with some probability. This latter issue might lead to undesirable situations especially when such a tolerance is not preferable. Alternatively, in robust optimization (RO) data are represented as uncertainty sets like polyhedral sets and the best decision is the one with the best worst-case performance, i.e., the one to handle the worst-case scenario in the uncertainty set in the most efficient way. Besides, RO is computationally tractable for some polyhedral or conic quadratic uncertainty sets and for many classes of optimization [36]. The interested reader can refer to [4, 6, 9–14, 33, 34, 40] for several examples of RO models and methodology. In network design, the most common component subject to uncertainty is the traffic matrix, i.e., the traffic demand between some node pairs in the network. In this chapter, we study the robust network loading problem under polyhedral uncertainty. A few polyhedral demand models have found acceptance in the telecommunications network design society. The initial efforts belong to Duffield et al. [18] and Fingerhut et al. [20], who propose the so-called hose model independently, for the design of virtual private networks (VPNs) and broadband networks, respectively. The hose model has become quickly popular since it handles complicated communication requests efficiently and scales well as network sizes continue to grow. This is mainly because it does not require any estimate for pairwise demands but defines the set of feasible demand realizations via bandwidth capacities of some endpoints called terminals in the customer network. Later, Bertsimas and Sim [13, 14] introduce an alternative demand definition where they consider lower and upper bounds on the uncertain coefficients and allow at most a fixed number of coefficients to assume their worst possible values. They suggest to use this quota as a measure for the trade-off between the conservatism and the cost of the final design. Their model, which we will refer to as the BS model in the rest of the chapter, has also gained significant adherence in several applications of network design. In our network design context, the Bertsimas–Sim uncertainty definition amounts to specifying lower and upper bounds on the pointto-point communication demands and to allowing at most a fixed number of pairs to exchange their maximum permissible amount of traffic. Finally, Ben-Ameur and Kerivin [8] propose to use a rather general demand polyhedron, which could be constructed by describing the available information for the specific network using a finite number of linear inequalities. The common point for all these demand models is that, no matter which definition you use, the concern is to determine the least costly link capacity configuration that would remain operational under the worst case. We will refer to the feasible demand realization which leads

A Hybrid Polyhedral Uncertainty Model

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to the most costly capacity configuration for the optimal routing as the worst-case scenario throughout this chapter. Against this background, our main contribution is to introduce a new demand model called the hybrid model, which specifies lower and upper bounds on the point-to-point communication demands as well as bandwidth capacities as in the hose model. In other words, the hybrid model aims to make the hose model more accurate by incorporating additional information on pairwise demands. The advantage of extra information in the form of lower and upper bounds is to avoid redundant conservatism. NLP is an important problem, which can be applied to different contexts like private network design or capacity expansion in telecommunications, supply chain capacity planning in logistics, or truck loading in transportation. Existing studies on the deterministic problem can be grouped under several classes. One source of variety is the number of facility alternatives. Single-facility [5, 16, 30, 32, 35] and two-facility [17, 21, 31] problems are the most common types. On the other hand, NLP with flow costs [17, 21, 35] and without flow costs [7, 16, 29–31] are also widely studied. Although static routing is always used, the multi-path routing [5, 7, 16, 17, 21, 29–31, 35] and single-path routing [5, 15, 22] lead to a technical classification of the corresponding literature. Although the deterministic NLP is widely studied, the Rob-NLP literature is rather limited. Kara¸san et al. [27] study DWDM network design under demand uncertainty with an emphasis on modelling, whereas Altın et al. [3] provide a compact MIP formulation for Rob-NLP and a detailed polyhedral analysis of the problem for the hose model as well as an efficient branch-and-cut algorithm. Atamtürk and Zhang [6] study the two-stage robust NLP where the capacity is reserved on network links before observing the demands and the routing decision is made afterwards in the second stage. Furthermore, Mudchanatongsuk et al. [33] study an approximation to the robust capacity expansion problem with recourse, where the routing of demands (recourse variables) is limited to a linear function of demand uncertainty. They consider transportation cost and demand uncertainties with binary capacity design variables and show that their approximate solutions reduce the worst-case cost but incur sub-optimality in several instances. In addition to introducing the hybrid model, we initially give a compact mixed integer programming model of Rob-NLP with an arbitrary demand polyhedron [3]. Then we focus on the hybrid model and discuss two alternative MIP models for the corresponding Rob-NLP. Next, we compare them in terms of the computational performance using an off-the-shelf MIP solver and mention the differences in terms of the polyhedral structures of their feasible sets. Moreover, we provide an experimental economic analysis of the impact of robustness on the design cost. Finally, we compare the final design cost for the hybrid model with those for the hose and BS models. In Section 2 we first define our problem briefly. Then in Section 2.1, we introduce the hybrid model as a new, general-purpose demand uncertainty definition and present alternative MIP models. We present results of computational tests in

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Section 3. Then, we conclude the chapter with Section 4, where we summarize our results and mention some future research directions.

2 Problem Definition For a given undirected graph G = (V, E) with the set of nodes V and the set of edges E, we want to design a private network among the set of customer sites W ⊆ V . Let Q be the set of commodities where each commodity q ∈ Q corresponds to a potential communication demand from the origin site o(q) ∈ W to the destination site t (q) ∈ W \ {o(q)}. In this chapter, our main concern is to allocate discrete number of facilities with different capacities on the edges of G to design a least-cost network viable for any demand realization in a prescribed polyhedral set. Let L be the set of facility types, C l be the capacity of each type l facility, pel be the cost of using one unit of type l facility on link e, and yel be the number of type l ∈ L facilities reserved on link e. Moreover, dq is the estimated demand from node q o(q) to node t (q) whereas f hk is the fraction of dq routed on the edge {h, k} ∈ E in the direction from h to k. Throughout the chapter, we will sometimes use {h, k} in place of edge e if we need to mention the end points of e. Using this notation, the MIP formulation for the traditional NLP is as follows: min

pel yel

(1)

e∈E l∈L

⎧ ⎨ 1 h = o(q) q q s.t. f hk − f kh = −1 h = t (q) ∀h ∈ V, q ∈ Q, ⎩ 0 otherwise k:{h,k}∈E q q ( f hk + f kh )dq ≤ C l yel ∀e = {h, k} ∈ E,

q∈Q

(2) (3)

l∈L

yel ≥ 0 integer q q f hk , f kh ≥ 0

∀l ∈ L , e ∈ E, ∀{h, k} ∈ E, q ∈ Q.

(4) (5)

The main motivation of the current work is to incorporate some robustness and flexibility into the capacity configuration decision. Accordingly, we consider the possibility of changes in demand expectations and determine the least-cost design based on a polyhedral set of admissible rather than a single matrix of average esti demands q mates. Let D = {d ∈ R|Q| : q∈Q az dq ≤ αz ∀z = 1, . . . , H, dq ≥ 0 ∀q ∈ Q} be the polyhedron containing non-simultaneous demand matrices, which are admissible given the available information about the network. The most significant impact of such an extension is observed in constraint (3), which has to be replaced with q∈Q

q

q

( f hk + f kh )dq ≤

l∈L

C l yel ∀d ∈ D, e = {h, k} ∈ E,

(6)


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since we want the final capacity configuration to support any feasible realization d ∈ D. However, this leads to a semi-infinite optimization problem since we need one constraint for each of the infinitely many feasible demand matrices in D. To overcome this difficulty, we use a common method in robust optimization [1, 11, 13] and obtain a compact MIP formulation of the problem. This method which was also used in [3] is now briefly summarized for the sake of completeness. First, observe that any one of the infinitely many non-simultaneous feasible communication request d ∈ D would be routed safely along each link if the capacity of each link is sufficient to route the most capacity consuming, i.e., the worst-case, admissible traffic requests. As a result, we can model our problem using the following semi-infinite MIP formulation (NLPpol ): min

pel yel

e∈E l∈L

s.t max d∈D

(2), (4), (5) q

q

( f hk + f kh )dq ≤

q∈Q

C l yel ∀e = {h, k} ∈ E,

(7)

l∈L

where we replace (3) with (7) to ensure (6). Notice that given a routing f , we can obtain the worst-case capacity requirement for each link e ∈ E by solving the linear programming problem on the left-hand side of (7). Hence for each link e = {h, k} ∈ E, we can apply a duality-based transformation to the maximization problem in (7) and reduce NLPpol to the following compact MIP formulation (NLP D ): min

pel yel

(8)

e∈E l∈L

(2), (4), (5)

s.t. H

αz λez ≤

C l yel ∀e ∈ E,

(9)

l∈L

z=1 q

q

f hk + f kh ≤

H

q

az λez ∀e = {h, k} ∈ E, q ∈ Q,

(10)

z=1

λez ≥ 0

∀z = 1, . . . , H, q ∈ Q,

(11)

where λ ∈ H |E| are the dual variables used in the transformation. The interested reader can refer to Altın et al. [2] for a more detailed discussion of this approach. The above duality transformation motivates two important contributions. First, we obtain a compact MIP formulation, which we can solve for small-to-mediumsized instances using off-the-shelf MIP solvers. Moreover, we get rid of the bundle constraints (3), which complicate the polyhedral studies on traditional NLP. In Altın et al. [3], we benefit from this single-commodity decomposition property and provide a thorough polyhedral analysis for the so-called symmetric hose model of demand uncertainty. In the next section, we will introduce the hybrid model as

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a new general-purpose polyhedral demand definition and study Rob-NLP for this uncertainty set.

2.1 The Hybrid Model Due to the dynamic nature of the current business environment, the variety of communication needs keeps increasing and hence it gets harder to accurately estimate point-to-point demands. Therefore, designing networks that are flexible enough to handle multiple demand scenarios efficiently so as to improve network availability has become a crucial issue. Although considering a finite number of potential scenarios is a well-known approach in stochastic optimization, Duffield et al. [18] and Fingerhut et al. [20] introduced the hose model as a first effort to use polyhedral demand uncertainty sets in telecommunications context. In its most general form, that is called the asymmetric hose model, outflow (bs+ ) and inflow (bs− ) capacities are set for each customer site s ∈ W as

dq ≤ bs+ ∀s ∈ W,

(12)

dq ≤ bs− ∀s ∈ W.

(13)

q∈Q:o(q)=s

q∈Q:t (q)=s

Then the corresponding polyhedron is D Asym = {dq ∈ R|Q| : (12), (13), dq ≥ 0 ∀q ∈ Q}. There are also the symmetric hose model with a single capacity bs for the total be incident to node s ∈ W , and a sum-symmetric hose model flow that can with s∈W bs+ = s∈W bs− . The hose model has several strengths. To name a few, the transmission capacities can be estimated more reliably and easily than individual point-to-point demands especially if sufficient amount of statistical information is not available. Moreover, it offers resource-sharing flexibility and hence improved link utilization due to multiplexing. Basically, the size of an access link can be smaller if we use hose model rather than point-to-point lines with fixed resource sharing. These and several other competitive advantages helped the hose model to prevail within the telecommunications society [1, 3, 19, 23–26, 28, 39]. On the other hand, Bertsimas and Sim [13, 14] proposed the BS model or the restricted interval model, which defines an applicable interval for each pairwise demand such that at most a fixed number of demands can take their highest values, simultaneously. For our problem, this implies that dq ∈ [d¯q , d¯q + dˆq ] for all q ∈ Q and at most of these demands differ from d¯q ≥ 0 at the same time. If we define ¯ ˆ each demand q ∈ Q as dq = dq + dq βq with βq ∈ {0, 1}, then the BS model requires q∈Q βq ≤ . Bertsimas and Sim [14] use to control the conservatism of the final design.


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In this section, we introduce the hybrid model. Although we study a private network design problem, the hybrid model can certainly be used in any context where parameter uncertainty is a point at issue. We call this new model as hybrid since a demand matrix d ∈ R|Q| has to satisfy the slightly modified symmetric hose constraint q∈Q:o(q)=s

2

dq ≤ bs

∀s ∈ W,

(14)

t (q)=s

as well as the interval restrictions dq ≤ u q ∀q ∈ Q, d¯q ≤ dq ∀q ∈ Q

(15) (16)

to be admissible. As a result, the corresponding demand polyhedron for the hybrid model is Dhyb = {d ∈ R|Q| : (14), (15), (16)}. We should remark here that (14) should not be considered as analogous to the conservatism level restriction in the BS model. Actually, the conservatism dimension is not articulated explicitly in the hybrid model, where the main purpose is to incorporate more information into the hose definition so as to avoid overly conservative designs taking care of unlikely worst-case demand realizations. Hence, notice that the hybrid model is a hybrid of the hose model and the interval uncertainty model but not the BS model. Finally, we are interested in the case where Dhyb = ∅ and hence q∈Q:o(q)=s 2 t (q)=s d¯q ≤ bs for all s ∈ W to have a meaningful design problem. Let Dsym = {d ∈ R|Q| : (14), dq ≥ 0 ∀q ∈ Q}. Notice that, Dhyb = Dsym if d¯q = 0 and u q ≥ min{bo(q) , bt (q) } for all q ∈ Q whereas Dhyb ⊆ Dsym otherwise. This is because we can avoid over-conservative designs and hence redundant investment by using additional information and our provisions about the specific topology.

2.2 Robust NLP with the Hybrid Model In this section, we will briefly discuss two alternative MIP models for Rob-NLP with the hybrid model of demand uncertainty. The first formulation follows directly from the discussions in Section 2. On the other hand, we slightly modify our Dhyb so as to express it in terms of deviations from the nominal values to obtain the second formulation. First, notice that NLP D reduces to the following compact MIP formulation (NLPhyb ) for the hybrid model:

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min

pel yel

e∈E l∈L

s.t. (2), (4), (5) q q s bs we + (u q λe − d¯q μe ) ≤ C l yel ∀e ∈ E, s∈W

q∈Q

o(q) we

(17)

l∈L

t (q) + we

q q + λe − μe q q λe , μe ≥ 0 wes ≥ 0

≥

q f hk

q

+ f kh

∀q ∈ Q, e = {h, k} ∈ E, (18) ∀q ∈ Q, e ∈ E, ∀s ∈ W, e ∈ E,

(19) (20)

where w, λ, and μ are the dual variables used in the duality transformation corresponding to constraints (14), (15), and (16), respectively.

2.3 Alternative Flow Formulation Given the hybrid model, we know that the best-case scenario would be the one where all demands are at their lower bounds. Then the total design cost would increase as the deviation from this best case increases. Consequently, we can restate the hybrid model in terms of the deviations from lower bounds, which requires us to modify NLPpol by replacing link capacity constraints (7) with

( f hk + f kh )d¯q + max q

q

ˆ Dˆ hyb d∈ q∈Q

q∈Q

( f hk + f kh )dˆq ≤ q

q

C l yel ∀e = {h, k} ∈ E,

l∈L

(21) 2 ˆ where Dˆ hyb = {dˆ ∈ R|Q| : 0 ≤ dˆq ≤ q ∀q ∈ Q; q∈Q:o(q)=s t (q)=s dq ≤ ˙bs ∀s ∈ W } such that q = u q − d¯q for all q ∈ Q and b˙ s = bs − 2 ¯ q∈Q:o(q)=s t (q)=s dq for all s ∈ W . This observation leads to the following result. Proposition 1 NLP D reduces to the following compact linear MIP formulation (NLPalt ) for the hybrid model: min

pel yel

e∈E l∈L

q

( f hk

(2), (4), (5) q ¯ q + f kh )dq + q ηe ≤ C l yel ∀e = {h, k} ∈ E, b˙s νes +

q∈Q o(q) νe

s∈W

q∈Q

t (q) q + νe + ηe o(q) t (q) νe , νe ,

q f hk

≥ q

l∈L

+

ηe ≥ 0

q f kh

∀q ∈ Q, e = {h, k} ∈ E, ∀q ∈ Q, e ∈ E.


165 |L|

Proof For link e = {h, k} ∈ E, the capacity assignment y = (ye1 , . . . , ye ) should be sufficient to route the worst-case demand as in (21). Then, we can model the maximization problem on the left-hand side of (21) as max

q q ( f hk + f kh )dˆq

(22)

q∈Q

s.t.

q∈Q:o(q)=s

2

dˆq ≤ b˙s ∀s ∈ W,

(23)

∀q ∈ Q,

(24)

∀q ∈ Q.

(25)

t (q)=s

dˆq ≤ q dˆq ≥ 0

Notice that for a given routing f , this is a linear programming problem. Since it is feasible and bounded, we can apply a duality transformation similar to Soyster [38]. q So, we associate the dual variables νes and ηe with (23) and (24), respectively, and obtain the equivalent dual formulation ⎛ min ⎝

b˙s νes +

s∈W

s.t.

o(q) νe

⎞ q ηe ⎠ q

(26)

q∈Q

t (q) q q + νe + ηe ≥ f hk q νes , ηe ≥ 0

q

+ f kh ∀q ∈ Q, ∀s ∈ W, q ∈ Q.

(27) (28)

Next, we can complete the proof by equally replacing the maximization problem in (21) with (26), (27), (28) and removing min since the facility capacities C l and the reservation costs pel are nonnegative for all l ∈ L. We show in Section 3, off-the-shelf MIP solvers can handle NLPalt better than NLPhyb in some instances.

