Distributions in Stochastic Network Models

DISTRIBUTIONS IN STOCHASTIC NETWORK MODELS DISTRIBUTIONS IN STOCHASTIC NETWORK MODELS G. SH. TSITSIASHVILI AND M. A...

Author: G. Sh. Tsitsiashvili | M. A. Osipova

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DISTRIBUTIONS IN STOCHASTIC NETWORK MODELS

DISTRIBUTIONS IN STOCHASTIC NETWORK MODELS

G. SH. TSITSIASHVILI AND M. A. OSIPOVA

Nova Science Publishers, Inc. New York

Copyright © 2008 by Nova Science Publishers, Inc.

All rights reserved. No part of this book may be reproduced, stored in a retrieval system or transmitted in any form or by any means: electronic, electrostatic, magnetic, tape, mechanical photocopying, recording or otherwise without the written permission of the Publisher. For permission to use material from this book please contact us: Telephone 631-231-7269; Fax 631-231-8175 Web Site: http://www.novapublishers.com NOTICE TO THE READER The Publisher has taken reasonable care in the preparation of this book, but makes no expressed or implied warranty of any kind and assumes no responsibility for any errors or omissions. No liability is assumed for incidental or consequential damages in connection with or arising out of information contained in this book. The Publisher shall not be liable for any special, consequential, or exemplary damages resulting, in whole or in part, from the readers’ use of, or reliance upon, this material. Independent verification should be sought for any data, advice or recommendations contained in this book. In addition, no responsibility is assumed by the publisher for any injury and/or damage to persons or property arising from any methods, products, instructions, ideas or otherwise contained in this publication. This publication is designed to provide accurate and authoritative information with regard to the subject matter covered herein. It is sold with the clear understanding that the Publisher is not engaged in rendering legal or any other professional services. If legal or any other expert assistance is required, the services of a competent person should be sought. FROM A DECLARATION OF PARTICIPANTS JOINTLY ADOPTED BY A COMMITTEE OF THE AMERICAN BAR ASSOCIATION AND A COMMITTEE OF PUBLISHERS. Library of Congress Cataloging-in-Publication Data

Distributions in stochastic network models / G. Sh. Tsitsiashvili, M.A. Osipova, editors. p. cm. ISBN 978-1-60692-599-7 1. Distribution (Probability theory) 2. Stochastic analysis. 3. Queuing theory. TSitsiashvili, G. Sh. (Gurami Shalvovich) II. Osipova, M. A. (Marina Anatol'evna) QA273.6.D567 2007 519.2'4--dc22 2007044152

Published by Nova Science Publishers, Inc.

New York

I.

Contents Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . § 1. Limit Distributions in Queueing Networks with Variable Structure . . . . . . . . . . . . . . . . . . . . . . . 1.1. Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2. Networks with completely variable structure . . . . . . . . . . 1.3. Networks with variable set of nodes . . . . . . . . . . . . . . 1.4. Networks with variable sets of transitions between nodes . . . 1.5. Interaction of networks with variable sets of nodes . . . . . . 1.6. Networks with variable state sets . . . . . . . . . . . . . . . . 1.7. Networks with variable types: opened and closed . . . . . . . 1.8. Additional algorithms . . . . . . . . . . . . . . . . . . . . . . § 2. Limit Distributions in Queueing Networks and Systems with Unreliable Elements . . . . . . . . . . . . . . . . . . . . . . 2.1. Unreliable servers and common queue to renewal . . . . . . . 2.2. Unreliable servers and their independent renewal . . . . . . . 2.3. Unreliable servers and their renewal in closed network . . . . 2.4. Unreliable transitions between nodes and their independent renewal . . . . . . . . . . . . . . . . . . . . . . . § 3. Limit Distributions in Queueing Networks with Different Types of Customers and Schemes of Their Transformations 3.1. Customers group transition between different sets of network nodes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2. Appearance and disappearance of customers in network nodes . . . . . . . . . . . . . . . . . . . . . . . . . 3.3. Decrease and increase in some times numbers of customers in network nodes and group exchange between nodes by customers 3.4. Unreliable customers . . . . . . . . . . . . . . . . . . . . . . . 3.5. Network with few types of customers . . . . . . . . . . . . . . 3.6. Network with “negative” customers flow . . . . . . . . . . . . § 4. Optimization of Queueing Network Ability to Handle Customers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1. Problem formulation . . . . . . . . . . . . . . . . . . . . . . . v

vii 1 1 5 6 6 7 8 9 10 13 13 14 16 19 21 21 23 24 25 26 27 29 29

vi

G. Sh. Tsitsiashvili and M. A. Osipova 4.2. Calculation of ability to handle customers . . . . . . . . . . . 4.3. Minimization of vector components maximum . . . . . . . . . 4.4. Maximization of ability to handle customers by route matrix § 5. Superposition of Queueing Networks . . . . . . . . . . . . . . 5.1. Product theorem . . . . . . . . . . . . . . . . . . . . . . . . . 5.2. Abilities to handle customers of opened networks . . . . . . . § 6. Embrechts–Veraverbeke Formula in Multiserver Queueing Models . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1. Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2. Systems with a competition of servers . . . . . . . . . . . . . 6.3. Systems with a competition of customers . . . . . . . . . . . 6.4. A generalized Lindley model . . . . . . . . . . . . . . . . . . 6.5. Proves of theorems . . . . . . . . . . . . . . . . . . . . . . . . § 7. Cooperative Effects in Queueing Systems with Rejection . 7.1. Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2. Stationary characteristics of an aggregated system . . . . . . 7.3. An aggregated system with a competition of servers . . . . . § 8. Asymptotic Analysis of Logical Systems with Unreliable Elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.1. Main characteristics . . . . . . . . . . . . . . . . . . . . . . . 8.2. Graphs with unreliable arcs . . . . . . . . . . . . . . . . . . . 8.3. Applications to lifetime models . . . . . . . . . . . . . . . . . 8.4. Calculation of graph characteristics . . . . . . . . . . . . . . . 8.5. Proves of main statements . . . . . . . . . . . . . . . . . . . . § 9. Cooperation and Competition in Risk Models . . . . . . . . . 9.1. Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.2. Cooperative effects . . . . . . . . . . . . . . . . . . . . . . . . 9.3. Individual and group risks . . . . . . . . . . . . . . . . . . . . 9.4. Competition and cooperation . . . . . . . . . . . . . . . . . . 9.5. Insurance of forward contracts . . . . . . . . . . . . . . . . . Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

30 32 33 35 35 38 39 39 40 43 44 47 53 53 54 56 57 57 57 58 58 61 63 63 64 65 66 66 69 73

Preface This monograph contains authors results obtained last four years in applied areas of queueing theory, risk theory, graph theory and reliability theory. Obtained results have been supported by FEB RAS grant (project 06-III-A-01-016), RFBR grants (projects 03-01-00512-a, 06-01-00063-a). Main part of these results have been published in Russian journals, international journals and proceedings of international conferences. Analyzed stochastic network models are aggregated systems of elements in random environment. To construct and to analyze a large number of different stochastic network models it is possible by a proof of new analytical results and a construction of calculation algorithms besides of an application of cumbersome traditional technique. Such constructive approach is in a prior detailed investigation of an algebraic model component and leads to an appearance of new original stochastic network models, algorithms and application to computer science and information technologies. Accuracy and asymptotic formulas, additional calculation algorithms have been constructed due to an introduction of control parameters into analyzed models, a reduction of multi dimensional problems to one dimensional problems, a comparative analysis, a graphic interpretation of network models, an investigation of new models characteristics, a choice of special distributions classes or principles of subsystems aggregation, proves of new statements. The authors thank V. A. Ivnitskij, S. G. Foss, I. A. Riabinin, E. D. Solojentsev, D. Daley for fruitful discussions and useful consultations.

§ 1. Limit Distributions in Queueing Networks with Variable Structure This paragraph is devoted to a calculation of limit distributions for exponential queueing networks in a random environment. Networks with a variable structure (a set of working nodes, service and input flow intensities, a route matrix, a state set, a set of transitions between nodes, a type - opened or closed) changes if environment parameters changes. Product formulas are obtained on a base of a main theorem and on a base of a dynamics scheme for considered models. Additional calculation algorithms are constructed.

1.1.

Preliminaries

Markov process ergodicity conditions. It is well known that an ordinary Markov process x(t), t ≥ 0, with a discrete state set X, transition intensities λij , i, j ∈ X, is ergodic if (A) a system of equations X X λji = ui λij , j ∈ X, (1.1) uj i∈X

i∈X

P has at least a single solution {ui } so that 0 < i∈X |ui | < ∞, (B) all states of this process are communicant ∀i, i∗ ∈ X ∃i1, . . . , ir ∈ X : λii1 > 0, λi1 i2 > 0, . . . , λir i∗ > 0

(1.2)

(C) a following regularity condition is true ∃Λ < ∞ : ∀j ∈ X

X

λji < Λ.

(1.3)

i∈X

In this case [26], [32] the limit distribution is defined from the system (1.1) and a normalization condition. If the state set is finite and all states are communicable the conditions (A) and (C) are true also. Opened and closed Jackson networks. Opened Jackson network G is a network [4, §2] with a Poisson input flow with the intensity λ. It consists of m ri-server queueing systems with exponentially distributed service times with the intensities µi , i = 1, . . ., m, and with customers circulating between them. A dynamics of the circulation is defined by the route matrix Θ = ||θij | |m i,j=0 , where θij is a probability of a customer transition after a service in the node i to the node j, θ00 = 0, the node with the number 0 is an internal source. Suppose that the route matrix is indivisible that is ∀i, j∈{0, 1, . . ., m} ∃ i1 , i2, . . . , ir ∈{1, . . . , m} : θii1 >0, θi1 i2 >0, . . . , θir j >0. 1

2

G. Sh. Tsitsiashvili and M. A. Osipova

Then the vector Λ = (λ, λ1, λ2, . . . , λm) is a single solution of the system Λ = ΛΘ,

(1.4)

and λ1 , λ2, . . . , λm > 0. A dynamics of the whole network (numbers of customers in service nodes) is described by the discrete Markov process y(t) with the state set Y = {n = (n1, . . . , nm ) : n1 , . . . , nm ≥ 0} and positive transition intensities (n ∈ Y ) L(n, n + e k ) = λθ0k , L(n + e k , n) = min(nk + 1, rk )µk θk0 ,

(1.5)

L(n + e k , n + e i ) = min(nk + 1, rk )µk θki , 1 ≤ k 6= i ≤ m.

(1.6)

Here e k is m-dimensional vector in which the component k equals 1 and all others - 0. If λi < 1, i = 1, . . ., m, (1.7) ri µi then the Markov process y(t) is ergodic and its limit distribution [4, § 2] π(n), n ∈ Y, is calculated by the formula π(n) = C −1 Ψ(n), Ψ(n) =

m Y

ai (ni ), C =

i=1

ai (0) = 1, ai (ni ) =

ni Y k=1

X

Ψ(n),

(1.8)

n ∈Y

λi , ni > 0. min(k, ri )µi

The condition (C) for the process y(t) is true as " # m X X L(n, n ∗ ) < λ + ri µi < ∞, i=1 n ∗ ∈Y the condition (B) is true because the route matrix is indivisible. Call the set of the parameters ({1, . . ., m}, λ, (µ1 , . . ., µm ), Θ, Y ) by the network G structure. b differs from an opened one by a suggestion that the fixed Closed network G number K of customers circulates in it and customers do not arrive and do not b = ||θbij ||m , depart. Displacements of customers are described by the rote matrix Θ i,j=1 b b b b λm) and the matrix Θ is indivisible. Then the solution of the system Λ = (λ1, λ2, . . ., b for arbitrary B > 0 B=

m X i=1

b=Λ bΘ b λbi , Λ

(1.9)

Limit Distributions in Queueing Networks with Variable Structure

3

exists and is single. A dynamics of this network (numbers of customers in service nodes) is described m X ni = K} by discrete Markov process yb(t) with the state set Yb = {n, n ∈ Y : i=1

and positive transition intensities (n + e k , n + e i ∈ Yb ) b + e k , n + e i ) = min(nk + 1, rk )µk θbki , 1 ≤ k 6= i ≤ m. L(n

The process yb(t) is ergodic (the condition (1.7) is not necessary) and its limit distribution π b (n), n ∈ Yb , [4, § 2] is calculated by the formula b C= π b (n) = C −1 Ψ(n), ai (ni ) = ai (0) = 1, b b