3 Experimental Results In this section, we focus on the single-facility multi-commodity problem where just one type of facility with C units of capacity is available. We perform our analysis in two stages. First, we compare the performance of ILOG Cplex for the two compact MIP formulations NLPhyb and NLPalt in terms of solution times and bounds they provide at the end of 2-h time limit. The instances polska, dfn, newyork, france, janos, atlanta, tai, nobel-eu, pioro, and sun are from the SNDLIB web site [37] whereas the remaining are used in Altın et al. [1] for a virtual private network design problem. For the SNDLIB instances [37], we have the average demand estimates d˜q . In order to generate the bandwidth values as well as the lower and upper bounds on pairwise communication demands, we have used the following relations:

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A. Altın et al.

q∈Q:o(q)=s

d˜q 1+ p ;

2

t (q)=s

d˜q ;

• u q = (1 + p)d˜q . In our tests, we choose p = 0.2. This parameter can be determined based on the available information about the demand pattern, past experience, etc. We should note that Dhyb would not get smaller as p increases and hence the optimal design would never get less conservative. By defining bs , u q , and d¯q as a function of p, we can interpret the trade-off between the conservatism of a design and its cost. The interested reader can refer to Altın et al. [3] for an analogous parametric analysis of the symmetric hose model. We have used AMPL to model the formulations and Cplex 11.0 MIP solver for numerical tests and set a 2-h solution time limit for all instances. We present the results of the initial comparison for two MIP models in Table 1 where we provide the following information: • • • • • • • • • • •

zhyb : best total design cost for NLPhyb at termination, tcp : solution time in CPU seconds for NLPhyb , G hyb : the gap at termination for NLPhyb , #hyb : number of B&C nodes for NLPhyb , which is 0 if no branching takes place, z alt : best total design cost for NLPalt at termination, tcp : solution time in CPU seconds for NLPalt , G alt : the gap at termination for NLPalt , #alt : number of B&C nodes for NLPalt , which is 0 if no branching takes place, ∗ indicates the best upper bound at termination, INF means that we have no integer solution at termination, ‘–’ under the z columns shows that even the LP relaxation cannot be solved in 2-h time limit.

We could solve both NLPhyb and NLPalt for 7 out of 17 instances to optimality within 2-h time limit. In addition to that, Cplex could also solve NLPalt for polska with C = 1000. For the same instance, Cplex could reduce the integrality gap to 2.16% with NLPhyb . We will analyze our results in two stages. Initially, in Fig. 1, we show the reduction in solution times when NLPalt rather than NLPhyb is solved for the first seven t −t instances in Table 1. We measure this improvement as hybthyb alt ×100 and thus positive values show the instances that are easier to solve using the alternative formulation. We see that except bhv6c and pdh, NLPalt is easier to solve for Cplex. On the other hand, for the remaining 10 instances, we see that Cplex achieved better upper bounds with NLPalt in 6 cases. Figure 2 displays the termination gaps for both models. Note that for newyork, nobel-eu, and sun, Cplex could not solve even the LP relaxation of NLPhyb whereas we have some upper bounds for NLPalt . We let gaphyb = 105% in Fig. 2 for these three instances just for convenience. On the other hand, we see that the upper bounds on total design cost at termination are smaller with NLPhyb for tai and janos whereas there is a tie for polska (C = 155)


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Table 1 A comparison of the projected formulation and the alternative flow formulation Instance (|V |,|E|,|W |,C) z hyb thyb (G hyb ) #(hyb) z alt talt (G alt ) #(alt) metro nsf1b at-cep1 pacbell bhv6c bhvdc pdh polska polska dfn newyork france atlanta tai janos nobel-eu sun

(11,42,5,24) (14,21,10,24) (15,22,6,24) (15,21,7,24) (27,39,15,24) (29,36,13,24) (11,34,6,480) (12,18,12,155) (12,18,12,1000) (11,47,11,155) (16,49,16,1000) (25,45,14,2500) (15,22,15,1000) (24,51,19,504k) (26,42,26,64) (28,41,28,20) (27,51,24,40)

768 86,600 47,840 10,410 810,368 952,664 2,467,983 44,253∗ 7478∗ 51,572∗ – 21,600∗ 458,020,000∗ 28,702,323.54∗ 1,289,931,888 – –

23.02 246 1.61 49.1 669.09 657.09 318.41 (0.77%) (2,16%) (3.85%) INF (3.22%) (0.11%) (20.37%) (99.8%) INF INF

428 2293 66 1216 13,148 1210 4796 25,871 12,742 4993 0 915 16,023 140 0 0 0

768 86,600 47,840 10,410 810,368 952,664 2,467,983 44,253∗ 7478 51,572∗ 1,318,400∗ 22,600∗ 458,040,000∗ 27,611,428.86∗ 1,289,911,204∗ 14,718,917,910∗ 62,938,898.76∗

13.28 134.95 1.41 28.06 725.19 149.52 827.58 (1.14%) 4591.14 (7.32%) (54.10%) (9.40%) (0.36%) (17.85%) (99.8%) (99.97%) (99.99%)

486 1470 90 1199 10,960 481 10,961 19,144 7961 281 100 151 7620 157 37 26 1

100%

50%

0% metro

nsf1b

at-cep1

pacbell

bhv6c

bhvdc

pdh

–50%

–100%

–150%

–200%

Fig. 1 Reduction in solution times when we solve NLPalt rather than NLPhyb with Cplex

and dfn. Based on the overall comparison of the two models that we show in Fig. 3, we can say Cplex can solve NLPalt more efficiently since solution times, termination gaps, and upper bounds are better with NLPalt . On the other hand, we suppose that we can make a better use of NLPhyb so as to develop efficient solution tools like a branch-and-cut algorithm. We suppose NLPhyb to be more advantageous than NLPalt since the latter does not have the nice single commodity decomposition property.

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A. Altın et al. 100%

alt

hyb

80% 60% 40% 20% 0%

Fig. 2 Comparison of termination gaps when we solve NLPalt and NLPhyb with Cplex

upper bound

solution time hyb 29%

22%

gap

22%

tie

hyb

hyb

40% alt 60%

alt 71%

alt

56%

Fig. 3 A general comparison of solving NLPalt and NLPhyb with Cplex

Next, we display how the design cost has changed according to the demand uncertainty model in Fig. 4. To this end, we consider three models: the interval uncertainty model with Dint = {dq ∈ R|Q| : d¯q ≤ dq ≤ u q ∀q ∈ Q}, the symmetric hose model with Dhose = {dq ∈ R|Q| : (14), dq ≥ 0 ∀q ∈ Q}, and the hybrid model. Notice that the interval model is a special case of the BS model with = |Q| and thus the corresponding worst case would be dqworst = u q for all q ∈ Q. We consider six instances, which we could solve to optimality in reasonable times for all demand models. We should remark here that we had to terminate the test for the bhvdc and bhv6c instances under interval uncertainty model after 60,000 CPU seconds with 0.21% and 0.3% gaps since the best solutions have not changed for a long while and the gaps are relatively small. Let z det be the total design cost for the deterministic case if we consider the best-case scenario with dq = d¯q for all q ∈ Q. Then for the three demand models, z −z we show the percent of increase in design cost, which is hybzdet det ∗ 100 for the hybrid model and similar for the other models, in Fig. 4. For each instance such


169

60% 50% 40% 30% 20% 10% 0% metro

nsf1b

at-cep1 int

hose

pacbell

bhv6c

bhvdc

hyb

Fig. 4 Increase in design cost with respect to the deterministic case for different demand models

an increase can be interpreted as the cost of robustness or the price that we should be ready to pay so as to have a more flexible network and hence increased service availability. We see that as we shift from the interval model to the hose model and then to the hybrid model, the total design cost decreases significantly, namely, the average increase rates are 44.89%, 29.33%, and 18.31% for these six instances with three models, respectively. Given that these instances are constructed using the same |W |+2∗|Q| ¯ u) ∈ R+ , we can interpret this decreasing trend in cost as parameters (b, d, a consequence of using more informative demand uncertainty sets and hence being protected against practically and technically more realistic worst-case scenarios. Our worst-case definition over a polyhedron is clearly quite different from simply determining the worst-case scenario a priori. Since we exploit the hybrid model information, we can avoid over-conservative designs. Suppose that we have not done so and we determine a worst case that can happen using the available infor|W |+2∗|Q| ¯ u) ∈ R+ mation (b, d, . Obviously, the safest approach would be to set dq = min{bo(q) , bt (q) , u q } for all q ∈ Q and then solve the nominal problem (1), (2), (3), (4), and (5) to get the optimal design cost z worst . When we compare the design cost z hyb with z worst for the six instances we have mentioned above, we see that the design costs have reduced by 18.27% on the average. On the other hand, the average savings is around 10.61% for the hose model. Figure 5 displays the percentage of savings in cost for each instance with both models. We also compare the design costs for the hybrid model and the BS model for = #0.1|Q|$ and = #0.15|Q|$. We show the relative savings the hybrid model provide in Fig. 6. We see that when = #0.1|Q|$, using the hybrid model yields a less costly design for all instances except nsf1b, where it is only 0.12% worse. On the average, the hybrid model provides 6.53% and 10.69% savings, respectively, for these six instances and the difference increases rapidly as grows larger. Finally, we consider the metro, at-cep1, and pacbell instances so as to compare the robust designs for the BS model ( = #0.15|Q|$) and the hybrid model in

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hyb

20% 15% 10%

hose

5% 0% metro

nsf1b

at-cep1

pacbell

bhv6c

bhvdc

Fig. 5 Reduction in design cost with respect to the worst-case scenario determined without exploiting the demand model

20%

Γ = ⎡0.1⏐Q⏐⎤

20%

15%

15%

10%

10%

5%

50%

0%

0%

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Fig. 6 Savings in design cost by using the hybrid model rather than the BS model

terms of their routing performances. For this purpose, we first generate 20 demand j matrices d˙1 , d˙ 2 ,..,d˙ 20 for each instance where the demand d˙q for each commodity q ∈ Q is normally distributed with mean d˜q and standard deviation 0.5d˜q . Then given the optimal capacity configurations y(B S) and y(hyb), we determine the maximum total flow F j (B S) and F j (hyb) we can route for the demand matrix d˙ j for all j = 1, . . . , 20 by solving a linear programming problem. For each demand matrix, the fraction routed for both demand models we calculate of demand j j as F j (B S)/ q∈Q d˙q and F j (hyb)/ q∈Q d˙q , respectively. Finally, we take the average over the 20 demand matrices to evaluate the two robust designs. We present our test results in Table 2 where Rhyb and R B S are the average routing rates for the hybrid and BS models, respectively, whereas cost shows the increase in design cost if the BS model rather than the hybrid model is used. We see that the average routing rates are quite close for metro and at-cep1, whereas they are equal for pacbell. On the other hand, y(B S) is clearly more costly than y(hyb) in all instances. Hence, we can suggest the hybrid model to provide almost the same level of availability at a much lower cost.


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Table 2 Routing rate and cost comparison between the hybrid model and the BS model Instance Rhyb (%) R B S (%) cost (%) metro at-cep1 pacbell

95.7 96.6 99.9

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4 Conclusion In this chapter, we introduced the hybrid model as a new demand uncertainty definition. It inherits the strengths of the two well-known and frequently used demand models: it is easy to specify like the hose model and it avoids over-conservatism like the BS model. We provided two compact MIP formulations, i.e., NLPhyb and NLPalt , for robust NLP under the hybrid model and compared them in terms of their computational performances. Finally, we discussed how the optimal design cost changes for different demand models. When compared with the interval model, the hose model, and the BS model, we observed that the hybrid model provides significant cost savings by exploiting additional information to exclude overly pessimistic worst-case scenarios. Our test results are encouraging for undertaking further studies on robust network design problems.

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Analytical Modelling of IEEE 802.11e Enhanced Distributed Channel Access Protocol in Wireless LANs Jia Hu, Geyong Min, Mike E. Woodward, and Weijia Jia

1 Introduction The IEEE 802.11-based wireless local area networks (WLANs) have experienced impressive commercial success owing to their low cost and easy deployment [8]. The basic medium access control (MAC) protocol of the IEEE 802.11 standard is distributed coordination function (DCF) [8], which is based on the carries sense multiple access with collision avoidance (CSMA/CA) protocol and binary exponential backoff (BEB) mechanism. The DCF is designed for the best-effort traffic only and does not provide any support of priorities and differentiated quality of service (QoS). However, due to the rapid growth of wireless multimedia applications, such as voice-over-IP (VoIP) and video conferencing, there is an ever-increasing demand for provisioning of differentiated QoS in WLANs. To support the MAC-level QoS, an enhanced version of the IEEE 802.11 MAC protocol, namely IEEE 802.11e [9], has been standardized. This protocol employs a channel access function called hybrid coordination function (HCF) [9], which comprises the contention-based enhanced distributed channel access (EDCA) and the centrally controlled hybrid coordinated channel access (HCCA). The EDCA is Jia Hu Department of Computing, School of Informatics, University of Bradford, Bradford, BD7 1DP, UK e-mail: [email protected] Geyong Min Department of Computing, School of Informatics, University of Bradford, Bradford, BD7 1DP, UK e-mail: [email protected] Mike E. Woodward Department of Computing, School of Informatics, University of Bradford, Bradford, BD7 1DP, UK e-mail: [email protected] Weijia Jia Department of Computer Science, City University of Hong Kong, 83 Tat Chee Ave, Hong Kong e-mail: [email protected] N. Gülpınar et al. (eds.), Performance Models and Risk Management in Communications Systems, Springer Optimization and Its Applications 46, C Springer Science+Business Media, LLC 2011 DOI 10.1007/978-1-4419-0534-5_8,

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the fundamental and mandatory mechanism of 802.11e, whereas HCCA is optional and requires complex scheduling algorithms for resource allocation. This chapter focuses on the analysis of the EDCA, which is an extension of DCF for provisioning of QoS. The EDCA classifies the traffic flows into four access categories (ACs), each of which is associated with a separate transmission queue and behaves independently. These ACs are differentiated through adjusting the parameters of arbitrary inter-frame space (AIFS), contention window (CW), and transmission opportunity (TXOP) limit [9]. The AIFS and CW schemes control the channel access time, while the TXOP scheme controls the channel occupation time after accessing the channel. Analytical performance evaluation of the DCF and EDCA has been extensively studied in recent years [1–3, 5–7, 10, 12–36]. Since the traffic loads are unsaturated in the realistic network environments, the development of analytical models for the DCF and EDCA in unsaturated conditions is an important and open research issue. With the aim of obtaining a thorough and deep understanding of the performance of EDCA, we propose a comprehensive analytical model to incorporate the three QoS differentiation schemes including the AIFS, CW, and TXOP schemes simultaneously in IEEE 802.11e WLANs under unsaturated traffic loads. First, we develop a novel three-dimensional Markov chain to analyse the backoff process of each AC. Afterwards, the transmission queue at each AC is modelled as a bulk service queueing system with finite capacity to address the difficulties of queueing analysis arising from the TXOP burst transmission scheme. The accuracy of the proposed model is validated by comparing the analytical results to those obtained from extensive NS-2 simulation experiments. The rest of the chapter is organized as follows. Section 2 introduces the MAC protocols. Section 3 describes a detailed survey of the related work on modelling of the DCF and EDCA. Section 4 presents the derivation of the analytical model for EDCA. After validating the accuracy of the model in Section 5, we conclude the chapter in Section 6.

2 Medium Access Control 2.1 Distributed Coordination Function (DCF) The DCF is the fundamental channel access scheme in the IEEE 802.11 MAC protocol [8]. A station with backlogged frames first senses the channel before transmission. If the channel is detected idle for a distributed inter-frame space (DIFS), the station transmits the frame. Otherwise, the station defers until the channel is detected idle for a DIFS, and then starts a backoff procedure by generating a random backoff counter. The value of the backoff counter is uniformly chosen between zero and W , which is initially set to Wmin and doubled after each unsuccessful transmission until it reaches a maximum value Wmax . It is reset to Wmin after the successful transmission or if the unsuccessful transmission attempts reach a retry limit. The backoff counter is decreased by 1 for each time slot when the channel is idle, halted when

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the channel is busy, and resumed when the channel becomes idle again for a DIFS. A station transmits a frame when its backoff counter reaches zero. Upon the successful reception of the frame, the receiver sends back an ACK frame immediately after a short inter-frame space (SIFS). If the station does not receive the ACK within a timeout interval [8], it retransmits the frame. Each station maintains a retry counter that is increased by 1 after each retransmission. The frame is discarded after an unsuccessful transmission if the retry counter reaches the retry limit. The above-mentioned procedure is referred to as the basic access method. The hidden terminal problem [18] occurs when a station is unable to detect a potential competitor for the channel because they are not within the hearing range of each other. To combat the hidden terminal problem, DCF also defines an optional four-way handshake scheme whereby the source and destination exchange requestto-send (RTS) and clear-to-send (CTS) messages before the transmission of actual data frame.

2.2 Enhanced Distributed Channel Access (EDCA) The EDCA was designed to enhance the performance of the DCF and provide the differentiated QoS [9]. As shown in Fig. 1, traffic of different classes is assigned to one of four ACs, which is associated with a separate transmission queue and behaves independently of others in each station. The QoS of these ACs are differentiated through assigning different EDCA parameters including AIFS values, CW sizes, and TXOP limits. Specifically, a smaller AIFS/CW results in a larger probability of winning the contention for the channel. On the other hand, the larger the TXOP limit then the longer are channel holding times of the station winning the contention. The operation of channel access in EDCA is similar to the DCF, as shown in Fig. 2. Before generating the backoff counter, the EDCA function must sense the channel to be idle for an AIFS instead of a DIFS in the DCF. The AIFS for a given AC is defined as AC0

AC1

AC2

AC3

Backoff AIFS[AC0] CW[AC0]




Virtual Collision Handler Transmission during TXOP[AC]

Fig. 1 The IEEE 802.11e MAC with four ACs

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AIFS[AC] SIFS

Contention Window from 0 to CWmin[AC]

Backoff Slots

Frame Transmission

Slot time Defer Access

Decrease backoff slots as long as channel stays idle

Fig. 2 The timing diagram of the EDCA channel access

AIFS[AC] = SIFS + AIFSN [AC] × aSlotTime, where AIFSN [AC] (AIFSN [AC] ≥ 2) is an integer value to determine AIFSN [AC] and aSlotTime is the duration of a time slot. When the EDCA function generates the backoff counter before transmission, the AC-specific Wmin and Wmax are used to differentiate the CW sizes of backoff counters. Upon winning the contention for the channel, the AC transmits the frames available in its queue consecutively provided that the duration of transmission does not exceed the specific TXOP limit [9]. Each frame is acknowledged by an ACK after an SIFS. The next frame is transmitted immediately after it waits for an SIFS upon receiving this ACK. If the transmission of any frame fails the burst is terminated and the AC contends again for the channel to retransmit the failed frame. When the backoff counters of different ACs within a station decreases to zero simultaneously, the frame from the highest priority AC among the contending ones is selected for transmission on the channel, while the others suffer from a virtual collision and invoke the backoff procedure with a doubled CW value. The backoff rule of EDCA is slightly different from that of DCF. In DCF, the backoff counter is frozen during the channel busy period, resumed after the channel is sensed idle for a DIFS, and decreased by 1 at the end of the first slot following DIFS. However, the backoff counter in EDCA is resumed one slot time before the end of AIFS. It means that the backoff counter has already been decremented by 1 at the end of AIFS. In addition, after the backoff counter decrements to zero, the AC has to wait for an extra slot before transmitting.

3 Related Work 3.1 Overview of Analytical Models of the DCF Performance modelling of the DCF has attracted considerable research efforts [1, 2, 13, 15, 17, 18, 20, 27, 31, 33, 34, 36]. For instance, Bianchi [1] proposed a bi-dimensional discrete-time Markov chain to derive the expressions of the saturation throughput for the DCF, under the assumption that all the stations have frames for transmission anytime. This simplified assumption excludes any need to


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consider queuing dynamics or traffic models for performance analysis. Many subsequent studies of the DCF have built upon Bianchi’s work. For instance, Ziouva and Antonakopoulos [36] have extended Bianchi’s model by taking account of the busy channel conditions for invoking the backoff procedure. Wu et al. [31] have modified Bianchi’s model to deal with the retry limit. Kumar et al. [13] have studied the fixed point formulation based on the analysis of Bianchi’s model and showed that the derivation of transmission probability can be significantly simplified by viewing the backoff procedure as a renewal process. However, realistic network conditions are non-saturated as only very few networks are in a situation where all nodes have frames to send all the time. Consequently, there are many research works devoted to developing analytical models for DCF under unsaturated working conditions [15, 18, 20, 27, 33, 34]. For example, Zhai et al. [33] showed that exponential distribution is a good approximation for the MAC service time distribution of the DCF and then presented an M/M/1/K queueing model to analyse the DCF under non-saturated conditions. Malone et al. [17] extended Bianchi’s model to a non-saturated environment in the presence of heterogeneous loads at the nodes, assuming that each MAC buffer has only one frame at most. Medepalli and Tobagi [18] presented a unified analytical model for the DCF where the transmission queue is modelled as an M/M/1 queuing system. Tickoo and Sikdar [27] proposed a discrete-time G/G/1 queue for modelling stations in the IEEE 802.11 WLAN.

3.2 Overview of Analytical Models of the EDCA Significant research efforts have been devoted to developing the analytical performance model for the AIFS and CW differentiation schemes defined in EDCA [3, 6, 7, 10, 12, 16, 22–26, 32, 35]. Most of these studies were based on the extension of Bianchi’s model [1] under the assumption of saturated traffic loads. For instance, Xiao [32] extended [1] to study the CW differentiation scheme of EDCA. Kong et al. [12] analysed the AIFS and CW differentiation by a three-dimensional Markov chain. Tao and Panwar [26] developed another three-dimensional Markov chain model where the third dimension represents the number of time slots after the end of the last AIFS period. Robinson and Randhawa [23] adopted a bi-dimensional Markov chain model where the collision probability is calculated as a weighted average of the collision probabilities in different contention zones during AIFS. Zhu and Chlamtac [35] proposed a Markov model of EDCA to calculate the saturation throughput and access delay. Huang and Liao [7] analysed the performance of saturation throughput and access delay, taking into account the virtual collisions among ACs inside each EDCA station. Tantra et al. [24] introduced a simple model which adopts two different types of Markov chains to model various ACs in EDCA. As another QoS scheme specified in the EDCA, the TXOP scheme has also drawn much research attention [5, 6, 10, 14, 19, 21, 27–29]. However, the existing analytical models were primarily focused on the saturation performance

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[14, 21, 28, 29]. More specifically, Tinnirello and Choi [28] compared the system throughput of the TXOP scheme with different acknowledgement (ACK) policies. Vitsas et al. [29] presented an analytical model for the TXOP scheme to derive the throughput and average access delay. Li et al. [14] analysed the throughput of the TXOP scheme with block ACK policy under noisy channel conditions. Peng et al. [21] evaluated the throughput of various ACs as a function of different TXOP limits. The majority of existing models for the EDCA were derived based on the assumption of saturated working conditions. However, since the traffic loads in the realistic network environments are unsaturated, there are also a number of analytical models for the EDCA under unsaturated conditions [3, 5, 6, 10, 16, 19, 25, 27]. For example, Tantra et al. [25] introduced a Markovian model to compute the throughput and delay performance metrics of EDCA with the CW scheme, assuming that each station has a transmission queue size of one frame. Engelstad and Osterbo [3] analysed the end-to-end delay of EDCA with the AIFS and CW schemes through modelling each AC as an M/G/1 queue of infinite capacity. Liu and Niu [16] employed an M/M/1 queueing model to analyse the EDCA protocol, considering the AIFS and CW schemes. They also assumed an infinite capacity of the transmission queue. As for the unsaturated model of the TXOP scheme, Hu et al. [5] analysed and compared the performance of the TXOP scheme with different ACK policies under unsaturated traffic loads. Tickoo and Sikdar [27] extended the G/G/1 discrete-time queueing model for the DCF to analyse the TXOP scheme under non-saturated traffic loads. Although the performance of the AIFS, CW, and TXOP schemes has been studied separately in non-saturated conditions, to the best of our knowledge, there has been very few analytical models [10] reported in the current literature for the combination of these three schemes in non-saturated conditions. Recently, Inan et al. [10] proposed an analytical model to incorporate these three schemes under unsaturated traffic loads. They modelled the unsaturated behaviour of EDCA through a three-dimensional Markov chain. Since the model introduced the third dimension of the Markov chain to denote the number of backlogged frames in the transmission queue, the complexity of the solution will become very high with a large buffer size. Different with the model in [10], we employ a method combining the queueing theory and Markov analysis. Consequently, the proposed model can handle a large buffer size without heavily increasing the complexity of the solution.

4 Analytical Model In this section, we propose a comprehensive analytical model of the EDCA under unsaturated traffic loads. We assume a network with n stations using the EDCA of IEEE 802.11e as the MAC protocol. We use the basic access scheme for channel contention. The analysis can be readily extended for the RTS/CTS scheme. The wireless channel is assumed to be ideal, thus the transmission failures are only caused by collisions and there is no hidden terminal problem.


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The ACs from the lowest priority to the highest one are denoted by subscripts 0, 1, 2, . . ., N . The transmission queue at each AC is modelled as a bulk service queueing system where the arrival traffic follows a Poisson process with rate λv (frames/s, v = 0, 1, 2, . . ., N ). The service rate of the queueing system, μv , is derived by analysing the backoff and burst transmission procedures of AC v . As shown in Fig. 3, a novel three-dimensional Markov chain is introduced to model the backoff procedure.

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Fig. 3 The three-dimensional Markov chain for modelling the backoff procedure of EDCA: (a) the three-dimensional Markov chain; (b) the sub-Markov chain for modelling the deferring period

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4.1 Modelling of the Backoff Procedure A discrete and integer timescale is adopted [1] where t and (t + 1) correspond to the starts of two consecutive time slots, which is the variable time interval between the starts of two successive decrements of the backoff counter, or a fixed time interval specified in the protocol [8], namely the physical time slot. Let s(t) and b(t) denote the stochastic processes representing the backoff stage (i.e., retry counter) and backoff counter for a given AC at time t, respectively. The newly introduced dimension, c(t), denotes the number of remaining time slots to complete the deferring period in AIFS of the AC (AIFSv ) after the minimum AIFS (AIFSmin ). Since only the head-of-burst (HoB) frame needs contend for the channel, the term frame refers to as the HoB frame in this section, if without any specifications. Assuming that the collision probability of frames transmitted from the AC v , pv , is independent of the number of retransmissions that a frame has suffered [1], the three-dimensional process {s(t), b(t), c(t)} can be modelled as a discrete-time Markov chain as shown in Fig. 3a, with a dashed line box shown in detail in the sub-Markov chain of Fig. 3b. The state transition probabilities in the three-dimensional Markov chain for the ACv are described as follows: P{i, j, 0|i, j + 1, 0} = pbv , P{i, j, dv |i, j, 0} = 1 − pbv , P{i, j, 0|i, j, 1} = ptv , P{i, j, k|i, j, k + 1} = ptv , P{i, j, dv |i, j, k} = 1 − ptv , P{i, j, dv |i − 1, 0, 0} = pv /Wiv , P{0, j, dv |i, 0, 0} = (1 − pv )/W0v , P{0, j, dv |m, 0, 0} = 1/W0v ,

0 ≤ i ≤ m, 0 ≤ j ≤ Wiv − 2, 0 ≤ i ≤ m, 1 ≤ j ≤ Wiv − 1, 0 ≤ i ≤ m, 0 ≤ j ≤ Wiv − 1,

(1a) (1b) (1c)

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(1h)

where m is the retry limit and Wiv is the contention window size after i retransmissions. pbv is the probability that the channel is idle in a time slot after the AIFS period of ACv , and ptv is the probability that the channel is idle in a time slot during the deferring period in AIFS of ACv after AIFSmin . dv denotes the difference in the number of time slots between AIFSmin and AIFSv , i.e. dv = AIFSN v − AIFSN min . These equations account, respectively, for the following (1a) the backoff counter is decreased by 1 after an idle time slot; (1b) the backoff counter is frozen and the AC starts deferring; (1c) the backoff counter is activated and decreased by 1 after the AIFS period; (1d) the remaining number of time slots for activating the backoff counter is decreased by 1 if the channel is detected idle in a time slot; (1e) the AC has to go through the AIFS period again if the channel is sensed busy during the deferring procedure; (1f) the backoff stage increases after an unsuccessful transmission and the AC starts deferring before activating the backoff counter; (1g) after a successful transmission, the contention window is reset to Wmin ; (1h) once


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the backoff stage reaches the retry limit, the CW is reset to Wmin after the frame transmission. Let bi, j,k be the stationary distribution of the three-dimensional Markov chain. First, the steady-state probabilities, bi,0,0 , satisfy bi,0,0 = pvi b0,0,0 ,

0 ≤ i ≤ m.

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Because of the chain regularities, for each j ∈ [0, Wiv − 1], we have 3 bi, j,0 = bi,0,0 (Wiv − j) Wiv ,

0 ≤ i ≤ m.

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From the balance equations in the sub-Markov chain, we have the following relations: dv −k , bi, j,k = bi, j,dv ptv

1 ≤ k ≤ dv ,

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d v −1 pv bi−1,0,0 + bi, j,k (1 − ptv ), Wiv k=1

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)

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( ' W0v 1 − pv − pv (2 pv )m + 2m pvm+1 (2 pv − 1) 2(1 − pv )(1 − 2 pv )

−

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(8)

i=0

The probability that an ACv transmits in a randomly chosen time slot, τv , given that its transmission queue is non-empty, can be written as τv = b0,0,0

m

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b0,0,0 (1 − pvm+1 ) . (1 − pv )

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Note that an AC can transmit only when there are pending frames in its transmission queue. Therefore, the transmission probability, τv , that the ACv transmits under unsaturated traffic loads can be derived by τv = τv (1 − P0v ),

(10)

where P0v is the probability that the transmission queue of the ACv is empty, which will be derived in the following sections. Taking into account virtual collisions, an AC will only collide with frames from other stations, or from the higher priority ACs in the same station. Therefore, the collision probability, pv , of the ACv is given by pv = 1 −

N "

(1 − τx )n−1

N "

(1 − τx ),

(11)

x>v

x=0

where n is the number of stations. The probability, pbv , that the channel is idle in a time slot after the AIFS period of the ACv , is the probability that no station is transmitting in the given slot pbv =

N "

(1 − τx )n .

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x=0

During the deferring period between AIFSmin and AIFSv , the ACs with priorities lower than or equal to ACv will not transmit in a time slot. As a result, the probability, ptv , that the channel is sensed idle in a time slot during the deferring period between AIFSmin and AIFSv , is given by


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ptv =

N "

(1 − τx )n .

(13)

x>v

4.2 Analysis of the Service Time The service time of the queueing system is defined as the time interval from the instant that an HoB frame starts contending for the channel to the instant that the burst is acknowledged following successful transmission or the instant that the HoB frame is dropped due to transmission failures. The service time is composed of two components: channel access delay and burst transmission delay. The former is the time interval from the instant the frame reaches to the head of the transmission queue, until it wins the contention and is ready for transmission, or until it is dropped due to transmission failures. The latter is defined as the time duration of successfully transmitting a burst (note that it equals zero if the HoB frame is dropped). Let E[Ssv ], E[Av ], and E[Bsv ] denote the mean of the service time, channel access delay, and burst transmission delay, respectively, where v represents that the burst is transmitted from ACv and s denotes the number of frames transmitted in the burst. E[Av ] is given by E[Av ] = Tcv ϕv + σv δv ,

(14)

where Tcv is the average collision time, σv is the average length of a time slot, ϕv and δv account for the average number of collisions and the average number of backoff counter decrements before a successful transmission from the ACv , respectively. According to the Markov chain in Fig. 1, ϕv and δv can be computed as ϕv =

m

δv =

i=0 m

i(1 − pv ) pvi ,

(15)

pvi Wiv /2,

(16)

i=0

where pvi is the probability that the backoff counter reaches stage i, and Wiv /2 is the average value of the backoff counter generated in the ith backoff stage. Let P T denote the probability that at least one AC transmits in a time slot. P T is given by PT = 1 −

N "

(1 − τx )n .

(17)

x=0

The probability, P Sv , that an AC v successfully transmits can be expressed as P Sv = nτv

N " x=0

(1 − τx )n−1

N " x>v

(1 − τx ).

(18)

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Since the channel is idle with probability (1 − P T ), a successful transmission from anAC x occurs with probability P Sx ; a collision happens with probability N P Sx ); the average length of a time slot, σv , can be calculated as (P T − x=0 σv

= (1 − P T )σ +

N

& P Sx Tsx +

PT −

x=0

N

) P Sx Tcv + E[X v ]P T,

(19)

x=0

where σ is the length of a physical time slot. E[X v ] is the total time spent on deferring the AIFS period of the AC v . Recall that an AC has to go through the AIFS period again if the channel is sensed busy during the deferring procedure, as shown in the sub-Markov chain of Fig. 3, E[X v ] may consist of several attempts for deferring the AIFS period of the ACv , which is given by E[X v ] =

∞

dv dv u−1 ptv (1 − ptv ) uTav ,

(20)

u=1 dv where u is the number of attempts for deferring the AIFS period of the ACv , ptv is the probability that a deferring attempt is successful, and Tav is the average time spent on each attempt, respectively. Tav is given by

Tav =

N

& P Sx Tsx

+

PT −

x>v

N

) P Sx

Tcv +

x>v

d v −1

s sσ ptv ,

(21)

s=1

where the first and second terms correspond to the “frozen time” of the backoff counter of the ACv caused by the transmission from the higher priority ACs with the smaller AIFS. The third term is the time spent on a failed attempt for down-counting the remaining time slots during the deferring period between AIFSmin and AIFSv . P T and P Sx denote the probabilities that at least one AC transmits and the AC x successfully transmits in a time slot, respectively, given that an AC v is deferring. P T and P Sx can be calculated as PT = 1 −

N "

(1 − τx )n ,

(22)

x>v

P Sx = nτx

N " y>v

(1 − τ y )n−1

N "

(1 − τ y ).

(23)

y>max{x,v}

Note that only the HoB frame is involved in the collision, Tcv is given by Tcv = TL + TH + TSIFS + TACK + AIFSv .

(24)


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On the other hand, the average time, Tsv , for a successful burst transmission from the ACv can be expressed as Tsv =

Fv

E[Bsv ]L sv , 1 − P0v

s=1

(25)

where Fv denotes the maximum number of frames that can be transmitted in a TXOP limit, the denominator (1 − P0v ) means that the occurrence of burst transmission is conditioned on the fact that there is at least one frame in the transmission queue, L sv (1 ≤ s ≤ Fv ) is the probability of having s frames transmitted within the burst, and E[Bsv ] is the burst transmission delay given by E[Bsv ] = AIFSv + s(TL + TH + 2TSIFS + TACK ) − TSIFS .

(26)

4.3 Queueing Model The transmission queue at the ACv can be modelled as an M/G[1,Fv ] /1/K queueing system [11], where the superscript [1, Fv ] denotes that the number of frames transmitted within a burst ranges from 1 to Fv and K represents the buffer size. The server becomes busy when a frame reaches to the head of the transmission queue. The server becomes free after a burst of frames are acknowledged by the destination following successful transmission or after the HoB frame is dropped due to transmission failures. The service time is dependent on the number of frames transmitted within a burst and the class of the transmitting AC. Thus, the service time of a burst with s frames transmitted from the ACv can be modelled by an exponential distribution function with mean E[Ssv ], then the mean service rate, μsv , is given by μsv = 1/E[Ssv ]. The state transition rate diagram of the queuing system at the AC v can be found in our previous work [5] where each state denotes the number of frames in the system. The transition rate matrix, Gv , of the Markov chain for the ACv can be obtained by the state transition rate diagram. The steady-state probability vector, Pv = (Pr v , r = 0, 1, . . . , K ), of the Markov chain satisfies the following equations: Pv Gv = 0 and Pv e = 1,

(27)

where e is a unit column vector. Solving these equations yields the steady-state vector as [4] Pv = u(I − v + eu)−1 ,

(28)

where r = I+Gr / min{Gr (ρ, ρ)}, u is an arbitrary row vector of r , and I denotes the unitmatrix. After obtaining Pv , we have L sv as L sv = Psv , 1 ≤ s < Fv and L sv = rK=Fv Pr v , s = Fv .

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The end-to-end delay of a frame is the time interval from the instant that the frame enters the buffer of the ACv to the instant that the frame leaves the buffer. Its mean value, E[Dv ], can be given by virtue of Little’s Law [11]: E[Dv ] =

E[Mv ] , λv (1 − PK v )

(29)

where E[Mv ] = rK=0 r Pr v is the average number of frames in the queueing system. λv (1 − PK v ) is the effective rate of the traffic entering into the transmission queue since the arriving frames are dropped if finding the queue full. Given the loss probability PK v , the throughput, v , of the ACv can be computed by v = λv E[P](1 − PK v )(1 − pvm+1 ),

(30)

where E[P] is the frame payload length and pvm+1 is the probability that the frame is discarded due to (m + 1) transmission failures.