X n ∈Yb

ni Y k=1

b b Ψ(n), Ψ(n) =

m Y i=1

ai (ni ), b

(1.10)

bi λ , 0 < ni ≤ K. min(k, ri )µi

The conditions (B), (C) for the process yb(t) are true as the route matrix is indivisible and the state set is finite. Call the set of the parameters b Yb ) ({1, . . ., m}, K, (µ1 , . . . , µm ), Θ, b structure. by the closed network G Product theorem. Denote X1, X2, finite or numerable sets (they may be multi dimensionable). Define the set X ⊆ X1 × X2 , so that its projection X on Xk coincides with Xk and the sets \ J(x1 ) = { (x1 , j) : j ∈ X2} X, x1 ∈ X1 . Then X =

[

J(x1 ). Consider the Markov process x(t) with the state set X and

x1 ∈X1

the matrix of the transition intensities ||λ(x, x ∗)||x ,x ∗ ∈X , satisfied the conditions (B), (C). Theorem 1.1. Suppose that for ∀x = (x1, x2) ∈ X the following equalities are true X X λ(x, x∗) = π(x∗2; x∗1)λ(x∗, x), (1.11) π(x2; x1) x∗ ∈J(x1 ) x∗ ∈J(x1 ) X X λ(x, x∗) = A(x∗1 )π(x∗2; x∗1)λ(x∗, x), (1.12) A(x1)π(x2; x1) x∗ ∈X\J(x1) x∗ ∈X\J(x1)

4


where

X

A(x1) > 0, x1 ∈ X1 :

A(x1 ) < ∞,

x1 ∈X1

π(x2; x1) > 0, (x1, x2) ∈ X : sup x1 ∈X1

X

π(x2; x1) < ∞,

x ∈J(x1 )

then the Markov process x(t) is ergodic and its limit distribution Π(x), x=(x1, x2)∈X, has the form X A(x1 )π(x2; x1). (1.13) Π(x) = C −1 A(x1)π(x2; x1), C = x ∈X Proof. From the equalities (1.11), (1.12) obtain that ∀x ∈ X X X λ(x , x ∗) + A(x1)π(x2; x1) λ(x , x ∗ ) = A(x1 )π(x2; x1) x ∗ ∈J(x1 ) x ∗ ∈X\J(x1) X X π(x∗2; x∗1)λ(x ∗ , x ) + A(x∗1 )π(x∗2; x∗1)λ(x ∗ , x ) = = A(x1) x ∗ ∈J(x1 ) x ∗ ∈X\J(x1) X X A(x∗1)π(x∗2; x∗1)λ(x ∗ , x ) + A(x∗1)π(x∗2; x∗1)λ(x ∗ , x ), = x ∗ ∈X\J(x1) x ∗ ∈J(x1 ) consequently X

A(x1)π(x2; x1)

x and

X

λ(x , x ∗) =

x

∗ ∈X

X

A(x∗1)π(x∗2; x∗1)λ(x ∗, x ),

∗ ∈X

Π(x ) = 1.

x ∈X So the discrete Markov process x(t) satisfies sufficient ergodicity conditions (A) (C) and its limit distribution has the form (1.13). Corollary 1.1. Suppose that X = X1 × X2 and transition intensities satisfied the conditions (B), (C) are defined, by the equalities (xi, x∗i ∈ Xi , i = 1, 2, I(A) is the event A indicator function) λ(x, x∗ ) = Lx1 (x2 , x∗2)I(x1 = x∗1) + ν(x1, x∗1)I(x2 = x∗2 ). Then if π(x2)

X

Lx1 (x2 , x∗2) =

x∗2 ∈X2

A(x1 )

X

X

π(x∗2)Lx1 (x∗2, x2), x1 ∈ X1,

(1.14)

(1.15)

x∗2 ∈X2

ν(x1 , x∗1) =

x∗1 ∈X1

X

A(x∗1 )ν(x∗1, x1), x2 ∈ X2,

x∗1 ∈X1

(1.16)

Limit Distributions in Queueing Networks with Variable Structure X A(x1 ) = 1,

5

x1 ∈X1

X

where π(x2) > 0, x2 ∈ X2 , and

π(x2) = 1, so the process x(t) is ergodic and

x2 ∈X2

its limit distribution has the form Π(x) = A(x1)π(x2), x = (x1 , x2) ∈ X.

1.2.

(1.17)

Networks with completely variable structure

Consider the opened Jackson network G with oneserver nodes. Denote S the set of all nonempty subsets s (with ordered by an increase elements ) of the set {1, . . . , m}. Suppose that the network G changes its structure as follows: in the state s it has the structure (s, λ(s), (µ1 (s), . . ., µm (s)), Θ(s), Y (s)) with Y (s) = {n=(ni , i ∈ s) : ni ≥ 0, i ∈ s}. Denote 0 s = (ni =0, i ∈ s) the zero element of the set Y (s). The network dynamics in the state s is described by the Markov process with the transition intensities Ls (n, n ∗), n, n ∗ ∈ Y (s), defined by the formulas (1.5), (1.6), and with the limit distribution π(n; s), n ∈ Y (s), defined by (1.8). Suppose that a dynamics of a considered network with a variable [ structure is J(s), J(s) = described by the discrete Markov process x(t) with the state set X= s∈S

{(s, n) : n ∈ Y (s)} and transition intensities ( Ls (n, n ∗ ), s = s∗ , n, n ∗ ∈ Y (s), λ((s, n), (s , n ))= ν(s, s∗ ), s6=s∗ , n=0 s ∈ Y (s), n ∗ =0 s∗ ∈ Y (s∗). ∗

∗

Fix A(s) > 0, s ∈ S,

X

A(s) = 1, and suppose that the transition intensities

s∈S

ν(s, s∗ ), s, s∗ ∈ S, satisfy the communication condition and are solutions of the equations system A(s)π(0 s ; s)

X s∗ ∈S

ν(s, s∗ ) =

X

A(s∗ )π(0 s∗ ; s∗)ν(s∗ , s), s ∈ S.

s∗ ∈S

Then by the theorem 1.1 the process x(t) is ergodic and its limit distribution has conditionally product form Π(s, n) = A(s)π(n; s), (s, n) ∈ X.

(1.18)

6


1.3.

Networks with variable set of nodes

Consider the opened network G in which onesrever nodes may fail and may be repaired. Suppose that S is a set of all nonempty subsets of the set {1, . . ., m} (with ordered by an increase elements). Suppose that an opened network in the state s ∈ S has the structure (s, λ, (µ1, . . . , µm ), Θ(s), Y ) and an exchange by customers may be only between nodes from the set s, customers from another nodes do not move. Put that for fixed λi < µi , i = 1, m, the indivisible transition matrixes Θ(s) satisfy the conditions (1.19) (λ, λi, i ∈ s) = (λ, λi, i ∈ s)Θ(s), s ∈ S. X A(s) = 1. If a dynamics of the considered network with Fix A(s) > 0, s ∈ S, s∈S

a variable structure is described by the discrete Markov process x(t) with the state set X = S × Y and the transition intensities ( Ls ((ni , i ∈ s), (n∗i , i ∈ s)), s = s∗ , λ((s, n), (s∗, n ∗)) = ν((s, n), (s∗, n), s 6= s∗ , n = n ∗ , with ν((s, n), (s∗, n)), s, s∗ ∈ S, n ∈ Y, satisfying the conditions (B), (C) and X X ν((s, n), (s∗, n)) = A(s∗ )ν((s∗, n), (s, n)), (1.20) A(s) s∗ ∈S

s∗ ∈S

then the process x(t) is ergodic by the theorem 1.1 and its limit distribution has the form Π(s, n) = A(s)π(n), (s, n) ∈ X.

(1.21)

Here π(n), n ∈ Y, is defined by the formula (1.8). Remark 1. If λi = λ < µi , i = 1, . . ., m, then arbitrary symmetric route matrixes Θ(s), s ∈ S, satisfy the formulas (1.19).

1.4.

Networks with variable sets of transitions between nodes

Consider an opened network with a cut-in and a cut-off of transitions between oneserver nodes. Denote S the set of all subsets from Φ = {(i, j), 0 ≤ i, j ≤ m, (i, j) 6= (0, 0)}, satisfying the conditions: (i ) ∃ i, j, 1 ≤ i 6= j ≤ m : (0, i), (0, j) ∈ s;


7

(ii ) ∀ i, 1 ≤ i ≤ m, (i, i) ∈ s; (iii ) (i, j) ∈ s ⇐⇒ (j, i) ∈ s; (iiii ) ∀ (i, j) ∈ s ∃ i1, . . . , , ik : (i, i1), (i1 , i2), (ik−1, ik ), (ik , j) ∈ s. Suppose that the network G in the state s has the structure (s, λ, (µ1 , . . ., µm ), Θ(s), Y ), s ∈ S. Assume that the route matrixes Θ(s) satisfy the formulas (1.19) and θij (s) > 0 ⇐⇒ (i, j) ∈ s.

(1.22)

Analogously fix the constants A(s). From the theorem 1.1 obtain that the discrete Markov process x(t) with the state set X = S ×Y and the transition intensities ( Ls (n, n ∗ ), s = s∗ , n, n ∗ ∈ Y, λ((s, n), (s∗, n ∗ )) = ν((s, n), (s∗, n)), s 6= s∗ , n = n ∗ , describing a dynamics of the network G with a variable structure is ergodic and its limit distribution is calculated by the formula (1.21). Here the transition intensities ν((s, n), (s∗, n)), s, s∗ ∈ S, n ∈ Y, satisfy the conditions (B), (C) and are solutions of the equations (1.20). Remark 2. For any s ∈ S there is symmetric and indivisible route matrix Θ(s), satisfying the condition (1.22). In this case λi = λ, i = 1, . . ., m.

1.5.

Interaction of networks with variable sets of nodes

Here an interaction between two networks with variable sets of working oneserver nodes is considered. Changeovers may be for arbitrary numbers of customers in nodes of interacting networks. Divide the set {1, . . . , m} into nointersected subsets [ M1, M2, M1 M2 = {1, . . . , m}, and denote Sj the set of all nonempty subsets (with ordered by an increase elements) of the set Mj , j = 1, 2. Suppose that an opened network Gj , j = 1, 2, in the state sj , sj ∈ Sj , has the structure (sj , λ, (µ1 , . . . , µm ), Θ(sj ), Yj ) where Yj = {n j = (ni , i ∈ Mj ), ni ≥ 0, i ∈ Mj }. The route matrix Θ(sj ) defines a transition on the nodes set sj ∈ Mj and is indivisible. In the network Gj an exchange by customers is only between nodes of the set sj , and customers situated in the nodes of the set Mj /sj , do not move. Suppose that for fixed λ1 , . . . , λm : λi < µi , i = 1, . . ., m, the route matrixes Θ(sj ), sj ∈ Sj , j = 1, 2, satisfy formulas (λ, λi, i ∈ sj ) = (λ, λi, i ∈ sj )Θ(sj ).

8


Denote S=S1X ×S2 , Y =Y1 ×Y2 , s=(s1, s2) ∈ S, n=(n 1 , n 2 ) ∈ Y , and fix A(s), s∈S, A(s) = 1. Suppose that the networks G1, G2 with nodes cut-in A(s) > 0, s ∈S and cut-off interact so that their joint dynamics may be described by the discrete Markov process x(t) with the state set X=S×Y and the transition intensities ((s, n), (s ∗ , n ∗ )∈X)  ∗ ∗ ∗  Ls1 ((ni , i ∈ s1 ), (ni , i ∈ s1 )), s1 = s1 , n 2 = n 2 , ∗ ∗ λ((s, n), (s , n ))= Ls2 ((ni , i ∈ s2 ), (ni∗ , i ∈ s2 )), s∗2 = s2 , n 1 = n 1∗ ,   ν((s, n), (s ∗, n)), s 6= s ∗, n ∗ = n. Here Lsj ((ni , i ∈ sj ), (ni∗ , i ∈ sj )), j = 1, 2, is defined from (1.5), (1.6). If the intensities ν((s, n), (s ∗ , n)), s ∗, s ∈ S, n ∈ Y, satisfy the conditions (B), (C) and are the equations (1.20) solutions then from the theorem 1.1 the process x(t) is ergodic and its limit distribution has the form (1.21).

1.6.