5 Validation of the Analytical Model To validate the accuracy of the proposed model, we compare the analytical performance results against those obtained from the NS-2 simulation experiments based on the TKN implementation of the IEEE 802.11e EDCA [30]. We consider a WLAN with 10 stations located in a 100 m × 100 m rectangular grid where all stations are within the sensing and transmission range of each other. Each station generates four ACs of traffic with the identical arrival rates. The packet arrivals at each AC are characterized by a Poisson process. For the sake of clarity, the number of frames is adopted to be the unit of the TXOP in this study. Each simulation is executed once with 600 s simulation time, which is sufficiently long as the simulation results do not change with any further increment of simulation time. The system parameters used in the analytical model and simulations follow the IEEE 802.11b standard [8] using direct sequence spread spectrum (DSSS) as the physical-layer technology and are summarized in Table 1.

Frame payload

Table 1 System parameters 8000 bits PHY header 192 bits

MAC header Channel rate Slot time SIFS

224 bits 11 Mbit/s 20 µs 10 µs

ACK Basic rate Buffer size Retry limit

112 bits + PHY header 1 Mbit/s 50 frames 7

To investigate the accuracy of the model under various working conditions, we consider two scenarios with the different combinations of EDCA parameters, as shown in Table 2. Figures 4 and 5 depict the results of the throughput, end-to-end


Scenarios

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AC0 AC1 AC2 AC3 AC0 AC1 AC2 AC3

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Table 2 EDCA parameters AIFSN CWmin CWmax 6 2 2 2 7 4 2 2

32 32 16 8 64 32 16 16

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Load per AC (Mbps)

(c)

Fig. 4 Performance measures versus the offered load per AC in scenario 1. (a) Throughput; (b) end-to-end delay; and (c) frame loss probability

delay, and frame loss probability versus the offered loads per AC in scenarios 1 and 2, respectively. The close match between the analytical results and those obtained from simulation experiments demonstrates that the proposed model can produce the accurate prediction of the performance of the EDCA protocol with AIFS, CW, and TXOP schemes under any traffic loads. Moreover, it is worth mentioning that the maximum throughputs of AC1 and AC 0 are much larger than their saturation throughputs, which emphasize the importance of analysing the EDCA protocol

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under unsaturated traffic loads. We can also observe that the curves for AC1 and AC0 are close together, but differ widely from those of AC3 and AC2 . This is because that the EDCA parameters (AIFS, CW, TXOP) of AC1 and AC0 are close to each other, while they are very different with those of AC3 and AC2 . 5

0.45

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Fig. 5 Performance measures versus the offered load per AC in scenario 2. (a) Throughput; (b) end-to-end delay; and (c) frame loss probability

6 Conclusions In this chapter, we have presented a detailed literature review of the existing analytical models of the IEEE 802.11 DCF and the IEEE 802.11e EDCA protocols. We have then proposed a comprehensive analytical model to accommodate the QoS differentiation schemes in terms of AIFS, CW, and TXOP specified in the IEEE 802.11e EDCA protocol under unsaturated traffic loads. First, we develop a novel three-dimensional Markov chain to analyse the backoff procedure of each AC. Afterwards, to address the difficulties of queueing analysis arising from the TXOP scheme, the transmission queue at each AC is modelled as a bulk service queueing system. The QoS performance metrics including throughput, end-to-end delay, and


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frame loss probability have been derived and further validated through extensive NS-2 simulation experiments. The proposed analytical model is based on the assumption that each AC is under Poisson traffic. However, WLANs are currently integrating a diverse range of traffic sources, such as video, voice, and data, which significantly differ in their traffic patterns as well as QoS requirements. In future work, we intend to develop an analytical model for EDCA in WLANs with heterogeneous multimedia traffic. On the other hand, admission control is an important mechanism for the provisioning of QoS in WLANs. We plan to develop an efficient admission control scheme based on the proposed analytical model and the game-theoretical approach.

References 1. Bianchi G (2000) Performance analysis of the IEEE 802.11 distributed coordination function. IEEE J Select Areas Commun 18(3):535–547 2. Choi J, Yoo J, Kim CK (2008) A distributed fair scheduling scheme with a new analysis model in IEEE 802.11 wireless LANs. IEEE Trans Veh Technol 57(5):3083–3093 3. Engelstad PE, Osterbo ON (2006) Analysis of the total delay of IEEE 802.11e EDCA and 802.11 DCF. In: Proceedings of IEEE ICC’06, Istanbul, vol 2, pp 552–559 4. Fischer W, Meier-Hellstern K (1993) The Markov-modulated Poisson process (MMPP) cookbook. Perform Eval 18(2):149–171 5. Hu J, Min G, Woodward ME (2007) Analysis and comparison of burst transmission schemes in unsaturated 802.11e WLANs. In: Proceedings of IEEE GLOBECOM’07, Washington, pp 5133–5137 6. Hu J, Min G, Woodward ME, Jia W (2008) A comprehensive analytical model for IEEE 802.11e QoS differentiation schemes under unsaturated traffic loads. In: Proceedings of IEEE ICC’08, pp 241–245 7. Huang CL, Liao W (2007) Throughput and delay performance of IEEE 802.11e enhanced distributed channel access (EDCA) under saturation condition. IEEE Trans Wireless Commun 6(1):136–145 8. IEEE (1999) Wireless LAN Medium Access Control (MAC) and Physical Layer (PHY) specifications. IEEE Standard 802.11 9. IEEE (2005) Wireless LAN Medium Access Control (MAC) and Physical Layer (PHY) specifications: Medium Access Control (MAC) Quality of Service (QoS) enhancements. IEEE Standard 802.11e 10. Inan I, Keceli F, Ayanoglu E (2009) Analysis of the 802.11e enhanced distributed channel access function. IEEE Trans Commun 57(6):1753–1764 11. Kleinrock L (1975) Queueing systems: theory. Wiley 12. Kong Z, Tsang D, Bensaou B, Gao D (2004) Performance analysis of IEEE 802.11e contention-based channel access. IEEE J Select Areas Commun 22(10):2095–2106 13. Kumar A, Altman E, Miorandi D, Goyal M (2007) New insights from a fixed-point analysis of single cell IEEE 802.11 WLANs. IEEE/ACM Trans Netw 15(3):588–601 14. Li T, Ni Q, Xiao Y (2006) Investigation of the block ACK scheme in wireless ad-hoc networks. Wireless Commun Mobile Comput 6(6):877–888 15. Lin L, Fu H, Jia W (2005) An efficient admission control for IEEE 802.11 networks based on throughput analysis of (un)saturated channel. In: Proceedings of IEEE GLOBECOM’05, St. Louis, Missouri, vol 5, pp 3017–3021 16. Liu J, Niu Z (2007) Delay analysis of IEEE 802.11e EDCA under unsaturated conditions. In: Proceedings of IEEE WCNC’07, Hong Kong, pp 430–434 17. Malone D, Duffy K, Leith DJ (2007) Modeling the 802.11 distributed coordination function in nonsaturated heterogeneous conditions. IEEE/ACM Trans Netw 15(1):159–172

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18. Medepalli K, Tobagi FA (2006) Towards performance modelling of IEEE 802.11 based wireless networks: A unified framework and its applications. In: Proceedings of IEEE INFOCOM’06, Barcelona 19. Min G, Hu J, Woodward ME (2008) A dynamic IEEE 802.11e TXOP scheme in WLANs under self-similar traffic: Performance enhancement and analysis. In: Proceedings of IEEE ICC’08, Beijing, pp 2632–2636 20. Ozdemir M, McDonald AB (2006) On the performance of ad hoc wireless LANs: A practical queuing theoretic model. Perform Eval 63(11):1127–1156 21. Peng F, Alnuweiri HM, Leung VCM (2006) Analysis of burst transmission in IEEE 802.11e wireless LANs. In: Proceedings of IEEE ICC’06, Istanbul, vol 2, pp 535–539 22. Ramaiyan V, Kumar A, Altman E (2008) Fixed point analysis of single cell IEEE 802.11e WLANs: Uniqueness and multistability. IEEE/ACM Trans Netw 16(5):1080–1093 23. Robinson JW, Randhawa TS (2004) Saturation throughput analysis of IEEE 802.11e enhanced distributed coordination function. IEEE J Select Areas Commun 22(5):917–928 24. Tantra JW, Foh CH, Mnaouer AB (2005) Throughput and delay analysis of the IEEE 802.11e EDCA saturation. In: Proc. IEEE ICC’05, Seoul, vol 5, pp 3450–3454 25. Tantra JW, Foh CH, Tinnirello I, Bianchi G (2006) Analysis of the IEEE 802.11e EDCA under statistical traffic. In: Proceedings of IEEE ICC’06, Istanbul, vol 2, pp 546–551 26. Tao Z, Panwar S (2006) Throughput and delay analysis for the IEEE 802.11e enhanced distributed channel access. IEEE Trans Commun 54(4):596–603 27. Tickoo O, Sikdar B (2008) Modeling queueing and channel access delay in unsaturated IEEE 802.11 random access MAC based wireless networks. IEEE/ACM Trans Netw 16(4):878–891 28. Tinnirello I, Choi S (2005) Efficiency analysis of burst transmission with block ACK in contention-based 802.11e WLANs. In: Proceedings of IEEE ICC’05, Seoul, vol 5, pp 3455–3460 29. Vitsas V, Chatzimisios P, Boucouvalas AC, Raptis P, Paparrizos K, Kleftouris D (2004) Enhancing performance of the IEEE 802.11 distributed coordination function via packet bursting. In: Proceedings of IEEE GLOBECOM’04, Dallas, Texas, pp 245–252 30. Sven W, Emmelmann M, Christian H, Adam W (2006) TKN EDCA model for ns-2. Technical University of Berlin, Technical Report TKN-06-003 31. Wu H, Peng Y, Long K, Cheng S, Ma J (2002) Performance of reliable transport protocol over IEEE 802.11 wireless LAN: analysis and enhancement. In: Proceedings of IEEE INFOCOM’02, New York, vol 2, pp 599–607 32. Xiao Y (2005) Performance analysis of priority schemes for IEEE 802.11 and IEEE 802.11e wireless LANs. IEEE Trans Wireless Commun 4(4):1506–1515 33. Zhai H, Kwon Y, Fang Y (2004) Performance analysis of IEEE 802.11 MAC protocols in wireless LANs. Wireless Commun Mobile Comput 4(8):917–931 34. Zhao Q, Tsang DHK, Sakurai T (2008) A simple model for nonsaturated IEEE 802.11 DCF Networks. IEEE Commun Lett 12(8): 563–565 35. Zhu H, Chlamtac I (2005) Performance analysis for IEEE 802.11e EDCF service differentiation. IEEE Trans Commun 4(4):1779–1788 36. Ziouva E, Antonakopoulos T (2002) CSMA/CA performance under high traffic conditions: Throughput and delay analysis. Comput Commun 25(3):313–321

Dynamic Overlay Single-Domain Contracting for End-to-End Contract Switching Murat Yüksel, Aparna Gupta, and Koushik Kar

1 Introduction The Internet’s simple best-effort packet-switched architecture lies at the core of its tremendous success and impact. Today, the Internet is firmly a commercial medium involving several competitive service providers and content providers. However, current Internet architecture allows neither (i) users to indicate their value choices at sufficient granularity nor (ii) providers to manage risks involved in investment for new innovative QoS technologies and business relationships with other providers as well as users. Currently, users can only indicate their value choices at the access/link bandwidth level not at the routing level. End-to-end QoS contracts are possible today via virtual private networks, but with static and long-term contracts. Further, an enterprise that needs end-to-end capacity contracts between two arbitrary points on the Internet for a short period of time has no way of expressing its needs. We envision an Internet architecture that allows flexible, fine-grained, dynamic contracting over multiple providers. With such capabilities, the Internet itself will be viewed as a “contract-switched” network beyond its current status as a “packetswitched” network. A contract-switched architecture will enable flexible and economically efficient management of risks and value flows in an Internet characterized by many tussle points [7] where competition for network resources takes place. Realization of such an architecture heavily depends on the capabilities of provisioning dynamic single-domain contracts which can be priced dynamically or based on intra-domain congestion. Implementation of dynamic pricing still remains a Murat Yüksel University of Nevada - Reno, Reno, NV 89557, USA e-mail: [email protected] Aparna Gupta Rensselaer Polytechnic Institute, Troy, NY 12180, USA e-mail: [email protected] Koushik Kar Rensselaer Polytechnic Institute, Troy, NY 12180, USA e-mail: [email protected]


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challenge, although several proposals have been made, e.g., [18, 25, 32]. Among many others, two major implementation obstacles can be defined: need for timely feedback to users about the price and determination of congestion information in an efficient, low-overhead manner. The first problem, timely feedback, is relatively hard to achieve in a wide area network such as the Internet. In [3], the authors showed that users do want feedback about charging of the network service (such as current price and prediction of service quality in near future). However, in [37], we illustrated that congestion control by pricing cannot be achieved if price changes are performed at a timescale larger than roughly 40 round-trip-times (RTTs). This means that in order to achieve congestion control by pricing, service prices must be updated very frequently (i.e., 2–3 s since RTT is expressed in terms of milliseconds for most cases in the Internet). In order to solve this timescale problem for dynamic pricing, we propose two solutions, which lead to two different pricing “architectures”: • By placing intelligent intermediaries (i.e., software or hardware agents) between users and the provider. This way it is possible for the provider to update prices frequently at low timescales, since price negotiations will be made with a software/hardware agent rather than a human. Since the provider will not employ any congestion control mechanism for its network and try to control congestion by only pricing, we call this pricing architecture as pricing for congestion control (PFCC). • By overlaying pricing on top of an underlying congestion control mechanism. This way it is possible to enforce tight control on congestion at small timescale, while performing pricing at timescales large enough for human involvement. The provider implements a congestion control mechanism to manage congestion in its network. So, we call this pricing architecture as pricing over congestion control (POCC). The big picture of the two pricing architectures PFCC and POCC are shown in Fig. 1. We will describe PFCC and POCC later in Section 4.

(a)

(b)

Fig. 1 Different pricing architectures with/without edge-to-edge congestion control: (a) Pricing for congestion control (PFCC) and (b) pricing over congestion control (POCC)

Dynamic Overlay Single-Domain Contracting

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The second problem, congestion information, is also very hard to solve in a way that does not require a major upgrade at network routers. However, in diff-serv [2], it is possible to determine congestion information via a good ingress–egress coordination. So, this flexible environment of diff-serv motivated us to develop a pricing framework on it. The chapter is organized as follows: In the next section, we position our work and briefly survey relevant work in the area. Section 3 introduces our contractswitching paradigm. In Section 4, we present PFCC and POCC pricing architectures motivated by the timescale issues mentioned above. In Section 5 we describe properties of distributed-DCC framework according to the PFCC and POCC architectures. Next in Section 6, we define a pricing scheme edge-to-edge pricing (EEP) which can be implemented in the defined distributed-DCC framework. We study optimality of EEP for different forms of user utility functions and consider effect of different parameters such as user’s budget, user’s elasticity. In Section 7, according to the descriptions of distributed-DCC framework and EEP scheme, we simulate distributed-DCC in the two architectures PFCC and POCC. With the simulation results, we compare distributed-DCC’s performance in PFCC and POCC architectures. We, then, extend the pricing formulations to an end-to-end level in Section 8. We finalize with summary and discussions in Section 9.

2 Related Work There have been several pricing proposals, which can be classified in many ways: static vs. dynamic, per-packet charging vs. per-contract charging, and charging a priori to service vs. a posteriori to service. Although there are opponents to dynamic pricing in the area (e.g., [21–23]), most of the proposals have been for dynamic pricing (specifically congestion pricing) of networks. Examples of dynamic pricing proposals are MacKie-Mason and Varian’s Smart Market [18], Gupta et al.’s priority pricing [11], Kelly et al.’s proportional fair pricing (PFP) [15], Semret et al.’s market pricing [24, 25], and Wang and Schulzrinne’s resource negotiation and pricing (RNAP) [32, 33]. Odlyzko’s Paris metro pricing (PMP) [20] is an example of static pricing proposal. Clark’s expected capacity [5, 6] and Cocchi et al.’s edge pricing [8] allow both static and dynamic pricing. In terms of charging granularity, smart market, priority pricing, PFP, and edge pricing employ per-packet charging, while RNAP and expected capacity do not employ per-packet charging. Smart market is based primarily on imposing per-packet congestion prices. Since smart market performs pricing on per-packet basis, it operates at the finest possible pricing granularity. This makes smart market capable of making ideal congestion pricing. However, smart market is not deployable because of its per-packet granularity (i.e., excessive overhead) and its many requirements from routers (e.g., requires all routers to be updated). In [35], we studied smart market and difficulties of its implementation in more detail.

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While smart market holds one extreme in terms of granularity, expected capacity holds the other extreme. Expected capacity proposes to use long-term contracts, which can give more clear performance expectation, for statistical capacity allocation and pricing. Prices are updated at the beginning of each long-term contract, which incorporates little dynamism to prices. Our work, distributed-DCC, is a middle-ground between smart market and expected capacity in terms of granularity. Distributed-DCC performs congestion pricing at short-term contracts, which allows more dynamism in prices while keeping pricing overhead small. In the area, another proposal that mainly focused on implementation issues of congestion pricing on diff-serv is RNAP [32, 33]. Although RNAP provides a complete picture for incorporation of admission control and congestion pricing, it has excessive implementation overhead since it requires all network routers to participate in determination of congestion prices. This requires upgrades to all routers similar to the case of smart market. We believe that pricing proposals that require upgrades to all routers will eventually fail in implementation phase. This is because of the fact that the Internet routers are owned by different entities who may or may not be willing to cooperate in the process of router upgrades. Our work solves this problem by requiring upgrades only at edge routers rather than at all routers.

3 Contract-Switching Paradigm The essence of “contract switching” is to use contracts as the key building block for inter-domain networking. As shown in Fig. 2, this increases the inter-domain architecture flexibilities by introducing more tussle points into the protocol design. Especially, this paradigm will allow the much needed revolutions in the Internet protocol design: (i) inclusion of economic tools in the network layer functions such as inter-domain routing while the current architecture only allows basic connectivity information exchange and (ii) management of risks involved in QoS technology investments and participation into end-to-end QoS contract offerings by allowing ISPs to potentially apply financial engineering methods.

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Fig. 2 Packet switching (a) introduced many more tussle points into the Internet architecture by breaking the end-to-end circuits of circuit switching into routable datagrams. Contract switching (b) introduces even more tussle points at the edge/peering points of domain boundaries by overlay contracts


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In addition to these design opportunities, the contract-switching paradigm introduces several research challenges. As the key building block, intra-domain service abstractions call for design of (i) single-domain edge-to-edge QoS contracts with performance guarantees and (ii) nonlinear pricing schemes geared toward cost recovery. Moving one level up, composition of end-to-end inter-domain contracts poses a major research problem which we formulate as a “contract routing” problem by using single-domain contracts as “contract links.” Issues to be addressed include routing scalability, contract monitoring, and verification as the inter-domain context involves large-size effects and crossing trust boundaries. Several economic tools can be used to remedy pricing, risk sharing, and money-back problems of a contract-switched network provider (CSNP), which can operate as an overlay re-seller ISP (or an alliance of ISPs) that buys contract links and sells end-to-end QoS contracts. In addition to CSNPs, the contract-switching paradigm allows more distributed ways of composing end-to-end QoS contracts as we will detail later.

4 Single-Domain Pricing Architectures: PFCC vs. POCC In our previous work [27], we presented a simple congestion-sensitive pricing “framework,” dynamic capacity contracting (DCC), for a single diff-serv domain. DCC treats each edge router as a station of a service provider or a station of cooperating set of service providers. Users (i.e., individuals or other service providers) make short-term contracts with the stations for network service. During the contracts, the station receives congestion information about the network core at a timescale smaller than contracts. The station, then, uses that congestion information to update the service price at the beginning of each contract. Several pricing “schemes” can be implemented in that framework. DCC models a short-term contract for a given traffic class as a tuple of price per unit traffic volume Pv , maximum volume Vmax (maximum number of bytes that can be sent during the contract), and the term of the contract T (length of the contract): Contract =< Pv , Vmax , T > .

(1)

Figure 3 illustrates the big picture of DCC framework. Customers can only access network core by making contracts with the provider stations placed at the edge routers. The stations offer contracts (i.e., Vmax and T ) to fellow users. Access to these available contracts can be done in different ways, what we call edge strategy. Two basic edge strategies are “bidding” (many users bid for an available contract) or “contracting” (users negotiate Pv with the provider for an available contract). Notice that, in DCC framework, provider stations can implement dynamic pricing schemes. In particular, they can implement congestion-based pricing schemes, if they have actual information about congestion in network core. This congestion information can come from the interior routers or from the egress edge routers depending on the congestion detection mechanism being used. DCC assumes that

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Edge Router

Edge Router

Edge Router

Customers

Network Core accessed only by contracts

Edge Router

Edge Router

Edge Router

Fig. 3 DCC framework on diff-serv architecture

the congestion detection mechanism is able to give congestion information in timescales (i.e., observation intervals) smaller than contracts. However, in DCC, we assumed that all the provider stations advertise the same price value for the contracts, which is very costly to implement over a wide area network. This is simply because the price value cannot be communicated to all stations at the beginning of each contract. We relax this assumption by allowing the stations to calculate the prices locally and advertise different prices than the other stations. We call this new version of DCC as distributed-DCC. We introduce ways of managing the overall coordination of the stations. As a fundamental difference between distributed-DCC and the well-known dynamic pricing proposals in the area (e.g., proposals by Kelly et al. [15] and Low et al. [17]) lies in the manner of price calculation. In distributed-DCC, the prices are calculated on an edge-to-edge basis, while traditionally it has been proposed that prices are calculated at each local link and fed back to users. In distributed-DCC, basically, the links on a flow’s route are abstracted out by edge-to-edge capacity estimation and the ingress node communicates with the corresponding egress node to observe congestion on the route. Then, the ingress node uses the estimated capacity and the observed congestion information in price calculation. However, in Low et al.’s framework, each link calculates its own price and sends it to the user, and the user pays the aggregate price. So, distributed-DCC is better in terms of implementation requirements, while Low et al.’s framework is better in terms of optimality. Distributed-DCC trades off some optimality in order to enable implementation of dynamic pricing. Amount of lost optimality depends on the closed-loop edge-to-edge capacity estimation.

4.1 Pricing for Congestion Control (PFCC) In this pricing architecture, provider attempts to solve congestion problem of its network just by congestion pricing. In other words, the provider tries to control congestion of its network by changing service prices. The problem here is


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that the provider will have to change the price very frequently such that human involvement into the price negotiations will not be possible. This problem can be solved by running intermediate software (or hardware) agents between end-users and the provider. The intermediate agent receives inputs from the end-user at large timescales and keeps negotiating with the provider at small timescales. So, intermediate agents in PFCC architecture are very crucial in terms of acceptability by users. If PFCC architecture is not employed (i.e., providers do not bother to employ congestion pricing), then congestion control will be left to the end-user as it is in the current Internet. Currently in the Internet, congestion control is totally left to end-users, and common way of controlling congestion is TCP and its variants. However, this situation leaves open doors to non-cooperative users who do not employ congestion control algorithms or at least employ congestion control algorithms that violate fairness objectives. For example, by simple tricks, it is possible to make TCP connection to capture more of the available capacity than the other TCP connections. The major problem with PFCC is that development of user-friendly intermediate agents is heavily dependent on user opinion, and hence requires significant amount of research. A study of determining user opinions is available in [3]. In this chapter, we do not focus development of intermediate agents.

4.2 Pricing Over Congestion Control (POCC) Another way of approaching the congestion control problem by pricing is to overlay pricing on top of congestion control. This means the provider undertakes the congestion control problem by itself and employs an underlying congestion control mechanism for its network. This way it is possible to enforce tight control on congestion at small timescales, while maintaining human involvement into the price negotiations at large timescales. Figure 1 illustrates the difference between POCC (with congestion control) and PFCC (without congestion control) architectures. So, assuming that there is an underlying congestion control scheme, the provider can set the parameters of that underlying scheme such that it leads to fairness and better control of congestion. The pricing scheme on top can determine user incentives and set the parameters of the underlying congestion control scheme accordingly. This way, it will be possible to favor some traffic flows with higher willingness to pay (i.e., budget) than the others. Furthermore, the pricing scheme will also bring benefits such as an indirect control on user demand by price, which will in turn help the underlying congestion control scheme to operate more smoothly. However, the overall system performance (e.g., fairness, utilization, throughput) will be dependent on the flexibility of the underlying congestion control mechanism. Since our main focus is to implement pricing in “diff-serv environment,” we assume that the provider employs “edge-to-edge” congestion control mechanisms under the pricing protocol on top. So, in diff-serv environment, overlaying pricing on top of edge-to-edge congestion control raises two major problems:

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1. Parameter mapping: Since the pricing protocol wants to allocate network capacity according to the user incentives (i.e., the users with greater budget should get more capacity) that change dynamically over time, it is a required ability to set corresponding parameters of the underlying edge-to-edge congestion control mechanism such that it allocates the capacity to the user flows according to their incentives. So, this raises need for a method of mapping parameters of the pricing scheme to the parameters of the underlying congestion control mechanism. Notice that this type of mapping requires the edge-to-edge congestion control mechanism to be able to provide parameters that tune the rate being given to edge-to-edge flows. 2. Edge queues: The underlying edge-to-edge congestion control scheme will not always allow all the traffic admitted by the pricing protocol, which will cause queues to build up at network edges. So, management of these edge queues is necessary in POCC architecture. Figure 1a and b compares the situation of the edge queues in the two cases when there is an underlying edge-to-edge congestion control scheme and when there is not.

5 Distributed-DCC Framework Distributed-DCC framework is specifically designed for DiffServ environments, because the edge routers can perform complex operations which are essential to several requirements for implementation of congestion pricing. Each edge router is treated as a station of the provider. Each station advertises locally computed prices with information received from other stations. The main framework basically describes how to preserve coordination among the stations such that stability and fairness of the overall network is preserved. We can summarize essence of distributed-DCC in two items: • Since upgrading of all routers is not possible to implement, pricing should happen on an edge-to-edge basis which only requires upgrades to edge routers. • Provider should employ short-term contracts in order to have ability to change prices frequently enough such that congestion pricing can be enabled. Distributed-DCC framework has three major components: Logical pricing server (LPS), ingress stations, and egress stations. Solid-lined arrows in the figure represent control information being transmitted among the components. Basically, ingress stations negotiate with customers, observe customer’s traffic, and make estimations about customer’s demand. Ingress stations inform corresponding egress stations about the observations and estimations about each edge-to-edge flow. Egress stations detect congestion by monitoring edge-to-edge traffic flows. Based on congestion detections, egress stations estimate available capacity for each edgeto-edge flow and inform LPS about these estimations.


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LPS receives capacity estimations from egress stations and allocates the network available capacity to edge-to-edge flows according to different criteria (such as fairness, price optimality).

5.1 Ingress Station i Figure 4 illustrates sub-components of ingress station i in the framework. Ingress i includes two sub-components: pricing scheme and budget estimator.

Fig. 4 Major functions of ingress i

Ingress station i keeps a “current” price vector pi , where pi j is the price for the flow from ingress i to egress j. So, the traffic using flow i to j is charged the price pi j . Pricing scheme is the sub-component that calculates price pi j for each edgeto-edge flow starting at ingress i. It uses allowed flow capacities ci j and other local information (such as bi j ), in order to calculate price pi j . The station, then, uses pi j in negotiations with customers. We will describe a simple pricing scheme edge-toedge pricing (EEP) later in Section 6. However, it is possible to implement several other pricing schemes by using the information available at ingress i. Other than EEP, we implemented another pricing scheme, price discovery, which is available in [1]. Also, the ingress i uses the total estimated network capacity C in calculating the Vmax contract parameter defined in (1). Admission control techniques can be used to identify the best value for Vmax . We use a simple method which does not put any restriction on Vmax , i.e., Vmax = C ∗ T where T is the contract length. Budget estimator is the sub-component that observes demand for each edge-toedge flow. We implicitly assume that user’s “budget” represents user’s demand (i.e.,

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willingness to pay). So, budget estimator estimates budget bî j of each edge-to-edge traffic flow.1

5.2 Egress Station j Figure 5 illustrates sub-components of egress station j in the framework: congestion detector, congestion-based capacity estimator, flow cost analyzer, and fairness tuner.

Fig. 5 Major functions of egress j

Congestion detector implements an algorithm to detect congestion in network core by observing traffic arriving at egress j. Congestion detection can be done in several ways. We assume that interior routers mark (i.e., sets the ECN bit) the data packets if their local queue exceeds a threshold. Congestion detector generates a “congestion indication” if it observes a marked packet in the arriving traffic.

1 Note

that edge-to-edge flow does not mean an individual user’s flow. Rather it is the traffic flow that is composed of aggregation of all traffic going from one edge node to another edge node.


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Congestion-based capacity estimator estimates available capacity cî j for each edge-to-edge flow exiting at egress j. In order to calculate cî j , it uses congestion indications from congestion detector and actual output rates μi j of the flows. The crucial property of congestion-based capacity estimator is that it estimates capacity in a congestion-based manner, i.e., it decreases the capacity estimation when there is congestion indication and increases when there is no congestion indication. This makes the prices congestion sensitive, since pricing scheme at ingress calculates prices based on the estimated capacity. Flow cost analyzer determines cost of each traffic flow (e.g., number of links traversed by the flow, number of bottlenecks traversed by the flow, amount of queuing delay caused by the flow) exiting at egress j. Cost incurred by each flow can be several things: number of traversed links, number of traversed bottlenecks, amount of queuing delay caused. We assume that number of bottlenecks is a good representation of the cost incurred by a flow. It is possible to define edge-to-edge algorithms that can effectively and accurately estimate the number of bottlenecks traversed by a flow [34]. LPS, as will be described in the next section, allocates capacity to edge-to-edge flows based on their budgets. The flows with higher budgets are given more capacity than the others. So, egress j can penalize/favor a flow by increasing/decreasing its budget bˆ i j . Fairness tuner is the component that updates bî j . So, fairness tuner penalizes or favors the flow from ingress i by updating its estimated budget value, i.e. bi j = f (bî j , rî j , < parameters >) where < parameters > are other optional parameters that may be used for deciding how much to penalize or favor the flow. For example, if the flow ingress i is passing through more congested areas than the other flows, fairness tuner can penalize this flow by reducing its budget estimation bî j . We will describe an algorithm for fairness tuner later in Section 5.2.1. Egress j sends cî j s (calculated by congestion-based capacity estimator) and bi j s (calculated by fairness tuner) to LPS.

5.2.1 Fairness Tuner We examine the issues regarding fairness in two main cases. We first determine these two cases and then provide solutions within distributed-DCC framework: • Single-bottleneck case: The pricing protocol should charge the same price (i.e., $/bandwidth) to the users of the same bottleneck. In this way, among the customers using the same bottleneck, the ones who have more budget will be given more rate (i.e., bandwidth/time) than the others. The intuition behind this reasoning is that the cost of providing capacity to each customer is the same. • Multi-bottleneck case: The pricing protocol should charge more to the customers whose traffic passes through more bottlenecks and cause more costs to the provider. So, other than proportionality to customer budgets, we also want to allocate less rate to the customers whose flows are passing through more bottlenecks than the other customers.

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For multi-bottleneck networks, two main types of fairness have been defined: max–min fairness [16] and proportional fairness [15]. In max–min fair rate allocation, all flows get equal share of the bottlenecks, while in proportional fair rate allocation flows get penalized according to the number of traversed bottlenecks. Depending on the cost structure and user’s utilities, for some cases the provider may want to choose max–min or proportional rate allocation. So, we would like to have ability of tuning the pricing protocol such that fairness of its rate allocation is in the way the provider wants. To achieve the fairness objectives defined in the above itemized list, we introduce new parameters for tuning rate allocation to flows. In order to penalize flow i to j, the egress j can reduce bî j while updating the flow’s estimated budget. It uses the following formula to do so: bi j = f (bî j , r (t), α, rmin ) =

bî j , rmin + (ri j (t) − rmin )α

where ri j (t) is the congestion cost caused by the flow i to j, rmin is the minimum possible congestion cost for the flow, and α is fairness coefficient. Instead of bî j , the egress j now sends bi j to LPS. When α is 0, fairness tuner is employing max–min fairness. As it gets larger, the flow gets penalized more and rate allocation gets closer to proportional fairness. However, if it is too large, then the rate allocation will move away from proportional fairness. Let α ∗ be the α value where the rate allocation is proportionally fair. If the estimation ri j (t) is absolutely correct, then α ∗ = 1. Otherwise, it depends on how accurate ri j (t) is. Assuming that each bottleneck has the same amount of congestion and capacity. Then, in order to calculate ri j (t) and rmin , we can directly use the number of bottlenecks the flow i to j is passing through. In such a case, rmin will be 1 and ri j (t) should be number of bottlenecks the flow is passing through.

5.3 Logical Pricing Server (LPS) Figure 6 illustrates basic functions of LPS in the framework. LPS receives information from egresses and calculates allowed capacity ci j for each edge-to-edge flow. The communication between LPS and the stations take place at every LPS interval L. There is only one major sub-component in LPS: capacity allocator. Capacity allocator receives cî j s, bi j s and congestion indications from egress stations. It calculates allowed capacity ci j for each flow. Calculation of ci j values is a complicated task which depends on internal topology. In general, the flows should share capacity of the same bottleneck in proportion to their budgets. Other than functions of capacity allocator, LPS also calculates total available network capacity C, which is necessary for determining the contract parameter Vmax at ingresses. LPS simply sums cî j to calculate C.


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Fig. 6 Major functions of LPS

5.3.1 ETICA: Edge-to-Edge, Topology-Independent Capacity Allocation First, note that LPS is going to implement the ETICA algorithm as a capacity allocator (see Fig. 6). So, we will refer to LPS throughout the description of ETICA below. At LPS, we introduce a new information about each edge-to-edge flow fi j . A flow f i j is congested if egress j has been receiving congestion indications from that flow recently (we will later define what “recent” is). Again at LPS, let K i j determine the state of f i j . If K i j > 0, LPS determines f i j as congested. If not, it determines f i j as non-congested. At every LPS interval t, LPS calculates K i j as follows: # K i j (t) =

k, congestion in t − 1 K i j (t − 1) − 1, no congestion in t − 1 K i j (0) = 0,

(2)

where k is a positive integer. Notice that k parameter defines how long a flow will stay in “congested” state after the last congestion indication. So, in other words, k defines the timeline to determine if a congestion indication is “recent” or not. According to these considerations in ETICA algorithm, Fig. 7 illustrates states of an edge-to-edge flow given that probability of receiving a congestion indication in the last LPS interval is p. Gray states are the states in which the flow is “congested,” and the single white state is the “non-congested” state. Observe that number of

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Fig. 7 States of an edge-to-edge flow in ETICA algorithm: the states i > 0 are “congested” states and the state i = 0 is the “non-congested” state, represented with gray and white colors, respectively

congested states (i.e., gray states) is equal to k which defines to what extent a congestion indication is “recent.”2 Given the above method to determine whether a flow is congested or not, we now describe the algorithm to allocate capacity to the flows. Let F be the set of all edgeto-edge flows in the diff-serv domain and Fc be the set of congested edge-to-edge flows. Let Cc be the accumulation of cî j s where f i j ∈ Fc . Further, let Bc be the accumulation of bi j s where f i j ∈ Fc . Then, LPS calculates the allowed capacity for f i j as follows: ci j =

bi j Bc C c , cî j ,

Ki j > 0 . otherwise

The intuition is that if a flow is congested, then it must be competing with other congested flows. So, a congested flow is allowed a capacity in proportion to its budget relative to budgets of all congested flows. Since we assume no knowledge about the interior topology, we can approximate the situation by considering these congested flows as if they are passing through a single bottleneck. If knowledge about the interior topology is provided, one can easily develop better algorithms by sub-grouping the congested flows that are passing through the same bottleneck.