Networks with variable state sets

Consider an opened Jackson network G with oneserver nodes. Denote Y (s) = S , n ∈ s, i = 1, m}, s ∈ S, where S is a finite set of finite subsets from s {n + e i\ Y (s) 6= ∅. Suppose that the network G in the state s, s∈S, has the Y so that s∈S

structure ({1, .., m}, λ, (µ1, . . . , µm ), Θ, Y (s)) and the matrix Θ is indivisible and numbers of customers in the network nodes are described by Markov process with the state set Y (s) and positive transition intensities Ls (n, n ∗ ), n, n ∗ ∈ Y (s), which are defined from (1.5), (1.6). So the G in the state s is defined by the nonoriented graph Γ(s) with nodes from the set Y (s) and arcs [n, n + e k ], [n + e k , n + e i ], n ∈ s, 1 ≤ k 6= i ≤ m. n2 6

n2 6

@ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ -

n1

@ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ -

n1

Fig. 1. Exammples of graphs Γ(s), m = 2. X A(s) = 1. If the graph Γ(s) is connected then the Fix A(s) > 0, s ∈ S, s∈S

Markov process, describing numbers of customers in the network G at the state s,


9

[48] is ergodic and its limit distribution π(n; s), n ∈ Y (s), is defined by the formula (1.8). Define a dynamics of the network with a variable and finite state set by the Markov process x(t) with the state set X = {(s, n) : s ∈ S, n ∈ Y (s)} and the transition intensities  Ls (n, n ∗), s = s∗ , n, n ∗ ∈ Y (s), \ ∗ ∗ λ((s, n), (s , n )) = ν((s, n), (s∗, n)), n ∗=n, n ∈ Y (s), s∗ 6= s.  s∈S

∗

∗

If the intensities ν((s, n), (s , n)), (s, n), (s , n) ∈ X, satisfy the conditions (B), (C) and are the equations (1.20) solutions then from the theorem 1.1 obtain that the process x(t) is ergodic and its limit distribution is calculated by the formula (1.18). Remark 3. The points 1.2 - 1.6 results may be spread onto closed networks with a variable structure. Particularly in the point 1.2 it is necessary to replace 0 s by the marked state n s .

1.7.

Networks with variable types: opened and closed

Suppose that the network G with the structure ({1, . . ., m}, λ, (µ1, . . . , µm ), Θ, Y ), in the state n set Yb , Yb ⊂ Y, may randomly stop an exchange by customers with its b with the structure ({1, . . ., m}, K, environment and become the closed network G b b (µ1 , . . . , µm), Θ, Y ). The closed network in the state Yb may become the opened b for fixed λ1, . . . , λm one and so on. Suppose that the indivisible route matrixes Θ, Θ satisfy the formulas b (λ, λ1, . . ., λm)=(λ, λ1, . . . , λm)Θ, (λ1, . . ., λm)=(λ1, . . . , λm)Θ.

(1.23)

Fix A(1), A(2) : A(1)+A(2) = 1, A(1), A(2) > 0. If the considered opened-closed network is described by the discrete Markov process x(t) with the state set X = J(1) ∪ J(2) = {(1, n) : n ∈ Y } ∪ {(2, n) : n ∈ Yb } and the transition intensities  ∗ ∗ ∗  L(n, n ), s = s = 1, n, n ∈ Y, b λ((s, n), (s∗, n ∗)) = L(n, n ∗ ), s = s∗ = 2, n, n ∗ ∈ Yb ,   ν((s, n), (s∗, n)), n ∗ = n, n ∈ Yb , s∗ 6= s, with the intensities ν((s, n), (s∗, n)) > 0, (s, n), (s∗, n) ∈ X, satisfying the equalities A(1)ν((1, n), (2, n)) = A(2)ν((2, n), (1, n)), n ∈ Yb ,

10


and the communicability condition then from the theorem 1.1 this process is ergodic and its limit distribution has the form X A(s)Ψ(n), Π(s, n) = C −1 A(s)Ψ(n), (s, n) ∈ X, C = (s,n )∈X where Ψ(n) for n ∈ Y or n ∈ Yb is defined by the formula (1.8). Remark 4. Replace the formulas (1.23) by the formulas (1.4), (1.9) in the openedclosed network definition and redefine the transition intensities as follows λ((s, n), (s∗, n ∗)) = ν(s, s∗ ), n

∗

= n = n 0 , s∗ 6= s, s∗ , s = 1, 2,

where n 0 ∈ Yb , is the denoted state. Then the limit distribution of the process x(t) is calculated for (s, n) ∈ X by the formula ( A(1)π(n), s = 1, Π(s, n) = A(2)b π(n), s = 2.

1.8.

Additional algorithms

Construction of route matrixes from the point 1.4. This algorithm is used for a constructive proof of the remark 2. Fix d0j = 1, 0 ≤ j ≤ m. Consider a recursion step k and choose θkj , k ≤ j ≤ m, from the conditions: X / s, 0 < θkj < dkj , j > k, θkj = dkk . (1.24) θkj = 0, (k, j) ∈ (k,j)∈s, j≥k

A possibility of such a choice is based for k = 0 on the condition (i ) and for k > 0 - on the condition (ii ) and on the inequality dkj > 0, j > k. Then define = dkj − θjk , 0 < dk+1 ≤ 1, k < j ≤ m, θjk = θkj , dk+1 j j

(1.25)

and begin (k + 1)−th step of a recursion. From the recurrent formulas (1.24), (1.25) obtain the conditions (1.22) and a fact that the matrix Θ is route and symmetric. The matrix Θ indivisibility may be obtained from the condition (iiii ). So the statement of the corollary 2 is proved. Check of transition graph connectivity in the point 1.6. A central moment in the point 1.6 is the condition of the graphs Γ(s), s ∈ S, connectivity. In the graph theory there are algorithms which allow to define a graph connectivity using its incidence matrix. But in a case of this matrix large dimension it is necessary to use a large volume of a computer memory. Here there is an algorithm of a nonoriented graph connectivity test Γ(s) = {[n, n + e k ], [n + e k , n + e i ], n ∈ s, 1 ≤ k 6= i ≤ m}


11

which takes into account a graph structure and so demands significantly smaller memory volume. It is clear that the graph Γ(s) connectivity is equivalent to the graph Γ0 (s) = {[n, n + e k ], n ∈ s, 1 ≤ k ≤ m} connectivity. Denote V (n) = {n, n + e k , n − e k , n + e k − e i , 1 ≤ k, i ≤ m} and fix an arbitrary vector n 1 ∈ s and put s1 = {n 1 }, sl+1 = s

\

[

!

V (n) , l ≥ 1.

n ∈sl Statement 1. The graph Γ0 (s) is connected if and only if sN = s with N = min(l : sl = sl+1 ). Proof. By a definition for ∀n ∗ ∈ V (n) the graph Γ0 ({n, n ∗}) is connected. It is possible to check by an induction for l ≥ 1 that s1 ⊆ s2 ⊆ . . . ⊆ s and all graphs Γ0 (sl ), l ≥ 1, are connected. Consequently from the equality sN = s obtain the graph Γ0 (s) connectivity. Vice versa, if sN 6= s then from the definition of V (n) the graph Γ0 (s) is not connected. So to check the graph Γ(s) connectivity it is necessary to construct the sequence sl , l ≥ 1. Show without a proof the formula which allows to avoid repetitions in a calculation of this sequence:    [ \ [ V (n) , l ≥ 1. s0 = ∅, s1 = {n 1 }, sl+1 = sl s  n ∈sl /sl−1

§ 2. Limit Distributions in Queueing Networks and Systems with Unreliable Elements In this paragraph multiserver queueing systems and networks with unreliable elements: nodes, ways between nodes, servers in multiserver nodes are considered. Obtained formulas for a calculation of limit distributions are corollaries of § 1product theorem and are based on on a choice of unreliable elements renewal scheme (an independent renewal, a renewal on a single repair place, a renewal network scheme), route algorithms and a customers service discipline. So an introduction of a control to a dynamics of analyzed models allow to connect constructively queuing networks theory and reliability theory.

2.1.

Unreliable servers and common queue to renewal

System M|M|r|0. Consider r - server queueing system with the Poisson input flow with the intensity λ, exponentially distributed service times with the parameter µ and a single repair place. Suppose that an each working server may fail with the intensity γ − and a failured server may be restored with the intensity γ + if the repair place is free. In an opposite case a failured server stands into a queue to a repair. If a number of customers in this system becomes larger than a number of working servers then the system stops a service and an acceptance of customers. If a number of customers becomes equal to a number of working servers then the system stops to accept new customers. This system is described by the Markov process x(t) with the state set X = {(s, n), s, n ∈ {0, . . ., r}} (here the component s characterizes a number of working servers and n - a number of customers in the system) and the transition intensities λ((s, n), (s∗, n∗)) of a view (1.14) where Ls (n, n + 1) = λ, 0 ≤ n < s, Ls (n, n − 1) = nµ, 0 < n ≤ s, ν(s, s + 1) = γ +, 0 ≤ s < r, ν(s, s − 1) = sγ − , 0 < s ≤ r. All other transition intensities equal zero. From the corollary 1.1 the Markov process x(t) is ergodic and its limit distribution has the form Π(s, n) = A(s)π(n), (s, n) ∈ X, !−1 n r X 1 λ n 1 λ , π(n) = n! µ n! µ n=0 s !−1 + s r X 1 γ+ 1 γ . A(s) = − s! γ s! γ −

(2.1)

(2.2)

s=0

Network with oneserver nodes. Consider the network G with the structure 13

14


({1, . . ., m}, λ, (µ1 , . . . , µm), Θ, Y ) (where Θ is indivisible) which has m unreliable oneserver nodes and a single repair place for their renewal. Suppose that a failured node (the failure intensity is γ −) is restored with the intensity γ + if the repair place is free. In an opposite case a failured server stands to a queue to the repair place. Denote S = {s = (s1 , . . . , sm) : 0 ≤ si ≤ m, i=1, m; si 6= sj , si , sj ≥ 1, i 6= j}. Describe this network by the Markov process x(t) with the state set X = S × Y (the component si = 0 if the node i is in a working phase, si = 1 if the node i is on a repair, si = k, k ≥ 2, if the node i is (k−1)-th in a queue to a repair; n = (n1 , . . . , nm ) is a vector of customers numbers in the network nodes) and the transition intensities λ((s, n), (s ∗ , n ∗ )) of the form (1.14) and ( + ∗ γ , s = (max{0, s1 − 1}, . . ., max{0, sm − 1}), ∗ ν(s, s ) = γ − , sk = 0, s∗k = 1 + max si , sj = s∗j , j 6= k, i=1,...,m

  λθ0k , n ∗ = n + e k , sk = 0, µ θ , n ∗ = n − e k , sk = 0, Ls (n, n ∗) =  k k0 ∗ µk θki , n = n + e i − e k , si = sk = 0.

(2.3)

By the corollary 1.1 the Markov process x(t) is ergodic and its limit distribution has the form (1.21) where C −1 A(s)= m!

γ+ γ−

Q(s ) m m X X , Q(s) = I(si=0), C= i=1

Q(s)=0

1 Q(s)!

γ+ γ−

Q(s) .

This result is true if the failure and the renewal intensities coincide. In an opposite case a calculation of the limit distribution in this network becomes much more complicated. Partially this problem may be solved for following models of nodes renewal.

2.2.

Unreliable servers and their independent renewal

System M|M|r|0. Consider a queueing system described in the subpoint 1 of the point 2.1. Suppose that an each server has its own repair place. Describe this system by the Markov process x(t) but with a difference that ν(s, s + 1) = (r − s)γ + . Then the limit distribution of the process x(t) will differ by the multiplyer + s r γ r! γ− , s ∈ S. A(s) = + − γ +γ s!(r − s)! γ − A network with oneserver nodes. Consider the opened queueing network G with m unreliable nodes, i-th node may fail with the intensity γi− and may be

Limit Distributions in Queueing Networks and Systems...

15

restored with the intensity γi+ . Describe this system by the Markov process x(t) with the state set X = S × Y, S = {s = (s1 , . . . , sm ) : si = 0, 1, i = 1, . . ., m} (the component si = 0 if the node i is in the working phase, the component si = 1 if the node i is restored, n = (n1, . . . , nm ) is the vector of customers numbers in the network nodes) and the transition intensities λ((s, n), (s ∗, n ∗ )) with the form (1.14) in which Ls (n, n ∗ ) are defined from the formulas (2.3) and ν((s1 , . . . , si−1, 1, si+1, . . . , sm), (s1, . . . , si−1 , 0, si+1, . . . , sm )) = γi+, ν((s1 , . . . , si−1, 0, si+1, . . . , sm), (s1, . . . , si−1 , 1, si+1, . . . , sm )) = γi−. The Markov process x(t) is ergodic and its limit distribution has the form (1.21) where Y Y γi+ γi− , A(s) = γi+ + γi− γi+ + γi− + − i∈Q (s ) i∈Q (s ) Q+ (s) = {i : si = 0}, Q− (s) = {i : si = 1}, s ∈ S. Network with multiserver nodes. Consider the opened network G with the structure ({1, . . . , m}, λ, (µ1 , . . . , µm ), Θ, Y ), Θ is indivisible and its nodes are ri - server queueing systems with a failure of servers. Denote the failure intensity of the server in the node i by γi− and the repair intensity - by γi+ . If a number of customers in a node is larger than a number of working servers then this node stops an acceptance and a service of customers. If a number of customers becomes equal to a number of working servers in a node then it stops an acceptance of new customers. Denote S = {0, 1, . . . , r1} × {0, 1, . . ., r2} × · · ·× {0, 1, . . ., rm}. Suppose that this network dynamics is described by the Markov process x(t) with the state set X = {(s, n), s, n ∈ S} (the vector s = (s1 , . . . , sm ) characterizes a number of working servers in nodes of the network and the vector n = (n1 , . . . , nm ) characterizes a number of customers in the network nodes) and the transition intensities λ((s, n), (s ∗, n ∗ )) of the form (1.14) where   ni θi0 µi , n ∗ = n − e i , nk , n∗k ≤ sk , k = 1, m, λθ0i, n ∗ = n + e i , nk , n∗k ≤ sk , k = 1, m, Ls (n, n ∗ ) =  ni θij µi , n ∗=n−e i +e j , i 6= j, nk , n∗k ≤ sk , k = 1, m, ∗

ν(s, s ) =

(ri − si )γi+ , s ∗ = s + e i , si γi−, s ∗ = s − e i .

16


If 0 < θ0k , θk0 < 1, 1 ≤ k ≤ m, and the following equalities (modifications of the system (1.4)) are true for some λi > 0, i = 1, m : λk θk0 = λθ0k , k = 1, m, λiθik = λk θki , k 6= i, k, i = 1, m,

(2.4)

then from the corollary 1.1 obtain that the discrete Markov process x(t) is ergodic and its limit distribution has the form (1.21) where π(n), n ∈ S, is defined by the formula (1.8) and A(s) =

m Y i=1

γi− γi+ + γi−

ri

ri ! si !(ri − si )!

γi+ γi−

si

, s ∈ S.

Remark 1. Suppose that the vector (λ, λ1, . . . , λm), λi > 0, i = 1, m, is the system (1.4) solution. Then the discrete Markov process x(t) with the state set X={(s, n)=((s1, . . . , sm ), (n1, . . . , nm )), 0 ≤ ni , si ≤ ri,

m X i=1

ni 6=

m X

ri }

i=1

and the transition intensities λ((s, n), (s ∗ , n ∗ )) of the form (1.14) which differ from the transition intensities described in this point by the summand   ni θi0 µi , n ∗ = n − e i , n∗k < sk , k = 1, m, ∗ λθ0i , n ∗ = n + e i , nk < sk , k = 1, m, Ls (n, n ) =  ni θij µi , n=n 0+e i , n ∗ =n 0+e j , i 6= j, n0k <sk , k = 1, m, is ergodic byX the theorem 1.1 and its limit distribution has the form C −1 A(s)Ψ(n) A(s)Ψ(n). where C = (s ,n )∈X

2.3.

Unreliable servers and their renewal in closed network

Network with oneserver nodes. Consider the opened queueing network G with oneserver nodes, a server in the node i fails with the intensity γi, i = 1, m, and is repaired in nodes m + 1, . . ., m + l with the intensities γi , i = m + 1, . . . , m + l. Call the set M1 = {1, . . ., m} by a working phase of the network G and the set b which M2 ={m + 1, . . ., m + l} - by a repair phase. Define the closed network G consists of m + l oneserver nodes and describes a failure and a repair of the network b G nodes in a working phase. Then the network G servers become the network G customers. b is Suppose that a motion of the network G servers in the closed network G m+l b b b controlled by the indivisible route matrix Θ = ||θi,j ||i,j=1 . In the network G nodes queues may appear. Servers of each node i ∈ M = M1 ∪ M2 may move to each node j ∈ M. A total number of servers in this network is K = m. Numbers of servers in


17

b nodes may be described by the Markov process s(t) with the state the network G set ) ( m+l X si = K (2.5) S = s = (s1, . . . , sm+l ) : si ≥ 0, i ∈ M, i=1

and positive transition intensities ν(s + E i , s + E j ) = γi θbij , s + E i , s + E j ∈ S, i 6= j

(2.6)

(here E k is (m + l)-dimensional vector in which k-th component equals 1 and all others - 0). The process s(t) is ergodic and its limit distribution (the point 1.2) has the form !s !si m+l X m+l Y λ Y b bi i λi −1 , s ∈ S, C = , (2.7) A(s) = C γi γi i=1 s ∈S i=1 b1, . . . , b b1, . . . , b b λm+l) = (λ λm+l)Θ. (λ

(2.8)

The formulas (2.6), (2.7) describe a model of nonload reserve when a server in a queue to a working place does not fail. An alternative model is a model of loaded reserve in which a server in a queue to a working place fails like at a working place. In this model for m + 1 ≤ i ≤ m + l the formula (2.6) is true and for 1 ≤ i ≤ m the formula ν(s + E i , s + E j ) = (si + 1)γiθbij , s + E i , s + E j ∈ S, i 6= j takes place and A(s) = C

−1

m+l m Y 1 Y si ! i=1

i=1

bi λ γi

!si

m m+l XY 1 Y , s ∈ S, C = s! s ∈S i=1 i i=1

bi λ γi

!si

.

Fix λ1, . . . , λm : λi < µi , i = 1, m, and for any nonempty set s ⊆ M1 choose the indivisible route matrix Θ(s) = ||θij (s)||i,j∈s which satisfies the formulas (1.19). Define the process x(t) = (s(t), y(t)) describing numbers of servers in the network b nodes and numbers of customers in the working nodes of the network G with the G state set X = S × Y and with the transition intensities Λ((s, n), (s ∗ , n ∗ )) of the form (1.14). Here ν(s, s ∗ ) is defined by the equalities (2.6) and   λθ0k , n ∗ = n + e k , sk > 0, ∗ µ θ , n ∗ = n − e k , sk > 0, Ls (n, n ) =  k k0 ∗ µk θki , n = n + e i − e k , si , sk > 0. From the corollary 1.1 obtain that the process x(t) is ergodic and its limit distribution has the form (1.21) where A(s) is defined by the formula (2.7). So it is possible to construct a network in which differences of failure intensities and differences of repair intensities are described (by a special choice of the route

18


b and to calculate its limit distribution without a consideration of complimatrix Θ) cated queues of different types elements as it was made in 2.1. Network with multiserver nodes Consider the opened network G in which the node i is ri-server queueing system with failures of servers, i = 1, . . . , m. Suppose that a server of the node i fails with the intensity γi , i ∈ M1 , and is restored in additional nodes m + 1, . . ., m + l with the intensities γi , i ∈ M2. Define the closed b in which in the nodes 1, . . ., m (working nodes) there are r1, . . . , rm network G working places for servers which process customers of the network G. These servers b and in the nodes m + 1, . . ., m + l (repair nodes) there are move in the network G rm+1 , . . . , rm+l repair places for them. b by the As in the previous point describe numbers of servers in the network G m X ri and the Markov process s(t) with the state set S defined by (2.5) where K = i=1

transition intensities ν(s + E i , s + E j ) = min(si + 1, ri)γi θbij , s + E i , s + E j ∈ S, i 6= j.

b = ||θbi,j ||m+l is indivisible and the formula (2.8) is true then If the route matrix Θ i,j=1 the process s(t) is ergodic and its limit distribution A(s), s ∈ S, has the form A(s) = C

−1

b Ψ(s), C=

ai (si ) = ai (0) = 1, b b

si Y k=1

X s ∈S

b b Ψ(s), Ψ(s) =

m+l Y i=1

b ai (si ),

(2.9)

bi λ , 0 < si ≤ K. min(k, ri )γi

b nodes and numbers of customers in the Numbers of servers in the network G working nodes of the network G are described by the Markov process x(t) with the state set X = S × J, J = {0, 1, . . ., r1} × {0, 1, . . ., r2} × · · · × {0, 1, . . ., rm}, and with the transition intensities λ((s, n), (s ∗, n ∗ )) of the form (1.14), (s, n), (s ∗, n ∗ ) ∈ X (the vector n = (n1 , . . . , nm) characterizes numbers of serving customers in the working nodes) where   ni θi0 µi , n ∗ = n − e i , ∗ λθ0i, n ∗ = n + e i , Ls (n, n )=  ni θij µi , n ∗ = n − e i +e j , i 6= j, for nk , n∗k ≤ min(sk , rk ), k = 1, m. If 0 < θ0k , θk0 < 1, k = 1, m, and for some λi > 0, i = 1, m, the formulas (2.4) are true then from the corollary 1.1 obtain that the discrete Markov process x(t) is ergodic and its limit distribution has form (1.21) where π(n) is defined by the formula (1.8) and A(s) - by the formula (2.9). Similar to the remark 1 a case when the vector Λ = (λ, λ1, . . . , λm), λi > 0, i = 1, m, is the system (1.4) solution is considered .


2.4.

19

Unreliable transitions between nodes and their independent renewal

In this point an opened network in which ways between nodes may fail and may be restored independently is considered. These transformations of a network structure take place for arbitrary numbers of customers in nodes. Calculation of limit distribution in network. Consider the set Φ ⊆ {(i, j), 0 ≤ i, j ≤ m, (i, j) 6= (0, 0)} of twosided ways between the network G nodes. Suppose that the set Φ satisfies the conditions (i ) ∀ i, 1 ≤ i ≤ m, (i, i) ∈ Φ, (ii ) (i, j) ∈ Φ ⇐⇒ (j, i) ∈ Φ, (iii ) ∀ (i, j) ∈ Φ ∃ i1, . . . , , ik : (i, i1), (i1, i2), (ik−1 , ik ), (ik , j) ∈ Φ. From these conditions it is easy to obtain that (i 0 ) ∃ i, j, 1 ≤ i 6= j ≤ m : (0, i), (0, j) ∈ Φ. Denote by S the set of all possible subsets s ⊆ Φ which satisfy the conditions (i ), (ii). Put that the network G in the state s has the structure (s, λ, (µ1 , . . . , µm ), Θ(s), Y ) for s ∈ S. Suppose that there λ1 < µ1 , . . . , λm < µm and the route matrix Θ(s) = ||θij (s)||m i,j=0 for fixed s ∈ S satisfies the condition (1.19) and θij (s) > 0 ⇐⇒ (i, j) ∈ s, s = Φ,

(2.10)

/ s, s ⊂ Φ. (2.11) θij (s) = 0, (i, j) ∈ X A(s) = 1 and assume that (i, j) way (which Fix A(s), A(s) > 0, s ∈ S : s∈S − , (i, j) ∈ Φ and may be connects the nodes i and j) may fail with the intensity γij + restored with the intensity γij . Describe this network dynamics by the discrete Markov process x(t) with the state set X = S × Y and the transition intensities λ((s, n), (s∗, n ∗ )) defined by the formula (1.14) where Ls (n, n ∗ ), satisfy (1.5), (1.6) and ν(s, s∗ ), s, s∗ ∈ S, satisfy the condition (B) and the equalities (1.16). An equality of all other intensities to zero means that in the state (s, n) for (i, j) ∈ / s a customer, directed to this way from one of two opposite directions, returns to an initial node. Then from the corollary 1.1 obtain that the Markov process x(t) is ergodic and its limit distribution is calculated by the formula (1.21) in which A(s) is defined by the formula ! ! + − Y Y γij γij , s ∈ S. (2.12) A(s) = + − + − γij + γij γij + γij (i,j)∈s

(i,j)∈s /

Algorithm of route matrixes construction. Consider a following algorithm of symmetric route matrixes Θ(s), s ∈ S, satisfying the condition (2.11) construction.