6 Single-Domain Edge-to-Edge Pricing Scheme (EEP) For flow f i j , distributed-DCC framework provides an allowed capacity ci j and an estimation of total user budget bî j at ingress i. So, the provider station at ingress i can use these two information to calculate price. We propose a simple price formula to balance supply and demand:

2 Note that instead of setting K to k at every congestion indication, more accurate methods can be ij used in order to represent self-similar behavior of congestion epochs. For simplicity, we proceed with the method in (2).


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pî j =

bî j . ci j

(3)

Here, bî j represents user demand and ci j is the available supply. The main idea of the EEP is to balance supply and demand by equating price to the ratio of users’ budget (i.e., demand) B by available capacity C. Based on that, we used the pricing formula: p=

Bˆ , Cˆ

(4)

where Bˆ is the users’ estimated budget and Cˆ is the estimated available network capacity. The capacity estimation is performed based on congestion level in the network, and this makes the EEP scheme a congestion-sensitive pricing scheme. We now formulate the problem of total user utility maximization for a multiuser multi-bottleneck network. Let F = {1, . . . , F} be the set of flows and L = {1, . . . , L} be the set of links in the network. Also, let L( f ) be the set of links the flow f passes through and F(l) be the set of flows passing through the link l. Let cl be the capacity of link l. Let λ be the vector of flow rates and λ f be the rate of flow f . We can formulate the total user utility maximization problem as follows: SYSTEM:

max λ

U f (λ f )

f

subject to λf

≤

cl ,

l = 1, . . . , L .

(5)

f ∈F(l)

This problem can be divided into two separate problems by employing monetary exchange between user flows and the network provider. Following Kelly’s [14] methodology we split the system problem into two. The first problem is solved at the user side. Given accumulation of link prices on the flow f ’s route, p f , what is the optimal sending rate in order to maximize surplus. FLOW f ( p f ): ⎧ ⎫ ⎨ ⎬ pl λ f max U f (λ f ) − ⎭ λf ⎩ l∈L( f )

over λ f ≥ 0.

(6)

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The second problem is solved at the provider’s side. Given sending rate of user flows (which are dependent on the link prices), what is the optimal price to advertise in order to maximize revenue. NETWORK(λ( p f )) :

max p

pl λ f

f l∈L( f )

subject to λf

≤

cl ,

l = 1, . . . , L

f ∈F(l)

over p ≥ 0. Let the total price paid by flow f be p f = F L O W f ( p f ) will be

(7)

l∈L( f )

f λ f ( p f ) = U −1 f ( p ).

pl . Then, solution to

(8)

When it comes to the NETWORK(λ( p f )) problem, the solution will be dependent on user flows utility functions since their sending rate is based on their utility functions as shown in the solution of FLOW f ( p f ). So, in the next sections we will solve the NETWORK(λ( p f )) problem for the cases of logarithmic and nonlogarithmic utility functions. We model customer i’s utility with the well-known function3 [15–17, 19]: u i (x) = wi log(x),

(9)

where x is the allocated bandwidth to the customer and wi is customer i’s budget (or bandwidth sensitivity). Now, we set up a vectorized notation, then solve the revenue maximization problem NETWORK(λ( p f )). Assume the network includes n flows and m links. Let λ be row vector of the flow rates (λ f for f ∈ F), P be column vector of the price at each link ( pl for l ∈ L). Define the n × n matrix P ∗ in which the diagonal element P j∗j is the aggregate price being advertised to flow j (i.e., p j = l∈L( j) pl ) and all the other elements are 0. Also, let A be the n × m routing matrix in which the element Ai j is 1 if ith flow is passing though jth link and the element Ai j is 0, if not, C be the column vector of link capacities (cl for l ∈ L). Finally, define the n ×n matrix λˆ in which the diagonal element λˆ j j is the rate of flow j (i.e. λˆ j j = λ j ) and all the other elements are 0.

3 Wang

and Schulzrinne introduced a more complex version in [33].


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Given the above notation, relationship between the link price vector P and the flow aggregate price matrix P ∗ can be written as A P = P ∗ e,

(10)

ˆ ˆ = e λ, λ = (λe)

(11)

T

T

where e is the column unit vector. We use the utility function of (9) in our analysis. By plugging (9) into (8) we obtain flow’s demand function in vectorized notation: λ(P ∗ ) = W P ∗ −1 ,

(12)

where W is row vector of the weights wi in flow’s utility function (9). Also, we can write the utility function (9) in vectorized notation as follows: U (λ) = W log(λˆ ).

(13)

The revenue maximization of (7) can be re-written as follows: max R = λA P P

subject to λA

≤

CT .

(14)

So, we write the Lagrangian as follows: L = λA P + (C T − λA)γ ,

(15)

where γ is column vector of the Lagrange multipliers for the link capacity constraint. Solving (15), we derive P: P = (C T )−1 W e.

(16)

Since P ∗ = (P ∗ )T , we can derive another solution: P = A−1 W T C −1 A T e.

(17)

Notice that the result in (16) holds for a single-bottleneck (i.e., single-link) network. In non-vectorized notation, this result translates to f ∈F w f p= . c

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The result in (17) holds for a multi-bottleneck network. This result means that each link’s optimal price is dependent on the routes of each flow passing through that link. More specifically, the optimal price for link l is accumulation of budgets of flows passing through link l (i.e., W T A T in the formula) divided by total capacity of the links that are traversed by the flows traversing the link l (i.e., A−1 C −1 in the formula). In non-vectorized notation, price of link l can be written as pl =

f ∈F(l) w f

f ∈F(l)

k∈L( f ) ck

.

Similar results can be found for non-logarithmic utility functions involving user’s utility-bandwidth elasticity [36].

7 Distributed-DCC: PFCC and POCC Architectures In order to adapt distributed-DCC to PFCC architecture, LPS must operate at very low timescales. In other words, LPS interval must be small enough to maintain control over congestion, since PFCC assumes no underlying congestion control mechanism. This raises practical issues to be addressed. For instance, intermediate agents between customers and ingress stations must be implemented in order to maintain human involvement into the system. Further, scalability issues regarding LPS must be solved since LPS must operate at very small timescales. Distributed-DCC operates on per edge-to-edge flow basis which means the flows are not on a per-connection basis. All the traffic going from edge router i to j is counted as only one flow. This actually relieves the scalability problem for operations that happen on per-flow basis. The number of flows in the system will be n(n − 1) where n is the number of edge routers in the diff-serv domain. So, indeed, scalability of the flows is not a problem for the current Internet since number of edge routers for a single diff-serv domain is very small. If it becomes so large in future, then aggregation techniques can be used to overcome this scalability issue, of course, by sacrificing some optimality. To adapt distributed-DCC framework to POCC architecture, an edge-to-edge congestion control mechanism is needed, for which we use Riviera [13] in our experiments. Riviera takes advantage of two-way communication between ingress and egress edge routers in a diff-serv network. Ingress sends a forward feedback to egress in response to feedback from egress, and egress sends backward feedback to ingress in response to feedback from ingress. So, ingress and egress of a traffic flow keep bouncing feedback to each other. Ignoring loss of data packets, the egress of a traffic flow measures the accumulation, a, caused by the flow by using the bounced feedbacks and RTT estimations. When a for a particular flow exceeds a threshold or goes below a threshold the flow is identified as congested or not-congested,


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respectively. The ingress node gets informed about the congestion detection by backward feedbacks and uses egress’ explicit rate to adjust the sending rate. We now provide solutions defined in Section 4.2, for the case of overlaying distributed-DCC over Riviera: 1. Parameter mapping: For each edge-to-edge flow, LPS can calculate the capacity share of that flow out of the total network capacity. Let γi j = ci j /C be the fraction of network capacity that must be given to the flow i to j. LPS can convey γi j s to the ingress stations, and they can multiply the increase parameter αi j with γi j . Also, LPS can communicate γi j s to the egresses, and they can multiply max_thr esh i j and min_thr esh i j with γi j . 2. Edge queues: In distributed-DCC, ingress stations are informed by LPS about allocated capacity ci j for each edge-to-edge flow. So, one intuitive way of making sure that the user will not contract for more than ci j is to subtract necessary capacity to drain the already built edge queue from ci j , and then make contracts accordingly. In other words, the ingress station updates the allocated capacity ci j for flow i to j by the following formula ci j = ci j − Q i j /T and uses ci j for price calculation. Note that Q i j is the edge queue length for flow i to j, and T is the length of the contract.

7.1 Simulation Experiments and Results We now present ns-2 [29] simulation experiments for the two architectures, PFCC and POCC, on single-bottleneck and multi-bottleneck topology. Our goals are to illustrate fairness and stability properties of the two architectures with possible comparisons of two. For PFCC and POCC, we simulate distributed-DCC’s PFCC and POCC versions which were described in Section 7. We will simulate EEP pricing scheme at ingress stations. The key performance metrics we will extract from our simulations are as follows: • Steady-state properties of PFCC and POCC architectures: queues, rate allocation • PFCC’s fairness properties: Provision of various fairness in rate allocation by changing the fairness coefficient α • Performance of distributed-DCC’s capacity allocation algorithm ETICA in terms of adaptiveness The single-bottleneck topology has a bottleneck link, which is connected to n edge nodes at each side where n is the number of users. The multi-bottleneck topology has n − 1 bottleneck links that are connected to each other serially. There are again n ingress and n egress edge nodes. Each ingress edge node is mutually connected to the beginning of a bottleneck link, and each egress node is mutually connected to the end of a bottleneck link. All bottleneck links have a capacity of 10 Mb/s and all other links have 15 Mb/s. Propagation delay on each link is 5 ms, and users send UDP traffic with an average packet size of 1000 B. To ease understanding the

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experiments, each user sends its traffic to a separate egress. For the multi-bottleneck topology, one user sends through all the bottlenecks (i.e., long flow) while the others cross that user’s long flow. The queues at the interior nodes (i.e., nodes that stand at the tips of bottleneck links) mark the packets when their local queue size exceeds 30 packets. In the multi-bottleneck topology they increment a header field instead of just marking. Figure 8a shows a single-bottleneck topology with n = 3. Figure 8b shows multi-bottleneck topology with n = 4. The white nodes are edge nodes and the gray nodes are interior nodes. These figures also show the traffic flow of users on the topology. The user flow tries to maximize its total utility by contracting for b/ p amount of capacity, where b is its budget and p is price. The flows’s budgets are randomized according to truncated normal [30] distribution with a given mean value. This mean value is what we will refer to as flows’s budget in our simulation experiments.

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Contracting takes place at every 4 s, observation interval is 0.8 s, and LPS interval is 0.16 s. Ingresses send budget estimations to corresponding egresses at every observation interval. LPS sends information to ingresses at every LPS interval. The parameter k is set to 25, which means a flow is determined to be non-congested at least after 25 LPS intervals equivalent to one contracting interval (see Section 5.3.1). The parameter δ is set to 1 packet (i.e., 1000 B), the initial value of cî j for each flow f i j is set to 0.1 Mb/s, β is set to 0.95, and %r is set to 0.0005. Also note that, in the experiments, packet drops are not allowed in any network node. This is because we would like to see performance of the schemes in terms of assured service. 7.1.1 Experiments on Single-Bottleneck Topology We run simulation experiments for PFCC and POCC on the single-bottleneck topology, which is represented in Fig. 8a. In this experiment, there are three users with budgets of 30, 20, 10, respectively, for users 1, 2, 3. Total simulation time is 15,000 s, and at the beginning only user 1 is active in the system. After 5000 s, user 2 gets active. Again after 5000 s at simulation time 10,000, user 3 gets active. For POCC, there is an additional component in the simulation: edge queues. The edge queues mark the packets when queue size exceeds 200 packets. So, in order to manage the edge queues in this simulation experiment, we simultaneously employ both of the two techniques described before.


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In terms of results, the volume given to each flow is very important. Figures 9a and 10a show the volumes given to each flow in PFCC and POCC, respectively. We see the flows are sharing the bottleneck capacity in proportion to their budgets. In comparison to POCC, PFCC allocates volume more smoothly but with the same proportionality to the flows. The noisy volume allocation in POCC is caused by coordination issues (i.e. parameter mapping, edge queues) investigated in Section 7.

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Figures 9b and 10b show the price being advertised to flows in PFCC and POCC, respectively. As the new users join in, the pricing scheme increases the price in order to balance supply and demand. Figures 9c and 10c show the bottleneck queue size in PFCC and POCC, respectively. Notice that queue sizes make peaks transiently at the times when new users gets active. Otherwise, the queue size is controlled reasonably and the system is stable. In comparison to PFCC, POCC manages the bottleneck queue much better because of the tight control enforced by the underlying edge-to-edge congestion control algorithm Riviera.

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7.1.2 Experiments on Multi-bottleneck Topology On a multi-bottleneck network, we would like to illustrate two properties for PFCC: • Property 1: Provision of various fairness in rate allocation by changing the fairness coefficient α of distributed-DCC framework (see Section 5.2.1) • Property 2: Performance of distributed-DCC’s capacity allocation algorithm ETICA in terms of adaptiveness (see Section 5.3.1) Since Riviera does not currently provide a set of parameters for weighted allocation on multi-bottleneck topology, we will not run any experiment for POCC on multi-bottleneck topology. In order to illustrate property 1, we run a series of experiments for PFCC with different α values. Remember that α is the fairness coefficient of distributed-DCC. Higher α values imply more penalty to the flows that cause more congestion costs. We use a larger version of the topology represented in Fig. 8b. In the multibottleneck topology there are 10 users and 9 bottleneck links. Total simulation time is 10,000 s. At the beginning, the user with the long flow is active. All the other users have traffic flows crossing the long flow. After each 1000 s, one of these other users gets active. So, as the time passes the number of bottlenecks in the system increases


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This variation in fairness is basically achieved by advertisement of different prices to the user flows according to the costs incurred by them. Figure 11b shows the average price that is advertised to the long flow as the number of bottlenecks in the system increases. We can see that the price advertised to the long flow increases as the number of bottlenecks increases. Finally, to illustrate property 2, we ran an experiment on the topology in Fig. 8b with small changes. We increased the capacity of the bottleneck at node D from

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10 to 15 Mb/s. There are four flows and three bottlenecks in the network as represented in Fig. 8b. Initially, all the flows have an equal budget of 10. Total simulation time is 30,000 s. Between times 10,000 and 20,000, budget of flow 1 is temporarily increased to 20. The fairness coefficient α is set to 0. All the other parameters (e.g., marking thresholds, initial values) are exactly the same as in the single-bottleneck experiments of the previous section. Figure 11c shows the volumes given to each flow. Until time 10,000 s, flows 0, 1, and 2 share the bottleneck capacities equally presenting a max–min fair allocation because α was set to 0. However, flow 3 is getting more than the others because of the extra capacity at bottleneck node D. This flexibility is achieved by the freedom given individual flows by the capacity allocation algorithm (see Section 5.3.1). Between times 10,000 and 20,000, flow 2 gets a step increase in its allocated volume because of the step increase in its budget. In result of this, flow 0 gets a step decrease in its volume. Also, flows 2 and 3 adapt themselves to the new situation by attempting to utilize the extra capacity leftover from the reduction in flow 0’s volume. So, flows 2 and 3 get a step decrease in their volumes. After time 20,000, flows restore to their original volume allocations, illustrating the adaptiveness of the scheme.

8 Pricing Loss Guarantee in End-to-End Service As stated in Section 3, in the contract-switched paradigm, it is possible to offer QoS guarantees beyond the best-effort service. In this section, we develop pricing for loss guarantees offered along with the base bandwidth service, where the price of the additional loss guarantee is a component of the overall price. Provision of a loss-based QoS-guaranteed service is inherently risky due to uncertainties caused by competing traffic in the Internet. Future outcomes of a service may be in favor of or against the provider, i.e., the provider may or may not deliver the loss-based QoS as promised. Uncertainty in quality of service is not unique to Internet services [28]. For example, an express delivery company may not always deliver customers’ parcels intact and/or on time; and when losses or delays occur, certain remedy mechanisms, such as money back or insurance, are employed to compensate the customers. On the other hand, the provider needs to take into account such uncertainty when pricing its services. In other words, prices should be set such that the provider will be able to recuperate the possible expenses it will incur for attempting to deliver the QoS, as well as the pay-offs to customers when the promised quality of service is not delivered. When further improving QoS deterministically gets too costly, the provider may be better off using economic tools to manage risks in QoS delivery, rather than trying to eliminate them. We use option pricing techniques to evaluate the risky nature of the loss-guaranteed service. In particular, we consider pricing from the provider’s perspective and evaluate the monetary “reward” for the favorable risks to the provider, which then becomes an added component to the base price of the


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contract. Pricing the risk appropriately lets the risk be fairly borne by the provider and the customer. If all ISPs employ an option-based pricing scheme for their intra-domain lossassured services (for details, see [10]), in this section, we consider the price of endto-end loss assurance. Here risk underlying a loss guarantee refers to whether the service the ith ISP delivers to a customer satisfies the loss guarantee specified in the contract or not. The price is set as the monetary “reward” the ith ISP gets when it is able to meet the loss guarantee, SiU . The price depends on the loss behavior of a customer’s traffic through the ISP’s domain, which is dictated by the characteristics of the network and traffic load. Option pricing techniques are used to value the service by utilizing the ISP’s preferences for different loss outcomes, captured in a state-price density (SPD) [10]. Let l be the end-to-end data loss rate of a customer’s data, li, contract be the data loss rate of the customer’s data in the ith ISP’s network, and l Nj, contract be the data loss rate of the customer’s data in the jth transit node, respectively. If the loss rates li, contract and l Nj, contract are less than 20%, the second-order terms in their product will be significant to one lower digit of accuracy, hence ignorable. If an ISP is causing a steady loss rate of higher than 20%, it practically does not make much sense to utilize their services for end-to-end loss guarantees. As loss rates are multiplicative, and all li, contract ’s and l Nj, contract ’s can safely be assume to be small, we have l≈

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N ⊆ V N are the set of ISP contracts and E contract ⊆ E are the set where Vcontract of transit nodes used in delivering a specific end-to-end contract, and V N , E are all the ISPs and transit nodes in a provider’s overlay. The intra-domain contract with each ISP is a guarantee that the loss rate within its domain does not exceed N Siu , i ∈ Vcontract . If the provider chooses a threshold value S u,N as the “assurance” for loss at transit nodes, the end-to-end loss guarantee can be approximately defined as Siu + S u,N . (18) Su ≈ N i∈Vcontract

To obtain the price of a contract, a payoff function, Yi , is defined that measures an ISP’s performance according to the contract definition. For example, the payoff function of a sample contract, ( A), defined in terms of loss rates starting at t = 3 pm for a duration of an hour may be given by Yi (li , Siu ) = 1{li ≤Siu } |li − Siu |,

(19)

where li is the loss outcome of the service. Therefore, in general a payoff function, Yi , can be created that (1) captures whether the loss outcome is within the contracted

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guarantee Siu and (2) how much better than Siu the provider is able to perform. The price of a contract Vi is then determined by the expectation of the payoff under a special probability measure Q, i.e., Vi = E Q (Yi ).