20


Remark that the algorithm which generates symmetric route matrixes Θ(s), s ∈ S, satisfying the condition (2.10) is in the point 1.8. At first fix d0j = 1, 1 ≤ j ≤ m, and define d00 = I(∃ i, 1 ≤ i ≤ m : (0, i) ∈ s). If k = 0 and d00 = 0 then put θ0i = θi0 = 0, i = 0, . . . , m and transit to the next recursion step with k = 1. And if k = 0, and d00 = 1, then put / s, θ0i > 0, (0, i) ∈ s, θ0i = 0, (0, i) ∈

m X

θ0i = 1,

i=1

θj0 = θ0j , d1j = d0j − θj0 , 0 ≤ d1j ≤ 1, 0 < j ≤ m, and transit to the recursion next step with k = 1. Suppose that the recursion step k ≥ 1 and choose θkj , k ≤ j ≤ m, from (a possibility of this choice is based on the condition (i )) : / s, θkj = 0, (k, j) ∈ 0 ≤ θkj ≤ dkj , (k, j) ∈ s, j > k :

X

θkj = dkk .

(2.13)

(k,j)∈s, j≥k

Then define = dkj − θjk , 0 ≤ dk+1 ≤ 1, k < j ≤ m, θjk = θkj , dk+1 j j

(2.14)

and begin the next recursion step with the number (k + 1). From the formulas (2.13), (2.14) obtain that the condition (2.11) is true and the matrix Θ is route and symmetric. Remark 2. For λi = λ < µi , i = 1, m, the matrix Θ(s) construction algorithm does not demand a complicated obtained information about connectivity components of the nonoriented graph with the node set {1, . . ., m} and with the arcs (i, j) ∈ s.

§ 3. Limit Distributions in Queueing Networks with Different Types of Customers and Schemes of Their Transformations In this paragraph product theorems receive their new development. Proved product theorem is spread onto opened queueing networks of Jackson type with different schemes of customers transformations. Models of a group arrival and a group displacement, models with customers: appearance and disappearance, division and aggregation and models with unreliable customers are considered. Partial balance conditions [42] are used to obtain formulas for limit distributions.

3.1.

Customers group transition between different sets of network nodes

Exchange with external source. Consider the opened network G with oneserver nodes and the structure s0 = ({1, . . ., m}, λ, (µ1, . . . , µm ), Θ, Y ) where Θ is indivisible. Suppose that in a random time moment the network s0 with the vector n = (n1 , . . . , nm) of customers numbers in different nodes may change a work principle. Namely, an each customer of the input flow transforms into m its copies which are directed to all network nodes. If n1 ≥ 1, . . . , nm ≥ 1 then a service of these customers is simultaneous and timing with the intensity µ. Then all these customers depart from the network. Denote the structure of this network by s1 = ({1, . . . , m}, λ, (µ, . . ., µ), Y ). A dynamics of this network with a variable structure may be described by the discrete Markov process x(t) with the state set X = S × Y,

(3.1)

S = {s0 , s1} and the transition intensities ( Li (n, n ∗ ), s = s∗ = si , λ((s, n), (s∗, n ∗ )) = ν(s, s∗), s 6= s∗ , n = n ∗,

(3.2)

where L0 (n, n ∗ ) are defined from the formulas (1.5), (1.6) and L1 (n, n + e) = λ, L1 (n + e, n) = µ, e =

m X

ek.

(3.3)

k=1

If A(s) > 0, s ∈ S,

X s∈S

21

A(s) = 1,

(3.4)

22

G. Sh. Tsitsiashvili and M. A. Osipova X X A(s) ν(s, s∗ ) = A(s∗ )ν(s∗, s), s ∈ S, s∗ ∈S

(3.5)

s∗ ∈S

λ=µ

m Y λk , µk

(3.6)

k=1

then from the corollary 1.1 obtain that the process x(t) is ergodic and its limit distribution has the product form Π(s, n) = A(s)π(n), (s, n) ∈ X.

(3.7)

Remark 1. This statement remains true if the formulas (3.3), (3.6) are rewritten in the form L1 (n, n + c) = λ, L1 (n + c, n) = µ.

λ=µ

m Y λk ck k=1

µk

,

(3.8)

where c = (c1, . . . , cm) ∈ Y, ci > 0, i = 1, . . . , m. It means that instead of a single copy ck copies arrive to the node k. If n1 ≥ c1 , . . ., nm ≥ cm then c1, . . . , cm customers are served in the network nodes simultaneously and timing with the intensity µ and then all these customers depart from the network. Exchange between subsets of nodes set. Consider the network G with oneserver nodes and the structure s0 and divide the nodes set M = {1, . . ., m} into nonintersected and nonempty subsets M + , M − . Denote the vector c = (c1, . . . , cm) ∈ Y :

X

ci > 0,

i∈M +

X

ci > 0.

(3.9)

i∈M −

In a random time moment the network s0 with the state n = (n1 , . . ., nm ) may change its dynamics principle: if ni ≥ ci , i ∈ M + (i ∈ M − ), then in an each node i ∈ M + (i ∈ M − ) ci customers are taken and are served simultaneously and timing with the intensity µ+ (µ− ). Then these customers depart from the network and in the nodes i ∈ M − (i ∈ M + ) ci new customers appear. Denote the structure of this network by s2 = (M + , M − , µ+ , µ− , Y ). Describe the network with a variable structure by the discrete Markov process x(t) with the state set (3.1) where S = {s0 , s2} and with the transition intensities (3.2). Here   X X ck e k , n + ck e k  = µ+ , L2 n + k∈M +

k∈M −

Limit Distributions in Queueing Networks with Different Types...   X X L2 n + ck e k , n + ck e k  = µ− . k∈M −

(3.10)

k∈M +

Suppose that the conditions (3.4), (3.5) and Y λi ci Y λi ci + µ = µ− µ µ i i + − i∈M

23

(3.11)

i∈M

are true. Then from the corollary 1.1 obtain that the process x(t) is ergodic and its limit distribution has the product form (3.7). Remark 2. It is possible to define on the set Y the transitions of N types putting for s ∈ {s1 , . . . , sN } the sets M + (s), M −(s) : M + (s) ∩ M − (s) = ∅, M + (s) ∪ M − (s), the intensities µ− (s), µ+(s) and the vectors (c1(s), . . . , cm(s)) from (3.9). If the conditions (3.4), (3.5) are true where S = {s0, s1 , . . . , sN } and Y i∈M + (s)

λi µi

ci (s) µ− (s) =

Y i∈M −(s)

λi µi

ci (s) µ+ (s), s ∈ S,

(3.12)

then the process x(t) with the state set (3.1) and the transition intensities (3.2) is ergodic and its limit distribution has the form (3.7). Here Li (n, n ∗ ), i = 1, . . . , N, are calculated by the type (3.10) formulas and the matrix ||ν(s, s∗)||s,s∗ ∈S is indivisible.

3.2.

Appearance and disappearance of customers in network nodes

Arbitrary number of customers. Consider the network with a variable structure with a dynamics described by the discrete Markov process x(t) with the state set (3.1) where S = {s0 , s3} and with the transition intensities (3.2). Suppose that L3 (n, n ∗ ), differ from zero only for the pairs (n, n ∗) of the set Z ⊆ Y × Y and the regularity condition X L3(n, n ∗ ) < ∞ sup n ∈Y n ∗ : (n ,n ∗ )∈Z is true. It means that in the network s3 = ({1, . . ., m}, Z) nodes customers may appear and disappear without their arrival to the network and departure from the network and without their service and displacement in the network. If the conditions (3.4), (3.5) and Ψ(n)L3(n, n ∗ ) = Ψ(n ∗ )L3 (n ∗ , n), (n, n ∗ ) ∈ Z,

24


are true then by the corollary 1.1 the process x(t) is ergodic and its limit distribution has the product form (3.7). Definite number of customers. Consider the network with a variable structure described by the Markov process x(t) with the state set (3.1) where S = {s0 , s4} and with the transition intensities ( L0 (n, n ∗ ), s = s∗ , ∗ ∗ λ((s, n), (s , n )) = ν((s, n), (s∗, n ∗)), s 6= s∗ which satisfy the communicability and regularity conditions. If the formulas (3.4) are true and A(s0)Ψ(n)ν((s0, n), (s4, n ∗ )) = A(s4)Ψ(n ∗ )ν((s4, n ∗ ), (s0, n)) then by the theorem 1.1 the process x(t) is ergodic and its limit distribution has the product form (3.7). Remark 3. If S = {s0, sa }, a ∈ Y, and the transition intensities are defined by the formulas ( L0(n, n ∗), s = s∗ , ∗ ∗ λ((s, n), (s , n )) = ν(s, s∗), s 6= s∗ , n ∗ = n + a and A(s0 )ν(s0 , sa ) = Ψ(a)A(sa )ν(sa , s0) then the process x(t) properties conserve.

3.3.

Decrease and increase in some times numbers of customers in network nodes and group exchange between nodes by customers

Denote S = {s0 , sr } where r is positive and rational number. Consider the network with a variable structure and with a dynamics described by the discrete Markov process x(t) and the state set (3.1) and the transition intensities ( Li (n, n ∗ ), s = s∗ = si , (3.13) λ((s, n), (s∗, n ∗ )) = ν(s, s∗), s 6= s∗ , n ∗ = rn. Here Lr (n, n ∗) are defined similar to L0(n, n ∗) by known λ, Θ, (µ1 (r), . . . , µm (r)). If the formulas (3.4), (3.5) and r λk λk = , k = 1, m, (3.14) µk µk (r)

Limit Distributions in Queueing Networks with Different Types...

25

are true then by the corollary 1.1 the process x(t) is ergodic and its limit distribution has the form (3.7). In considered network model at random time moment numbers of customers in nodes change in equal times and then network change its structure. A dynamics of network model with group customers exchange between nodes is described in the remark 4. Remark 4. If R = ||rij ||m i,j=1 is nondegenerate matrix with rational elements, S = {s0 , sR}, and in the formulas (3.13) the equality n ∗ = rn is replaced by n ∗ = nR (here LR (n, n ∗), n, n ∗ ∈ Y, is defined similar to L0(n, n ∗) by (µ1 (R), . . ., µm (R), λ, Θ)) and the formula (3.14) is replaced by rij m Y λj λi = µi µj (R) j=1

then the statement concerning the process x(t) and its limit distribution remains true.

3.4.

Unreliable customers

System M|M|1| ∞. Suppose that customers of the considered system may fail with the intensity α and may be restored with the intensity β at m repair places. Describe this system dynamics by the discrete Markov process x(t) with the state set X={(n, n0) : n ≥ 0, n0 ≥ 0} (here n is a number of working customers, n0 is a number of nonworking customers in the system) and with positive transition intensities λ((n, n0), (n + 1, n0)) = λ, λ((n, n0), (n − 1, n0)) = µ, λ((n, n0), (n − 1, n0 + 1)) = α, λ((n, n0), (n + 1, n0 − 1)) = β min(m, n0) where λ is the input flow intensity and µ is the service intensity. If ρ=

α λ < 1, ρρ0 < 1, ρ0 = µ β

then by the theorem 1.1 the process x(t) is ergodic and its limit distribution has the form ∞ X 0 −1 0 n A(n0), Π(n, n ) = C A(n )(1 − ρ)ρ , C = n0 =0 0

0

A(0) = 1, A(n ) =

n Y k=1

(ρρ0)k , n0 > 0. min(k, m)

Analogously a case of r-server system may be considered.

26


Network with oneserver nodes.Consider the opened network G with the structure s0 (here Θ is indivisible) with m oneserver nodes, each node has its own repair place. Customers failed during their service may by repaired at these places. Denote the failure intensity by αj and the repair intensity by βj for the node j and put ρj =

λj 0 αj , ρj = µj βj

where the vector (λ, λ1, . . . , λm) is single solution of the system (1.4). Numbers of customers in the considered network nodes are described by the discrete Markov process x(t) with the state set X = {(n, n 0) : (n, n 0 ) ∈ Y × Y } (here n is the vector of working customers numbers n 0 is the vector of nonworking customers numbers at the network nodes) and with the positive transition intensities  0 0  L0(n, l), l = n , λ((n, n 0 ), (l, l 0)) = αj , l = n − e j , l 0 = n 0 + e j ,   βj , l = n + e j , l 0 = n 0 − e j . If ρj < 1, ρj ρ0j < 1, j = 1, m, then by the theorem 1.1 the process x(t) is ergodic and its limit distribution has the form Π(n, n 0 ) = CA(n 0 )Ψ(n), A(n 0 ) =

m m Y Y 0 (ρi ρ0i)ni , C = (1 − ρj )(1 − ρj ρ0j ). i=1

3.5.

j=1

Network with few types of customers

Consider the network G with m oneserver nodes and different types of customers. This network in the state s, s ∈ S = {1, . . ., l}, processes customers of single type. Suppose that in the state s it has the structure ({1, . . ., m}, λ(s), (µ1 (s), . . ., µm (s)), Θ(s), Y ) where the vector Λ(s) = (λ(s), λ1(s), . . . , λm(s)) is defined in a single way by the system Λ(s) = Λ(s)Θ(s), Θ(s) indivisible. Suppose that λk (s) < µk (s), k = 1, m, and the formulas (3.4), (3.5) with indivisible matrix ||ν(s, s∗)||s,s∗∈S are true. Describe a dynamics of this network with few types of customers and changeovers from one type to another by the Markov process x(t) with the state set X = {(s, N ) : s ∈ S, N ∈ Y S },

(3.15)

Limit Distributions in Queueing Networks with Different Types...