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The probability measure Q captures the providers preference for loss outcomes and is termed the state price density (SPD). Note that Vi and Yi are time dependent just as li . Time is not explicitly indicated due to our assumption of identical time descriptions of all contracts. For the choice of payoff function given in (19), it can be shown that the price for a given li process, Vi (SiU ) = Vi (SiU ; li ), has the following properties: (1) Vi (SiU ) is a convex function and (2) Vi (0) = 0; and Vi (SiU ) is non-decreasing with SiU up to a certain SiU , after which Vi (SiU ) = 0. These properties are reasonable since the risk being priced is defined by the outcomes of an ISP’s performance against the contract without any account for efforts in providing the service. A lower SiU does not imply a greater effort on the provider’s part, neither does a higher SiU imply less effort. A higher SiU , up to SiU , however, does imply that the provider gets a greater benefit of risk bearing. With an increasing threshold, greater loss outcomes will be in favor of the provider; hence, the provider’s performance with respect to the contract specifications becomes better, therefore, gets a higher reward. It should be noted that this is only part of the picture. When the provider violates the contract, he also incurs a penalty. Clearly with an increasing SiU the benefits to the provider increase, but the penalty on violation must also simultaneously increase. The penalty determination, however, is beyond the scope of this chapter. Without the need to define a specific form of an ISP’s utility function for losses, we stipulate Q that captures the ISP’s preference structure for loss outcomes, by some general properties of the ISP’s preferences for loss outcomes in its domain. (1) The ISP would expect that losses in its domain are rare events during a contract term. (2) If losses do happen, they will more likely take small to moderate values. (3) The ISP will not be rewarded when large losses occur. In particular, we consider ISPs with two types of preference structures and the corresponding intra-domain prices: 1. An ISP has a strict preference for smaller losses over large losses, i.e., the loss free state is the most favorable outcome to the ISP. This results in performancebased prices. Price is higher when network is at low utilization and the provider is capable of performing in better accordance with the contract. 2. We also consider an alternative preference structure where the loss outcome most desirable to the ISP is at a small but positive level. This will be the case if customers of the ISP services can tolerate certain small losses, as the ISP is possibly able to accommodate more customers by allowing small losses to an individual customer’s data. This results in congestion-sensitive prices. Price is higher when


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network is at high utilization, which discourages customers from buying loss assurances when the network is congested. Using the assumptions described above, we develop pricing strategies for end-toend service with loss assurances. An end-to-end loss-assured service is characterized by its source and destination (s–d pair) location and the loss guarantee, along with other specifications of the service. Consider a contract for service between a certain s–d pair with an end-to-end loss guarantee S u . In the following an end-to-end contract (service) refers to a contract thus defined, and end-to-end pricing refers to the pricing of the end-to-end loss assurance, unless otherwise stated. The provider can acquire and concatenate intra-domain contracts in multiple ways to provide a service. Different customers’ traffic between an s–d pair may be routed differently within the overlay constructed from acquired contracts; on a given path, the provider may purchase from the same ISP intra-domain contracts with different loss assurances to achieve the end-to-end assurance S u , as long as (18) holds. The routing information and ISP contract types used, however, are invisible to customers. For a price, a customer simply expects to receive the same service from a certain contract. Therefore, at a certain time, prices are determined entirely by the specifications of the contract – the s–d pair and the loss guarantee S u . In particular, if the end-to-end services can be created in an oligopolistic competition, i.e., there are alternatives for how end-to-end services can be created, and no single provider exerts its monopoly power in creating such services, as well as there is efficiency in how such services can be created dynamically, the following is true about the price of an end-to-end contract. Proposition 1 Under the assumption that the market for end-to-end services is competitive and efficient, the price of an end-to-end contract is the lowest price over all possible concatenations of intra-domain services to deliver the end-to-end contract, denoted by V ∗ (S u ). Assume that there are R routes to construct a loss-guaranteed service between an s–d pair, where path r (r = 1, . . . , R) involves h r ISPs. On path r , the provider u at purchases from each constituent ISP, i, a service contract with loss guarantee Si,r u a price Vi (Si,r ). The provider assigns a price for the risk for loss at all h r − 1 transit nodes on path r , using a price function VN . In addition, assume that the provider will u assign a threshold value, Sru, N as an “assurance” to lrN , with {Si,r , Sru, N } satisfying u, N the condition of (18). VN depends on the loss assurance Sr and other characteristics, Θ, of the provider. To solve the end-to-end pricing problem, we first define price of a path. Definition 1 (Price of a path) The price of path r , V r (S u ), r = 1, . . . , R, is defined as the price of the end-to-end loss assurance if path r were the only path to deliver the end-to-end contract. The following proposition is obtained by the direct application of Proposition 1. Proposition 2 The price of a path r , V r (S u ), for an end-to-end contract with loss assurance S u is determined by the least costly concatenation of intra-domain

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contracts on path r , to eliminate arbitrage between different concatenations of intradomain contracts, i.e., u Vi (Si,r ) + VN (Sru,N ). (21) V r (S u ) = min u , S u,N } {Si,r r

i ISP i on path r

In our earlier work [12], we developed a categorization-based pricing scheme for end-to-end bandwidth, where prices are determined by the hop count h and number of bottlenecks (ISPs and/or transit nodes) b on the most likely path between an s–d pair. We will utilize this bh classification for pricing end-to-end loss assurance, so that the two price components, for bandwidth and loss-guarantee, are combined consistently to form the price of the contract. In addition, only simple cycle-free paths are considered, since presence of cycles artificially inflates traffic in a network, and thus increases the provider’s cost in providing the service. By applying Propositions 1 and 2, the provider’s pricing problem to determine V ∗ (S u ) is defined as follows: Problem 1 (End-to-end pricing problem) min

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This most general formulation of the pricing problem, where each ISP has a unique price function, can be complex to solve. This is especially true since the provider needs to solve a problem of this form for each s–d pair in the overlay. Estimating the price functions Vi (Siu ) of all ISPs also adds to the complexity of the pricing model. Furthermore, prices need to be recomputed when the provider changes the configuration of its overlay. Therefore, simplifying assumptions will be necessary for solving the end-to-end pricing problem. In Section 8.1, we will conduct a numerical simulation analysis of the above pricing problem.

8.1 Numerical Evaluation We consider the price of an end-to-end loss assurance S u between an s–d pair in an overlay with N = 10 ISPs. We will study the solutions to the end-to-end pricing problems with loss-free and loss-prone transit nodes, respectively. For simplicity, we assume that the provider and all ISPs use congestion-based pricing. Instead of a fixed overlay configuration, we consider all possible combinations of the numbers


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of ISPs (h r ) and of bottleneck ISPs (br ) on a path r when h r ≤ 10. The price functions of underlying ISPs can be described by quadratic functions [10]: V0 (g) = cg 2 , V1 (g) = ct cg 2 , ct > 1. We use representative intra-domain price functions with c = 8667 and ct = 1.074 (t = 3pm). ct indicates the difference between the price functions of congested and non-congested ISPs. The price function for transit nodes is also a quadratic function VN (g) = c N g 2 ; c N is varied to examine the effect of the price of transit nodes on the end-to-end price. 8.1.1 Solution to Loss-Free Nodes Figure 12 shows V r with different combinations of h r and br on the path, assuming all transit nodes are loss free. We see that V r decreases with h r . At the same time, V r increases with br , the number of bottleneck ISPs on the path, for a given choice of h r ; this is also shown in Fig. 13 which gives the relationship between the price of a path and the number of bottleneck ISPs on the path with h r = 10. It is further observed that between h r and br , h r seems to have a more significant effect on V r ; a longer path involving more ISPs is less expensive than a shorter path, regardless of how many bottleneck ISPs are encountered on the path. Therefore, in this case the end-to-end price V ∗ will be determined by the longest path that involves the least bottleneck ISPs. This resembles the high diversification principle of the financial portfolio theory. To obtain the end-to-end price V ∗ , the provider will need to (i) search for the longest path between the s and d pair in a bh class and (ii) among such paths, search for the one involving fewest bottleneck ISPs.

8.1.2 Solution to Leaky Nodes We now study the effect of introducing risks at transit nodes to the end-to-end price V ∗ . We assume the provider uses a congestion-based pricing for risk of loss,

Fig. 12 V r with ISP classification (beta SPD)

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Fig. 13 V r with # of bottlenecks (beta SPD): loss-free transit nodes, h r = 10

producing a quadratic price function VN (g; h) = c N ,h g 2 , where g, h are the loss assurance and number of ISPs on the path, respectively. Different price functions are studied by varying the coefficient c N ,h ; in particular, we study the cases when VN (g; h) > V1 (g), V1 (g) > VN (g; h) > V0 (g), and VN (g; h) < V0 (g), ∀h, respectively. c N ,h also depends on h; we set c N ,h to decrease linearly by 5% for every additional transit node involved. Figure 14 shows V r with different h r , br combinations, where λ = 0.3 is the constraint on the proportion of the loss assurance that can be assigned to transit nodes Sru,N , and VN (g; h) < V1 (g) (Fig. 14 (a)), VN (g; h) > V1 (g) (Fig. 14b), respectively. The scenario with V1 (g) > VN (g; h) > V0 (g) looks similar to the loss-free transit node case (Fig. 12) and is not shown here.

Fig. 14 V r with leaky transit nodes (beta SPD, λ = 0.3): (a) V1 (g) > V0 (g) > VN (g; h) and (b) VN (g; h) > V1 (g) > V0 (g)

Comparing Fig. 14 with Fig. 12, we can see that introducing risks at transit nodes decreases the overall price levels for V r , regardless of the relative relations between the price functions, although as expected, between these two scenarios the decrease in V r is more significant with a lower price function VN in Fig. 14a. Similar to the case of loss-free transit nodes, the primary effect on V r is of h r ; that is, the price of a longer path involving more ISPs and transit nodes is always lower. However, the relationship between V r and the number of bottlenecks on the path br becomes


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Fig. 15 V r with # of bottlenecks (beta SPD): leaky transit nodes, h r = 10

irregular, as is also seen in Fig. 15. Therefore, to obtain the exact end-to-end price, the provider will need to search for all the longest paths in a bh class and choose V ∗ as the price of the least expensive path. In all scenarios considered above, the longest paths involving more ISP contracts will be preferred in constructing the end-to-end loss assurance. This resembles the diversification technique to reduce risk in risk management [4]. The provider benefits from allocating risk in the end-to-end loss assurance to more ISP domains.

9 Summary In this chapter, we describe a dynamic congestion-sensitive pricing framework that is easy to implement and yet provides great flexibility in rate allocation. The distributed-DCC framework presented here can be used to provide short-term contracts between the user and the service provider in a single diff-serv domain, which allows the flexibility of advertising dynamic prices. We observe that distributedDCC can attain a wide range of fairness metrics through the effective use of an edge-to-edge pricing (EEP) scheme that we provide. Also, we introduce two broad pricing architectures based on the nature of the relationship between the pricing and congestion control mechanisms: pricing for congestion control (PFCC) and pricing over congestion control (POCC). We show how the distributed-DCC framework can be adapted to these two architectures and compare the resulting approaches through simulation. Our results demonstrate that POCC is better in terms of managing congestion in network core, while PFCC achieves wider range of fairness types in rate allocation. Since distributed-DCC is an edge-based scheme, it does not require upgrade of all routers of the network, and thus, existing tunneling techniques can be used to implement edge-to-edge closed-loop flows. An incremental deployment of distributed-DCC is possible by initially installing two distributed-DCC edge routers, followed by replacement of others in time.

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We also describe a framework for pricing loss guarantees in end-to-end bandwidth service, constructed as an overlay of edge-to-edge services from many distributed-DCC domains. An option-based pricing approach is developed for endto-end loss guarantees over and above the price for basic bandwidth service. This provides a risk-sharing mechanism in the end-to-end service delivery. The price of the end-to-end contract is determined by the lowest price over all valid intra-domain contract concatenations. Based on certain simplifying homogeneity assumptions about the available intra-domain contracts, numerical studies show the importance of diversification in the path chosen for end-to-end service. Acknowledgments This work is supported in part by National Science Foundation awards 0721600, 0721609, and 0627039. The authors wish to thank Shivkumar Kalyanaraman for his mentoring and Lingyi Zhang for excellent research assistance.

References 1. Arora GS, Yuksel M, Kalyanaraman S, Ravichandran T, Gupta A (2002) Price discovery at network edges. In: Proceedings of international symposium on performance evaluation of telecommunication systems (SPECTS), San Diego, CA, pp 395–402 2. Blake S et al. An architecture for differentiated services. IETF RFC 2475, December 2008 3. Bouch A, Sasse MA (2001) Why value is everything?: A user-centered approach to Internet quality of service and pricing. In: Proceedings of IEEE/IFIP IWQoS, Karlsruhe, Germany. 4. Chiu DM (1999) Some observations on fairness of bandwidth sharing. Tech. Rep. TR-99–80, Sun Microsystems Labs 5. Clark D (1997) Internet cost allocation and pricing. In: McKnight LW, Bailey JP (eds) MIT Press, Cambridge, MA 6. Clark D (1995) A model for cost allocation and pricing in the Internet. Tech. Rep., MITPress, Cambridge, MA 7. Clark DD, Wroclawski J, Sollins KR, Braden R (2005) Tussle in cyberspace: defining tomorrow’s Internet. IEEE/ACM Trans Netw 13(3): 462–475 8. Cocchi R, Shenker S, Estrin D, Zhang L (1993) Pricing in computer networks: motivation, formulation and example. IEEE/ACM Trans Netw 1:614–627 9. Crouhy M, Galai D, Mark R (2001) Risk management. McGraw-Hill, New York, NY 10. Gupta A, Kalyanaraman S, Zhang L (2006) Pricing of risk for loss guaranteed intra-domain Internet service contracts. Comput Netw 50:2787–2804 11. Gupta A, Stahl DO, Whinston AB (1997) Priority pricing of integrated services networks. In: McKnight LW, Bailey JP (eds) MIT Press, Cambridge, MA 12. Gupta A, Zhang L (2008) Pricing for end-to-end assured bandwidth services. Int J Inform Technol Decision Making 7(2):361–389 13. Harrison D, Kalyanaraman S, Ramakrishnan S (2001) Overlay bandwidth services: basic framework and edge-to-edge closed-loop building block. In: Poster in SIGCOMM, San Diego, CA 14. Kelly FP (1997) Charging and rate control for elastic traffic. Eur Trans Telecommun 8:33–37 15. Kelly FP, Maulloo AK, Tan DKH (1998) Rate control in communication networks: shadow prices, proportional fairness and stability. J Oper Res Soc 49: 237–252 16. Kunniyur S, Srikant R (2000) End-to-end congestion control: utility functions, random losses and ecn marks. In: Proceedings of conference on computer communications (INFOCOM) Tel Aviv, Israel


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17. Low SH, Lapsley DE (1999) Optimization flow control – I: basic algorithm and convergence. IEEE/ACM Trans Netw 7(6):861–875 18. MacKie-Mason JK, Varian HR (1995) Pricing the Internet. In: Public Access to the Internet, Kahin B, Keller J (eds), Cambridge, MA: MIT Press, 269–314 19. Mo J, Walrand J (2000) Fair end-to-end window-based congestion control. IEEE/ACM Trans Netw 8(5):556–567 20. Odlyzko AM (1997) A modest proposal for preventing Internet congestion. Tech. Rep., AT & T Labs 21. Odlyzko AM (1998), The economics of the Internet: utility, utilization, pricing, and quality of service. Tech. Rep., AT & T Labs 22. Odlyzko AM (2000) Internet pricing and history of communications. Tech. Rep., AT & T Labs 23. Paschalidis IC, Tsitsiklis JN (2000) Congestion-dependent pricing of network services. IEEE/ACM Trans Netw 8(2):171–184 24. Semret N, Liao RR-F, Campbell AT, Lazar AA (1999) Market pricing of differentiated Internet services. In: Proceedings of IEEE/IFIP international workshop on quality of service (IWQoS), London, England, pp 184–193 25. Semret N, Liao RR-F, Campbell AT, Lazar AA (2000) Pricing, provisioning and peering: dynamic markets for differentiated internet services and implications for network interconnections. IEEE J Select Areas Commun 18(12): 2499–2513 26. Shenker S (1995) Fundamental design issues for the future Internet. IEEE J Select Areas Commun 13:1176–1188 27. Singh R, Yuksel M, Kalyanaraman S, Ravichandran T (2000) A comparative evaluation of Internet pricing models: Smart market and dynamic capacity contracting. In: Proceedings of workshop on information technologies and systems (WITS), Queensland, Australia 28. Teitelbaum B, Shalunov S (2003) What QoS research hasn’t understood about risk. In: Proceedings of the ACM SIGCOMM 2003 workshops 148–150, Karlsruhe, Germany 29. UCB/LBLN/VINT network simulator – ns (version 2) (1997) http://wwwmash.cs.berkeley.edu/ns 30. Varian HR (1999) Estimating the demand for bandwidth. In: MIT/Tufts Internet Service Quality Economics Workshop, Cambridge, MA 31. Varian HR (1999) Intermediate microeconomics: a modern approach. W. W. Norton and Company, New York, NY 32. Wang X, Schulzrinne H (2000) An integrated resource negotiation, pricing, and QoS adaptation framework for multimedia applications. IEEE J Select Areas Commun 18(12):2514–2529 33. Wang X, Schulzrinne H (2001) Pricing network resources for adaptive applications in a differentiated services network. In: Proceedings of INFOCOM, Shanghai, China, pp 943–952 34. Yuksel M (2002) Architectures for congestion-sensitive pricing of network services. PhD thesis, Rensselaer Polytechnic Institute, Troy, NY 35. Yuksel M, Kalyanaraman S (2002) A strategy for implementing the Smart Market pricing scheme on Diff-Serv. In: Proceedings of IEEE GLOBECOM, pages 1430–1434, Taipei, Taiwan. 36. Yuksel M, Kalyanaraman S (2003) Elasticity considerations for optimal pricing of networks. In: Proceedings of IEEE symposium on computer communications (ISCC), Antalya, Turkey, pp 163–168 37. Yuksel M, Kalyanaraman S (2005) Effect of pricing intervals on congestion-sensitivity of network service prices. Telecommun Syst 28(1):79–99

Modelling a Grid Market Economy Fernando Martínez Ortuño, Uli Harder, and Peter Harrison

1 Introduction It has for some time been widely believed that computing services will, in due course, be provided similar to telephone, electricity and other utilities today. As a result, a market will develop with different Grid companies competing and cooperating to serve customers; there are already signs that this is beginning to happen. Companies will have to set prices to attract customers and make profits. Governments that may choose to provide GRID services will similarly have to set prices, perhaps with a different aim of maximizing utilization. This area of Grid research has been highlighted as a key area of research in the EU report on next-generation Grids. The development of the Grid has come to a stage where companies are beginning to sell Grid services. For example, both IBM and SUN are selling Grid access by the hour; already a market exists where companies compete for GRID customers. It will probably not take long until we see brokers (also called middlemen or suppliers to end-users in different environments) re-selling Grid services they bought in bulk from providers. This would constitute the usual 3-tier setup of markets seen, for instance, in the electricity market, where we have generators, suppliers and users. For the companies involved, who may wish to provide GRID access as an efficient alternative to buying computers, it is important to have a market model to be able to Fernando Martínez Ortuño Department of Computing, Imperial College London, Huxley Building, 180 Queens Gate, London SW7 2AZ, UK e-mail: [email protected] Uli Harder Department of Computing, Imperial College London, Huxley Building, 180 Queens Gate, London SW7 2AZ, UK e-mail: [email protected] Peter Harrison Department of Computing, Imperial College London, Huxley Building, 180 Queens Gate, London SW7 2AZ, UK e-mail: [email protected] N. Gülpınar et al. (eds.), Performance Models and Risk Management in Communications Systems, Springer Optimization and Its Applications 46, C Springer Science+Business Media, LLC 2011 DOI 10.1007/978-1-4419-0534-5_10,

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test various pricing schemes. This will become even more evident once futures and options for Grid access are sold. Future customers of this Grid market are expected to be banks and insurance, engineering and games companies, as well as universities and other research organisations. Although a great deal of effort has been aimed at modelling financial markets, much less is known about modelling commodity markets. Our aim has been to produce a high-level market model for Grid computing and a future Grid-based market economy, rather than the traditional use of economic methods for the micromanagement of future Grids. The approach bears some similarities and analogies to current structures in place for the airline, electricity and telephone industries, but is specifically focused on the needs and characteristics unique to the Grid. We mainly discuss and elaborate on results that the authors have already published in workshops [16, 17, 21], adding new material to make the paper coherent and up to date. First, we review the use of peer-to-peer (P2P) technology for commercial Grid computing. Using economic incentives to facilitate fair use of computing power in a group is not new. Greenberger [15] discusses this as far back as 1966. He mentions various ways and methods to use economic ideas to make queueing systems fairer. In 1968 Sutherland describes how access to a PDP-11 was organised at Harvard using an auction-based system [31]. In a very simple auction, users bid for exclusive access to the PDP-1. Their budgets were replenished every day and left-overs could not be saved. The more senior the position held in the department the larger the budget of an individual. This system allowed fair access by not excluding anyone and also maximised the utilisation of the system by refilling the budgets at the end of each day. Later, for instance, Nielsen [23] and Cotton [9] discuss shared access to resources in a multi-user environment. In general there are two main reasons to charge: for computer access: • Fair access: Using real or fake money a group organises access to facilities by means of charging for their use. • Profit: A company essentially hires out facilities to third parties for real money. The first case is basically what Sutherland describes in a very simple setting where only one user has access at a time. The money does not need to be real in this case but can be tokens. Priorities can be implemented by giving different groups of people a different budget. In this case micromanagement of resources might also make sense. The second case is actually not too different from mobile phone use, for example. A company is unlikely to charge users for detailed use of facilities but rather for accessing services over a certain period of time. The concept of the Grid [14] renewed interest in using economic incentives to manage computer systems. The idea of the Grid is to provide access to computing power in the same way as electricity is offered to end-users. Authors have suggested

1 An

18-bit computer made by Digital Equipment Corporation in the early 1960s,

http://en.wikipedia.org/wiki/PDP-1

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the use of economics in micro- and macro-management and the methods tend to be either auctions or commodity markets, see [36], Regev and Nisan [26], Waldspurger et al. [32] and Buyya [6]. One of the major problems with any kind of scheduling or access-providing infrastructure is to avoid centralised services since they are likely to become a bottleneck for a large number of users or transactions. An example of a centralised system is Tycoon [19], which has a central service locator service and experienced this problem. There is therefore now a trend to try to build future Grid architectures using P2P networks, which are de-centralised. Examples are the P-Grid, which is mainly a data Grid [1], and more generally Catallaxy [2]. Another approach is middleware for activating the global open Grid (MAGOG) [8, 27] which will be our main focus in this chapter. We present three different ways to model some aspects of the simplified MAGOG system. First, we describe an analytical model, which is essentially a mean field approximation of the system. Next we show how an agent-based simulation can be used to investigate the system. Agent-based simulations have in the past been used to model privatised electricity markets [3] and auctions like the Marseille fish market [18]. In fact our market of computing power is very similar to that of perishable commodities like fish as it is difficult to store. In our simulation model, the agents have only limited knowledge of the entire system and, rather than globally optimising agents, they are satisficing agents with bounded rationality [5, 30]. We find that the mean field approximation and simulation show good agreement. Lastly, we use Markov decision processes (MDP) introduced by Belman to investigate decision strategies and market behaviour for a good introduction to the topic see [25]. We next present a short review of P2P networks, together with some related, basic graph theory, and then describe the salient features of MAGOG before introducing our simplifications for their modelling.

1.1 P2P Networks and Graphs In recent years P2P networks have become very popular for disseminating content amongst users. In particular, there are networks that do not need a central server, like Freenet. One of the most popular applications of P2P networking is Skype which allows video calls via the Internet. Some of Skype’s services have to be paid for which needs minimal central bookkeeping. A very good review of P2P networking is [20]. Essentially P2P networks are about resource sharing and in the next section we show how this can be extended to resources other than storage to allow Grid computing via P2P technology. Another interesting and important aspect of P2P networks is the way their nodes are connected. Graph theory is a suitable means to describe this precisely. A graph consists of nodes (vertices) which are connected by links (edges). Nodes can be characterised by their degree which is the number of links they have with other nodes. In the case of a directed graph, where links have a sense of direction, one distinguishes between the in and out degree of a node. Graphs can either be connected

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or unconnected, depending on whether there is a continuous path from each node to all other nodes. To compare different graphs, one can, for instance, measure their node degree distribution. Other measures include the diameter, which is the longest of all shortest paths between all pairs of nodes in the network. This gives an indication of the size of the network. There is also the clustering coefficient, which is roughly the ratio of the numbers of existing links between nodes and the theoretical maximum number of links. The two types of graphs we have used in this study are the Erd˝os–Rényi (ER) graphs [12] and Barabási–Albert (BA) [4] graphs. ER graphs are “random” and so are characterised by the fact that their node degree is a binomial random variable n, which tends to Poisson in large graphs: P(n = k) ≈ K k

e−K , k!

where K is the average node degree. In contrast, the node degrees in a BA graph have a power law distribution: P(n = k) ≈ (k + c)−α , where c is a constant. The power law is often referred to as scale free. BA graphs are small world graphs as their nodes are highly clustered. Small world networks were first classified by Watts and Strogartz [35] and a fairly concise review of complex networks can be found in [13]. The P2P network structure of the overlay network has been studied for Freenet, which is scale free [24], and for Gnutella, which is small world and scale free [34]. The evidence that scale-free networks are in common use and that BA graphs reflect this property well is the reason we have chosen BA networks in our simulations, in addition to ER graphs.

1.2 MAGOG The MAGOG system has not yet been implemented and some of the details of its architecture will be omitted as they are not important for our investigation. In fact one can look at MAGOG as an example application of our analysis which is abstract enough to cover other different architectures with the same general structure. The architecture of MAGOG was first described in [27]. The global open Grid (GoG) is meant to be a P2P network of nodes that provide (seller) or require computing services (buyer). Nodes can be any computing device that can install the MAGOG software. The inventors of MAGOG believe this will be possible on a large range of devices, from smartphones to supercomputers. It is envisaged that nodes send out messages (called Ask or Bid bees) over the network to advertise that they either need or can supply computing services at a certain price; this is referred to as double message flooding. Using a micropayment system, nodes will


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be able to exchange real money for services. Each node has a list of nearest neighbours that will be seeded in an appropriate way when they (re-)join the system; therefore, nodes only have local knowledge of the system. Apart from the payment process there is no centralised point in the system. Therefore, the system should scale extremely well. Each node passes on messages from its neighbours to its neighbours until a counter attached to the message, the time to live (TTL), has been exceeded at which point the message is discarded. This way messages can only travel a certain number of hops. Nodes also keep a copy of messages in a buffer (the pub) and try to match up suitable needs or services advertised in the messages. If a match can be made, the node stops forwarding a matched message. Similarly, a message advertising services or needs that have been matched is discarded. The combination of pubs and double message flooding allows matches to be found between nodes up to twice the number of TTL hops apart. The design of MAGOG was inspired by Hajek’s model of market economies, Catallaxy [7], which contrasts with the model of Walras [31] that assumes global knowledge of all agents in the market.

2 Abstract MAGOG To be able to model MAGOG, we need a simplified abstraction of the system and so we specify the connections between and characteristics of (buying or selling) nodes in the system. In our initial model, the number of buyer and seller nodes is constant and nodes cannot change from being a buyer to a seller and vice versa. The nodes use their links bi-directionally for message transfer. The list of nearest neighbours is simply given by the nearest neighbours of the nodes of the chosen network. The pub of each node is finite and the older messages are discarded when the node runs out of pub memory. Buyer nodes have a finite budget which gets replenished after B time epochs. This stops the exponential price increases seen in [16]. There is also a smallest currency unit that can be used to pay for services, which prevents infinitesimally small prices. Nodes can only buy or sell one resource at a time. After a preset time, matched up nodes go back to an unsatisfied state. As in the real MAGOG, messages have a TTL and get discarded once they have hopped between TTL nodes, as do messages which already have been satisfied. We use a call auction algorithm for matching the orders in the pubs, similar to the one used to set the opening and closing prices in the stock market [11], as an approximation to the continuous trading that would take place in a real MAGOG deployment. All new unmatched messages are then forwarded to the nearest neighbours. Once matches have been found, nodes stay in the satisfied state for a while and after that start sending out messages advertising needs or services. In our simplified model the agents behave in a very simple way to change prices:

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• Buyers will try to acquire the same service for less the next time and therefore bid pb = (1 − ) pb , with 0 < < 1. • Similarly, sellers will try to sell the same resource for more the next time and ask ps = (1 + ) ps , with 0 < < 1. Another source of price variation is starvation, when sellers or buyers cannot find a match. In this case, sellers reduce their price and buyers increase their bids according to the same ratios in the equations above after TTL epochs. The bid prices buyers can make cannot exceed their remaining budget and if a buyer runs out of money, he stops bidding until his funds have been replenished. Nodes that currently have a match continue to operate as usual with respect to forwarding messages and finding matches for other nodes.

2.1 An analytical approximation If we assume that the P2P network is fully connected, we can approximate the system as a single central market place, where the orders of all nodes get together and match according to price constraints and supply/demand. Under these conditions, the average price of all deals in the system after n time steps is given by b 1−b n δ− ) , P(n) = P0 (δ+

(1)

where P0 is the initial price and δ± = 1 ± . The price change ranges between 0 and 1. The percentage of buyers b ranges from 0 to 1 inclusively. In order to derive expression (1), it is assumed that time is synchronous and discrete for all nodes in the system. At every time step, all nodes submit their orders for buying and selling with their respective bid and ask prices. All orders of a certain time step meet at the central market place, where deals are made. A deal is made between a buyer and a seller if the bid price of the buyer is greater than or equal to the ask price of the seller. The deal price for these two market participants is always the average between their ask and bid prices. The nodes that have found a deal in a certain time step change their ask/bid price in their own interest on the next time step: the bid price of the buyer for the next time step will be the one he has used in the present time step multiplied by δ− ; the ask price of the seller for the next time step will be the one he has used in the present time step multiplied by δ+ . The average deal price in the system at a certain time step is the average of all deal prices in that time step. Some nodes will not be able to find a deal in a particular time step. This may happen because their prices do not match or because there is a shortage of supply or demand (respectively for buyers and sellers). In this case, the unsatisfied nodes will change their respective bid/ask prices for the next time step against their own interest: the bid price of the buyer for the next time step will be the one he has used in the present time step multiplied by δ+ ; the ask price of the seller for the next time step will be the one he has used in the present time step multiplied by δ− .


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We further assume that initially (first time step: n = 0) all buyers are bidding P0 and all sellers are asking P0 . Following the evolution of time as explained above, one arrives at the expression for the average deal price in the system given by (1). Therefore, the price development in the system depends on the expression in the bracket of (1), which we define as F: b 1−b δ− . F(, b) = δ+

(2)

Depending on the value of F, the deal price in the system will evolve towards one out of three possible values: ⎧ ⎪ ⎨∞ if F > 1 P(n) → P0 if F = 1 . (3) ⎪ ⎩ 0 if F < 1 We can now calculate the critical value of b() for which F = 1: b() = log(1 − )/(log(1 − ) − log(1 + )).

(4)

Equation (4) is plotted in Fig. 1, and represents the combination of values of b and (which make F = 1) for which the deal price in the system will evolve towards P0 . A combination of b and situated above the line plotted in Fig. 1 (F > 1) will make the deal price evolve towards infinity, whereas a pair of b and situated below the line (F < 1) will make the deal price evolve towards 0. As one can appreciate from Fig. 1, the percentage of buyers has to be greater than 0.5 to achieve a stable price in this simple model, independently of the value of . In order to have a realistic market model, both a simulation and a real system should avoid prices that evolve towards extreme values (0 and infinity), achieving a bounded price fluctuation with a certain stability. This may be achieved by a 1 F=1 0.9 F>1

b(Δ)

0.8 0.7

F 1) and for > 0.3894 the price will tend to zero (because F < 1). We compared this theoretical result against a simulation using a BA graph with 16,384 nodes and show the results in Table 1. The simulation achieves a large price, bounded only by the limit on the budgets and the time the simulation has run (and

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therefore equivalent to the theoretical final price of infinity), for ≤ 0.4; and a price of 1.0, the minimum asking price in the simulation (and therefore equivalent to the theoretical final price of 0), for ≥ 0.5. The value for which the final price in the simulations reverts from tending to infinity to tending to zero is between 0.415 and 0.425. This fits fairly well with the prediction from the analytical model, taking into account the simplifications made to derive the latter. We also note that the final price in the simulations follows a similar pattern to the value of F 10 (, 0.6) (the greater the value of F 10 (, 0.6), the faster the final price in the simulations approaches infinity), up until = 0.3, as it is possible to see from Table 1. In summary the results from the analytical model and the simulation are not identical but show very similar results.

2.4 A Further Comparison with the Analytic Model Inspired by Fig. 1 we investigate whether the system evolves to the line of stability if it has the chance to adjust the for each node separately, and the nodes can go into hibernation, both by using local knowledge. We first note that the price evolution of the system2 appears to be as stable as one could expect, as depicted in Fig. 3. This system was started off with a percentage of buyers b = 0.5 and all nodes started with a = 0.7. Buyer nodes went into hibernation for 150 epochs when they estimated b ≥ 0.5, sellers hibernated the same length if they estimated b < 0.5. They estimated this percentage by looking at the messages they had forwarded from their pubs in each epoch. This ratio was continually recalculated every epoch by averaging the information of the new epoch with that of the previous epochs up to the past 10. After 10 epochs had elapsed, all stored information about the ratio was erased and the cycle starts again for the next 10 epochs. We have run simulations of this system with different values of initial b and and all systems that achieved stable deal prices ended up having b = 0.79 and an average of 0.8. As can be seen in Fig. 4 this is in line with the prediction of a stable model by the analytical approximation. We observed that initial combinations of b and close to the F = 1 line in Fig. 4 lead to a stable price. Further away from the line the price ends up being 1 (limited by the minimum ask price) or very large (limited by the finite budget). This instability also arises with initial points of b > 0.8. It appears that these initial conditions are too extreme to be accommodated by any length of hibernation time or . The choice of the hibernation length seems to influence how far the initial condition can be chosen away from the stable line and still result in a stable system. In Fig. 4 we show a number of hibernation times for several initial choices of b and that makes the system achieve a final stable price. The hibernation time increases for initial points that are below the line for a stable deal 2 In this mode the TTL is 7, the nodes’ pubs have a size of 100. Buyers and sellers start bidding/asking with an initial price of 1200. Buyers have a limited budget of 10,000, which is re-filled after they have been picked 10 times. A node that has been picked four times re-enters the market. The diameter of the 512 node BA network is 9.


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Table 1 Comparison of the simulation results with the analytical model for different values

Value of F 10 (, 0.6)

0.01 0.05 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0.99 0.99999

1.0197 1.0915 1.1623 1.2231 1.1589 0.9758 0.7119 0.4295 0.1955 0.0544 0.0047 6.2104×10 −7 6.3998×10−19

>1 >1 >1 >1 >1

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