27

where N = (n s , s ∈ S) is the vector containing vectors of customers numbers in the network nodes n s = (ns1 , . . ., nsm ) for all customers types, Y S = Y × . . . × Y (here a number of multiplies in direct product coincides with a number of elements in the set S) and with transition intensities ( Ls (n s , b s ), s = s∗ , (3.16) λ((s, N ), (s∗, B)) = ν(s, s∗), s 6= s∗ , N = B where Ls (n s , b s ), n s , b s ∈ Y, are defined from the equalities (1.5), (1.6). So a changeover from a processing of one customers type to another is independently on numbers of customers of different types in the network nodes. By the corollary 1.1 the process x(t) is ergodic and its limit distribution Π(s, N ), (s, N ) ∈ X, has the form X Y Y Ψs (n s ), C = Ψs (n s ), (3.17) Π(s, N ) = A(s)C −1 S s∈S s∈S N ∈Y m n Y λi (s) si s . Ψs (n ) = µi (s) i=1

Remark 5. Suppose that in the state l + 1 of the considered network there is a possibility to appear and to disappear for customers of some types from the set {1, . . ., l} without their arrival, departure, service and displacement between the network nodes. Then the formula (3.17) is true if the network dynamics is in an accordance of the first point in the subsection 3.2.

3.6.

Network with “negative” customers flow

Consider the network G with the structure s0 and with an additional Poisson input flow with the intensity λ− . A customer of the additional flow (call this customer m X δk = 1, and “negative”) arrives to the node k server with the probability δk , k=1

immediately departs from the network capturing a main flow customer if it is at this server. Denote this network structure by − − − s− 0 = ({1, . . ., m}, λ, (µ1 , . . . , µm ), Θ , Y ) − m − − where µ− j = µj + λ δj , j = 1, m, Θ = ||θij ||i,j=0 , and − θ0j

= θ0j , j=0, m,

− θij

m X µi − − = θij − , i, j=1, m, θi0 = 1 − θij , i=1, m. µi j=1

The matrix Θ− is indivisible and there is the single solution of the system − − − − − − (λ, λ− 1 , λ2 , . . . , λm ) = (λ, λ1 , λ2 , . . . , λm )Θ .

28


Numbers of customers in the network s− 0 nodes describe by the Markov process y(t) with the state set Y and the transition intensities (1.5), (1.6) which are defined by the network parameters. If − λ− i < µi , i = 1, m,

then the process y(t) is ergodic and its limit distribution (see preliminaries) has the form m − ni m Y Y λi λ− − i , n ∈ Y, C = 1− − . π (n) = C µ− µi i i=1 i=1

§ 4. Optimization of Queueing Network Ability to Handle Customers In this paragraph a problem of a choice of a route matrix which optimizes an ability to handle customers in an opened queueing network is considered. The ability to handle customers is defined here as a maximal possible input flow intensity for which the network is not overloaded. Conditions of a limitation by a probability for numbers of customers in the network nodes [9] allow to calculate the ability to handle customers and to represent it as a grainy function of mean service times in the network nodes and of mean numbers of visits of the network nodes by a customer. A route matrix which maximizes the ability to handle customers is calculated as a permissible solution of a special type transportation problem. Note that in [6] a customer route optimization problem is solved but with a criterion of a mean delay of a customer in the network.

4.1.

Problem formulation

Consider the opened queueing network with m servers (oneserver queueing systems with the service discipline FCFS) and with the internal source in the node 0. Customers service times in the network nodes are independent and identically distributed (i.i.d.) random variables with the distribution functions (d.f.) Gi (t) and mean values gi > 0 and for some C > 0 the equalities Gi (C) = 1, 1 ≤ i ≤ m,

(4.1)

are true. The recurrent input flow to this network is 0 = t1 ≤ t2 = t1 + ξ1 ≤ t3 = t2 + ξ2 ≤ · · · where {ξn , n ≥ 1} is the sequence of i.i.d. random variables with d.f. F (t), Eξ1=f, 0 < f < ∞. The input flow intensity En(x) , n(x) = sup(k : tk < x), x→∞ x

λ = lim

is defined by the integral renewal theorem [7, chapter 9, § 4] and satisfies the equality λ = f −1 . A customer displacement in the network is defined by the route matrix Θ = ||θij | |m i,j=0 . Each customer in an accordance with the route matrix has from an arrival to departure moment its own random sequence of attended nodes 0, i1, i2, . . . , ir, 0. This sequence create a Markov chain with the transition probabilities matrix Θ. Suppose that these random ways are independent. Denote the service time of the input flow customer n at the node i during the j . Put the total time of the customer n service at the node i by attendance j by ζi,n 29

30


ηi,n and the total number of the n attendances at the node i by Ni,n , ENi,n = Mi . Then Ni,n X j ζi,n , (4.2) ηi,n = j=1

and the sequences of random variables {ξn , n ≥ 1}, {Ni,n, n ≥ 1}, 1 ≤ i ≤ m, j , n ≥ 1, j ≥ 1} are independent and each of them consists of i.i.d. random {ζi,n variables (with its own common distribution for each sequence). From the equality (4.2) and the Wald identity [7, chapter 14, § 2] obtain Ti = Mi gi.

(4.3)

Denote by Xi (t) the total residual service time after the time moment t at the node i of customers which arrive to the network up to the moment t, X(t) = m X Xi (t). Say that the random process X(t) is bounded by a probability if i=1

∀ε > 0 ∃ A > 0 : ∀ t > 0 P (X(t) > A) < ε. Fix d.f.‘s G1 (t), . . ., Gm (t) and the route matrix Θ and by known F (t) define is the process X(t) bounded by a probability or not. At first prove that a respond to this question depends only on the mean f of d.f. F (t) and so from the input flow intensity λ = f −1 . Consequently it is correct to define an ability to handle customers λG = λG (G1, . . . , Gm, Θ) of the network G as follows: λG = sup(λ : X(t) is bounded by a probability). A second aim of this paragraph is an optimization of the ability to handle customers λG by the route matrix Θ defined by boundaries on mean numbers of attendances Mi , i = 1, . . . , m.

4.2.

Calculation of ability to handle customers

Opened network. Suppose that the route matrix Θ is indivisible. Then the vector Λ = (λ, λ1, λ2, . . . , λm), λi = λMi , i = 1, . . . , m, is [4] the single solution of the system Λ = ΛΘ. From the condition (4.1) and the statement of [9, theorem 1.1, (i)] (see also [19, theorem 2]) the following theorem is true. Theorem 4.1. If the condition λi gi < 1, 1 ≤ i ≤ m,

(4.4)

takes place then the random process X(t) is bounded by a probability. Theorem 4.2. If the condition (4.4) is not true and there is i, 1 ≤ i ≤ m, so that λi gi > 1 then the random process X(t) is not bounded by a probability.

Optimization of Queueing Network Ability to Handle Customers

31

Proof. Suppose that there is i, 1 ≤ i ≤ m, so that λigi > 1. Define the random process Yi (t) by the equalities Yi (t) = 0, t = t1 = 0, Yi (tn + 0) = Yi (tn ) + ηi,n , Yi (t) = [Yi (tn + 0) − (t − tn )]+ , tn < t ≤ tn+1 , n ≥ 1, where d+ = max(0, d) for a real number d. By a definition Xi (t) = 0, Yi (t) = 0, t = t1 = 0, then by an induction for n ≥ 1 obtain Xi (tn ) ≥ Yi (tn ), Xi (t) ≥ Yi (t), tn < t ≤ tn+1 , n ≥ 1, and so almost surely (a.s.) Xi (t) ≥ Yi (t), t ≥ 0.

(4.5)

Construct the random sequence Zi,n = Yi (tn ), Zi,1 = 0, Zi,n+1 = [Zi,n + ηi,n − ξn ]+ , n ≥ 1, and then the random process Zi (t) = 0, t = t1 = 0, Zi (t) = Zi,n+1 , tn < t ≤ tn+1 , n ≥ 1. By a definition of the process Zi (t) and the inequality (4.5) obtain that a.s. Xi (t) ≥ Yi (t) ≥ Zi (t), t ≥ 0.

(4.6)

As f > 0 so from the strong law of large numbers a.s. tn /n → f, n → ∞, and so a.s. (4.7) lim tn = ∞. n→∞

Also from the strong law of large numbers and the equalities λ = f −1 , (4.3) and the condition λi gi = Mi gi /f = Ti/f > 1 obtain that a.s. lim Zi,n = ∞ and so from n→∞

(4.6), (4.7) the following equalities are true a.s. lim Zi (t) = lim Xi(t) = ∞.

t→∞

t→∞

(4.8)

From (4.8) and known statements of the theorem which connects a.s. and weak tendencies [43, ch. 3, § 8] obtain that for any A > 0 lim P (Xi (t) > A) = 1 =⇒ lim P (X(t) > A) = 1.

t→∞

t→∞

The theorem 4.2 is proved. Corollary 4.1. The ability to handle customers of the opened network G satisfies the equality −1 . (4.9) λG = max Mi gi 1≤i≤m

32

G. Sh. Tsitsiashvili and M. A. Osipova This result is a corollary of the theorems 4.1, 4.2 and the formulas (4.3), (4.4).

Remark 1. If the node i is ri-server queueing system then in the formula (4.9) it is necessary to replace gi by gi /ri. Queueing system. Consider the system H1 = M |M |∞ with Poisson input flow with the intensity λ and infinite number of servers, with exponentially distributed service times with the parameters ν1 , ν2, . . . Customers of this system fill the servers 1, 2, . . . consequently. A number of customers in the system H1 is defined by the birth and death process x(t) with the death and birth intensities µ1 = ν1 , µ2 = µ1 +ν2 , . . . and λ. A number of customers of the system H2 = M |M |1|∞ with Poisson input flow with the intensity λ and with exponentially distributed service times with the intensities µn , depending on a number of customers n ≥ 0, is described by the death and birth process x(t) with the death and birth intensities µ1 , µ2, . . . and λ. Define abilities to handle customers of the systems H1 , H2 by the equality λi = sup(λ : the process x(t) is ergodic), i = 1, 2.

(4.10)

The death and birth process x(t) with the state set {0, 1, . . .}, and the death and birth intensities µ1 , µ2, . . . and λ is ergodic [28], [29] if and only if the conditions ∞ X λ0 . . . λk−1 k=1

µ1 . . . µk

< ∞,

∞ X µ1 . . . µk k=1

λ1 . . . λk

=∞

(4.11)

are true. From the formulas (4.10), (4.11) and the Koshi-Adamar theorem [18] obtain λi = lim inf (µ1 . . . µk )1/k , i = 1, 2. k→∞

4.3.

Minimization of vector components maximum

Consider a problem of a minimization of vector components maximum − → − → Φ(M) = max Mi gi =⇒ min, M = (M1 , . . ., Mm ) ∈ M, 1≤i≤m

(4.12)

− → where the set M consists of the vectors M = (M1, . . . , Mm ) so that M1 > 0, . . ., Mm > 0,

m X

Mk ≥ 1,

(4.13)

k=1

and satisfies the conditions: a) the set M is convex, closed and has smooth boundary, − → − → − → b) ∃ M ∗ ∈ M so that the minimum Φ(M ) by all M ∈ M satisfying the equalities − → M1 g1 = . . . = Mm gm is in the point M ∗.

Optimization of Queueing Network Ability to Handle Customers

33

Theorem 4.3. Suppose that the set M satisfies the conditions a), b). If the plain − → − → − → L tangent to M at the point M ∗ may be described by the equation M · A = d in → − which the vector A = (A1, . . . , Am) satisfies the conditions A1 > 0, . . . , Am > 0,

A1 Am + ...+ =1 g1 gm

(4.14)

− → then the single solution of the problem (4.12) is the point M ∗ . − → − → Proof. Put Γd = {M : Φ(M) ≤ d}. As the set M is convex and the plain L is − → tangent to the boundary M at the point M ∗ then from the conditions (4.14) m

− → − → Ai − → − → X Mi gi ≤ Φ(M ), M ∈ M. d≤M· A = gi i=1

− → − → − → Suppose that M 0 6= M ∗ , Φ(M 0) = d, then there is the index j for which Mj gj < d. So from the conditions (4.14) obtain that → − → − M0 · A =

X 1≤i≤m, i6=j

Mi gi

Ai Aj + Mj gj < d. gi gj

− → − → / M and so M ∗ is the single From the last two inequalities chains establish that M 0 ∈ solution of the problem (4.12). The theorem 4.3 is proved. Remark 2. The theorem 4.3 defines a case which is important for queueing networks and when a maximum of an ability to handle customers reaches for equal load coefficients in different network nodes. A described algorithm of the problem (4.12) solution is assumed in [36] for a deterministic vector optimization problem.

4.4.

Maximization of ability to handle customers by route matrix

Suppose that the set M satisfies the conditions a), b) and the problem (4.12) has − → the single solution M (k) . For the route matrix Θ with positive elements (besides of − → − → θ00 ) define the vector M = M (Θ) = = (M1 , . . ., Mm ) as the single solution of the system (4.15) (1, M1, M2, . . . , Mm) = (1, M1, M2, . . . , Mm)Θ − → and put Θ(M) = {Θ : M (Θ) ∈ M}. Consider a problem of an ability to handle customers of the network G maximization by the route matrix λG (G1, . . . , Gm, Θ) =⇒ max, Θ ∈ Θ(M),

(4.16)

and the equivalent problem − → Φ(M(Θ)) =⇒ min, Θ ∈ Θ(M).

(4.17)

34

G. Sh. Tsitsiashvili and M. A. Osipova (k)

Denote M0

= 1 and (k)

ψi,j = Mi θij , 0 ≤ i, j ≤ m.

(4.18)

Theorem 4.4. The matrix Θ is the problem (4.16) solution if and only if the matrix ||ψi,j ||m i,j=0 is a permissible solution of the transportation problem m X j=0

ψi,j =

m X

(k)

ψj,i = Mi , 0 ≤ i, j ≤ m.

j=0

From the equalities (4.9), (4.15), (4.18) and the problems (4.16), (4.17) equivalence obtain that the theorem 4.4 becomes a direct corollary of the theorem 4.3. Remark 3. The most suitable for a maximization of an ability to handle customers in the network G is a choice of boundaries not on the route matrix Θ but on the mean numbers of visits M1 , . . . , Mm of the network G nodes.

§ 5. Superposition of Queueing Networks In this paragraph a problem of a calculation of limit distributions in a superposition of queueing networks is solved. A superposition of queueing network is an opened or closed network in which some nodes are replaced by opened networks. Limit distributions in these superpositions are defined by limit distributions in initial networks using product formulas. Such constructions allow to calculate their distributions in a few steps of networks superposition. In this paragraph abilities to handle customers in a superposition of networks are calculated. Network superpositions may be interpreted as recursively defined structures. Such structures now are widely used in a nanotechnology.

5.1.

Product theorem

Consider opened Jackson networks G, G0 described in § 1 with the sets of oneserver nodes {g0 , g1, . . . , gm}, {g00 , g10 , . . . , gr0 }, with the input Poisson flows (with single intensities) and with the intensities µ1 , . . . , µm and µ01 , . . ., µ0r . Denote by service 0 0 r 0 Θ = ||θij | |m i,j=0 , Θ = |θij |i,j=0 the route matrixes of the networks G, G . Define m

the superposition G = G ⊗ G0 of the networks G, G0 by a replacement of the node gm in G by the network G0 . Here an input flow (output flow) of the network G0 is created from customers arriving (departing) to the node (from the node) gm . In the network G the input flow is Poisson with the single intensity, the nodes set is {g0 , g1 , . . . , gm+r−1 } = {g0, g1, . . . , gm−1, g10 , . . ., gr0 } is defined from the formulas and the route matrix Θ = |θ ij |m+r−1 i,j=0 θ ij = θij , i, j = 0, 1, . . ., m − 1, 0 θm−1+i m−1+j = θij , i, j = 1, . . ., r, 0 θi m−1+j = θim θoj , i = 0, 1, · · · − 1, j = 1, . . ., r,

0 θm−1+i j = θio θmj , i = 1, . . . , r, j = 0, 1, . . ., m − 1. satisfies properties of a Lemma 5.1. If θmm = 0 then the matrix Θ = |θij |m+r−1 i,j=0 route matrix and is indivisible.

Proof. The matrix Θ is route as all its elements are nonnegative and m+r−1 X j=0

θ ij =

m−1 X j=0

θij +

r X

0 θim θ0j = 1 − θim + θim = 1, i = 0, 1, . . ., m − 1,

j=1

35

36


and for i = 1, . . . , r m+r−1 X

θ m−1+i j =

j=0

m−1 X

0 θio θmj

j=0

+

r X

0 0 0 θij = θio (1 − θmm ) + 1 − θi0 = 1.

j=1

Show that for ∀ i, j ∈ {0, 1, . . ., m + r − 1} ∃ k1, . . . , ks ∈ {1, . . ., m + r − 1} : θik1 , θk1 k2 , . . . , θks j > 0.

(5.1)

If i, j ∈ {0, 1, . . . , m − 1} (i, j ∈ {m, . . . , m + r − 1}) then the formulas (5.1) may be obtained from the matrix Θ (the matrix Θ0 ) indivisibility. If i ∈ {0, 1, ..., m−1}, j = {m, . . ., m + r − 1} then from the matrix Θ indivisibility ∃ i1, . . . , is ∈ {1, . . ., m − 1} : θii1 = θii1 > 0, . . ., θis−1 is = θis−1 is > 0, θism > 0, and from the matrix Θ0 indivisibility ∃ j1, . . . , jn ∈ {1, . . ., r} : θ00 m−1+j1 >0, θ m−1+j1 m−1+j2 = θj0 1 j2 >0, . . ., θm−1+jn j = θj0 n j−m+1 >0. As θ is m−1+j1 = θis m θ00 m−1+j1 then θii1 >0, . . . , θis−1 is >0, θis m−1+j1 >0, θm−1+j1 m−1+j2 >0, . . . ., θm−1+jn j >0. The case i ∈ {m, . . ., m + r − 1}, j ∈ {0, 1, . . ., m − 1} is considered similar. The lemma 1 is proved. From the lemma 1, [4, § 2] there is the single vector Λ so that Λ = Λ Θ, Λ=(1, λ1 , . . . , λm+r−1 ).

(5.2)

Lemma 5.2. If θmm = 0 then λi = λi , i = 1, . . . , m − 1, λm−1+i = λm λ0i, i = 1, . . . , r. Here Λ = (1, λ1, λ2, . . . , λm), Λ0 = (1, λ01, λ2, . . . , λ0r ) are solutions of the systems Λ = ΛΘ, Λ0 = Λ0Θ0 . Proof. Indeed if i ∈ {1, . . ., m − 1} then from the formula (5.2) obtain λi = λi = θ 0i +

m+r−1 X j=1

λj θ ji = θ0i +

m−1 X

λj θji +

j=1

r X

0 λm λ0j θj0 θmi = λi .

j=1

If i ∈ {1, .., r} then λmλ0i = λm−1+i = θ0 m−1+i +

m+r−1 X j=1

λj θ j m−1+i =

Superposition of Queueing Networks =

0 θ0i θ0m

+

m−1 X

0 λj θ0i θjm

+

r X

j=1

=

0 θ0m θ0i

+

0 θ0i (λm

37

0 λm λ0j θji =

j=1

0 − θ0m − λm θmm ) + λmλ0i − λmθ0i = λm λ0i.

The lemma 2 is proved. Describe a dynamics of the network G by the discrete Markov process y(t) with the state set Y = {y = (n1, . . . , nm−1 , n01, . . . , n0r ) : n1 , . . . , nm−1, n01 , . . ., n0r ≥ 0}. Theorem 5.1. If θmm = 0 and ρi = λi/µi < 1, i = 1, . . ., m − 1, ρ0i = λm λ0i/µ0i < 1, i = 1, . . ., r, then the Markov process y(t) is ergodic and its limit distribution P (y), y ∈ Y , has the form m−1 r Y Y P (x) = ai (ni ) a0i (n0i ), i=1

ai (ni ) = (1 −

ρi )ρni i ,

i ∈ {1, . . ., m},

i=1

a0i (n0i )

0

= (1 − ρ0i)(ρ0i)ni , i ∈ {1, . . ., r}.

Proof. The network G is Jackson network and so the theorem statement is a corollary of the lemmas 1, 2 and [4, theorem 2.1]. b described in § 1, with the nodes set Consider now the closed network G, b = |θbij |m b1 , . . . , µ bm and the route matrix Θ {g1, . . . , gm}, the service intensities µ i,j=1 m

b ⊗ G0 and the constant number K of customers. Define now the superposition Gc =G 0 b of the closed network G and the network G by a replacement of the node gm in b by the network G0 . A dynamics of the network Gc describe by the the network G discrete Markov process y c (t) with the state set ( ) m−1 r X X 0 0 0 Y c = y c =(n1 , . . . , nm−1 , n1, . . . , nr ) ∈ Y : ni + ni = K . i=1

i=1

It is simple to prove for the network Gc the lemmas 1,2 analogs and the following theorem. Theorem 5.2. If θmm = 0 then the process yc (t) is ergodic and its limit distribution P c (yc ), y c ∈ Y c , has the form P c (yc ) = C −1

m−1 Y i=1

ai (ni ) b

r Y

a0i (n0i ), C =

i=1

X m−1 Y y c ∈Y c i=1

ai (ni ) b

r Y i=1

ρini , ρbi = b λi/b µi , i ∈ {1, . . . , m}. b ai (ni ) = (1 − ρbi )b

b1, . . ., b λm) is the system (1.9) solution. Here (λ

a0i (n0i ),

38

5.2.


Abilities to handle customers of opened networks

Construct the following sequence of the opened networks. Replace each node of the network G by the network G and denote this network by G(1). In the network G(1) replace each node by the network G and obtain the network G(2) and so on. After n steps obtain the network G(n) with mn+1 nodes among which there are mn nodes with service intensity µk . Denote the route matrix of the network G(i) by Θ(i) and suppose that the single solution of the system Λ(i) = Λ(i) Θ(i) is Λ(i) = (i) (i) (1, λ1 , . . ., λmi+1 ). If θii = 0, i = 1, . . ., m, then from the lemma 2 the network G(1) satisfies the coinditions (1)

(1)

(1)

λk = λk λ1, λm+k = λk λ2, . . . , λm2−m+k = λk λm , k = 1, . . . , m, and the network G(2) satisfies the coinditions (2)

(1)

(2)

(1)

(2)

(1)

λk = λk λ1 , λm+k = λk λ2 , . . . , λm3−m+k = λk λm2 , k = 1, . . . , m, and so on and the network G(n) , k = 1, . . . , m, satisfies the coinditions (n)

(n−1)

λk = λk λ1

(n)

(n−1)

, λm+k = λk λ2

(n)

(n−1)

, . . ., λmn+1 −m+k = λk λmn

.

So in the network G(n) nodes with the service intensities µk correspond to the vector Λ(n) components of the form (n−1)

λk λj

(n−1)

, where λj

= λh1 1 λh2 2 · · · λhnn , h1 + · · · + hn = n.

Calculate now the ability to handle customers an of the network G(n) . From the statement 4.1 obtain µk µk 1 n min = min . min an = 1≤j≤n λj 1≤k≤m, 1≤j≤mn λ λ(n−1) 1≤k≤n λk k j 1 1 < 1 then an → 0, n → ∞, if min > 1 then an → ∞, n → ∞, if 1≤j≤n λj 1≤j≤n λj 1 µk = 1 then an ≡ min . min 1≤j≤n λj 1≤k≤n λk

If min

§ 6. Embrechts–Veraverbeke Formula in Multiserver Queueing Models In this paragraph mathematical models of multiserver queueing systems with a competition of servers, with a competition of customers and with a linear dependence of service and interarrival times on waiting times are considered. These models are analyzed using the Embrechts–Veraverbeke formula and its multivariate analog. A comparison of tails of stationary waiting times distributions for a system with a competition of servers and without it shows that a competition may significantly improve a quality of a multiserver system. An analogous result is obtained for an ability to handle customers in these systems. A comparison analysis allow to get over a complicated procedure of accuracy estimates for the tails of stationary waiting times distributions in a classical multiserver queueing system [20]. Upper and low bounds of tails of stationary waiting times distributions are obtained in a queueing system with a linear dependence of service and interarrival times on waiting times. An asymptotic equivalence of these bounds is proved. This result is generalized onto a multiserver queueing system with the same dependence by a multivariate analog of the Embrechts–Veraverbeke formula.

6.1.

Preliminaries

Suppose that F (x) = P (f ≤ x) is a distribution function (d.f.) of a nonnegative random variable f and F (x) = 1 − F (x). Denote S=

F (x), x ≥ 0 :

Z

x

F (x − u)dF (u) ∼ 2F (x), x → ∞ 0

the class of subexponential distributions [2] and introduce the following classes of d.f‘s [23]: Z ∞ F (t)dt < ∞, S∗ = {F (x), x ≥ 0 : F (x) < 1, µ = 0

Z

x

F (x − u)F (u)du ∼ 2µF (x), x → ∞}, 0

L∗ = {F (x) : ∀M > 0 lim

t→∞

F (t + M ) = 1} , F (t)

0 xl (x) < ∞} , R(a) = {xa l(x), l(x) ∈ L}. L1 = {l(x) ∈ L : lim sup l(x) x→∞ Here L is the class of positive, continuously differentiable on [0, ∞) and slowly varying functions, ∞ < a < ∞. List [16], [23] main properties of these classes of d.f.‘s. 39

40


Property 1. The inclusion S∗ ⊂S⊂L∗ is true. If F (x)∈R(a), a x) = G (x), x ≥ 0, Eζ1 = bm . Denote by wn the waiting time of n-th customer and put W (x) = lim P (wn > x), n ≥ 1. n→∞

The function W (x) is defined for the systems A, B by d.f.‘s G(x), H(x). Fix G(x) and determine ) ( −1 Z ∞ H(t)dt by all H(x) : lim W (x) = 0 Λ = Λ(G) = sup λ = x→∞

0

an ability to handle customers of the systems A, B. A concept of the ability to handle customers is based on the integral renewal theorem from which it is possible to obtain that Z ∞ −1 1 En(x) = = H(t)dt , n(x) = sup(k : tk < x). lim x→∞ x Eξ1 0 For the system B the following statement is true: lim W B (x) = 0 if and only if x→∞

bm < Eξ1. So for B the formula Λ =

Z

∞

m

−1

G (t)dt

(6.2)

takes place. The system A is described

0

by the Kiefer-Wolfowitz Markov chain [31] (wn, 1 , wn, 2, . . . , wn, m ), n ≥ 1, where wn, i is an interval between the moment tn and the moment when i servers become free of 1-st,. . . , (n − 1)-th customers of the input flow. A necessary and sufficient condition of the equality lim W A (x)=0, W A (x) = lim P (wn, 1 > x),

x→∞

n→∞

is the inequality Eη1 < mEξ1 −1 Z ∞ G(t)dt . from which it is possible to obtain Λ = m 0

(6.3)

42


Our problem is to make asymptotic investigation of the ratio of the systems A, B abilities to handle customers R∞ m m 0 G (t)dt (6.4) Ω= R∞ 0 G(t)dt and to make a comparison asymptotic analysis of the functions W A (x), W B (x), x→∞. Theorem 6.1. Suppose that for some a > 0, b > 0 the formula G(x) ∼ axb , x → 0,

(6.5)

Ω = O(m1−1/b), m → ∞.

(6.6)

is true then for 0 < b ≤ 1

Theorem 6.2. Suppose that for the system B the condition (6.2) is true and d.f. G(t) satisfies the condition a) or the condition b) of the lemma 6.1. Then for any m > 0 the formula R∞ m G (t)dt W B (x) ∼ x , x → ∞, (6.7) Eξ1 − bm is true. Remark 1. Consider the following two d.f.‘s G(x), x ≥ 0, satisfying the condition (6.5) and the conditions of the theorem 6.2: the Weibull distribution G(x) = exp(−axb ), 0 < b < 1, and the Burr distribution G(x) = (1 + cxb )−a/c , 0 < a, 0 < b ≤ 1, 0 < c < ab. Remark 2. In [20, remark 2] for the system A it is proved that if d.f. G(x) ∈ S and the condition Eη1 < Eξ1 is true then Z ∞ m G(t)dt , x > 0. (6.8) ∃ K > 0 : W A (x) ≥ K x

The replacement ξi → ξi /m and the equality P (ξi ≤ mξi ) = 1 lead to the fact that the inequality (6.8) is true in the condition (6.3) also. Then it is possible to show that the tail of the stationary waiting time distribution in the system B is lighter then in the system A. Namely if the condition a) of the lemma 6.1 is true then integrating by parts obtain from the formula (6.1) Z ∞ −1 m m , G (t)dt ∼ xG (x) ma + 1 x m Z ∞ −1 m m m G(t)dt ∼ x G (x) , x → ∞, a+1 x Analogously if the condition b) of the lemma 6.1 is true then Z ∞ m Z ∞ m m G (x) G (x) m , , x → ∞. G (t)dt ∼ G(t)dt ∼ m mq(x) q (x) x x

Embrechts–Veraverbeke Formula in Multiserver Queueing Models

43

Remark 3. In the theorems 6.1, 6.2 a comparison of the abilities to handle customers in the systems A and B is defined by the function G(x) behavior for x → 0. And a comparison of the tails of stationary waiting times is defined by the function G(x) behavior for x → ∞.

6.3.

Systems with a competition of customers

In this section a multiserver queueing system with a customers competition for a minimal waiting time is considered. This consideration is compared with the previous section consideration and it is shown that the servers competition decreases significantly the tail of the waiting time limit distribution in a comparison with the customers competition. Consider a queueing system D|GI|m|∞ with the input flow 0 = t1 ≤ t2 = t1 + 1 ≤ t3 = t2 + 1, . . . and a group arrival. An arriving group n consists of m customers with numbers 1, . . . , m. The customer with the number i of the group n is directed to the server (i) (i) i and has the service time ηn . The random variables ηn , n ≥ 1, 1 ≤ i ≤ m, are i.i.d. with the common d.f. G(t) satisfying the condition a) or the condition b) of (i) the lemma 6.1, b = Eηn < 1. Suppose that at each server there is FCFS service (i) discipline. Denote wn the waiting time of n-th arriving customer in i-th server, (i) w1 = 0, 1 ≤ i ≤ m. Then the following recurrent formulas are true: (i)

wn+1 = max(0, wn(i) + ηn(i) − 1), 1 ≤ i ≤ m, n ≥ 1,

(6.1)

(i)

and {wn , n ≥ 1}, 1 ≤ i ≤ m, are i.i.d. random sequences. Put un = min(wn(1), . . ., wn(m)) and suppose that a customer with a minimal waiting time in the group n is a winner among this group customers. Denote a described system of customers competition by C and put W (t) = lim P (wn(i) > t), 1 ≤ i ≤ m, n→∞

m

W C (t) = lim P (un > t) = W (t). n→∞

(6.2)

Using the formula (6.7) for m = 1 and (6.2) obtain the following statement. Theorem 6.3. For customers-winners the waiting time limit distribution satisfies the following asymptotic formula: m GI (t) W C (t) ∼ , t → ∞. (6.3) 1−b Remark 4. The tail (6.7) of the waiting time limit distribution in the system B is significantly lighter than the tail (6.3) of the customers-winners waiting time limit distribution in the system C.

44

6.4.


A generalized Lindley model

In this section a generalized Lindley model is considered in an assumption of heavy tailed service time distribution. This model and its asymptotic analysis allow to investigate a linear dependence of service and interarrival times on waiting times in oneserver queueing system. An asymptotic analysis based on the Embrechts Veraverbeke formula gives simple formulas for tails of waiting times limit distributions. These formulas are expanded onto multiserver generalized Lindley model with group arrival and a competition of customers in groups for a minimal waiting time. A comparison of the tails asymptotic is made. A onedimensional model. Consider a generalization of Lindley‘s recursion ( (bn − an + wn )+ , yn = 1, + (6.1) wn+1 = (bn − an + yn wn ) = (bn − an − wn )+ , yn = −1. Here (yn , an , bn), n ≥ 0, is the sequence of i.i.d. random vectors with independent components, I(A) is the indicator function of the event A, a+ = max(0, a). The nonnegative random variables an , bn have d.f.‘s A(t), B(t) relatively and for some p, 0 ≤ p < 1, the equalities P (yn = 1) = p, P (yn = −1) = 1 − p are true. The equation (6.1) reduces to the classical Lindley recursion [34] for p = 1 in which wn is the waiting time, bn is the service time of customer n and an is the interarrival time between customers n and n + 1. If p = 0 then (6.1) describes the waiting time of the server in an alternating service model with two service points [38], [51]. If 0 < p < 1 then the equation (6.1) may be interpreted as a special case of a queueing model in which service bn and interarrival an times depend on waiting wn times as follows [8], [52]: wn+1 = (bn − an + wn )+ , ( ( an , yn = 1, bn , yn = 1, an = bn = an + awn , yn = −1, bn + (a − 2)wn, yn = −1. Then an , bn are called the nominal service time of customer n and the nominal interarrival time between customers n and n + 1 and the constant a > 2. Then an asymptotic formula for limit distribution of waiting time is obtained in a case of heavy tailed service time distribution by means of the Embrechts– Veraverbeke formula. Suppose that w0 ≥ 0 has an arbitrary deterministic initial meaning. Theorem 6.4. If P (y1 ≤ 0, b1 − a1 ≤ 0) > 0 and d.f. B(t) ∈ S then W (t) ∼

B(t) , t → ∞, 1−p

where lim Wn (t) = W (t), Wn (t) = P (wn ≤ t). n→∞

(6.2)


45

Remark 5. If the recurrent formula (6.1) is replaced by the formula wn+1 = f (wn , an , bn, yn ), n ≥ 1, in which f (w, a, b, y) = I(y = 1)f1 (w, a, b) + I(y = −1)f2 (w, a, b), w + b − a ≤ f1 (w, a, b) ≤ w + b, b − a − w ≤ f2 (w, a, b) ≤ b, then lim inf W n (t) ∼ lim sup W n (t) ∼ n→∞

n→∞

B(t) , t → ∞. 1−p

This asymptotic formula is true without the condition of the function W (t) existence. Its proof is similar to the formula (6.2) proof. A multivariate Embrechts–Veraverbeke formula. Suppose that i.i.d. random (1) (m) vectors with independent and nonnegative components b n =(bn , . . . , bn ), n ≥ 1, have the common d.f. (t =(t1 , . . . , tm ) ≥ 0) B(t ) = B (1) (t1 ) · . . . · B (m) (tm ), B (i) (ti ) = P (b(i) n ≤ ti ).

(6.3)

Denote (1)

(m)

Bk (t ) = Bk1 (t1 ) · . . . · Bkm (tm ), k ∈ Q, Q = {k = (k1, . . . , km) : ki = 1, 2, . . ., i = 1, m},   X (i)  b(i) where Bki (ti ) = P  n ≤ ti . Designate for d.f. C(t ) of the random vector 1≤n≤ki

c = (c1, . . . , cm) C(t ) = P (c > t ) (an inequality between vectors c, t means that the same inequalties are true for their components). Theorem 6.5. If d.f.‘s B (1) (t1), . . . , B (l)(tm ) ∈ S and for some a > 0 X

Π(k) exp(aH(k)) < ∞, H(k) =

V (t) ∼ B(t)

X k∈Q

where V (t) =

X k∈Q

ki,

(6.4)

i=1

k∈Q then

m X

K(k)Π(k), t → ∞, K(k) =

m Y

ki ,

i=1

Π(k)Bk (t) and Π(k) is probability distribution.

(6.5)

46


Corollary 6.2. Suppose that Π(k) = π(k) if k = (k, . . . , k) ∈ Q where π(k), k = 1, 2, . . ., is probability distribution with final moment m then V (t) ∼ B(t)

∞ X

km π(k), t → ∞.

k=1

A multidimensional model. Consider a multidimensional generalization of the model (6.1). Suppose that m-dimensional vectors of waiting times in this model are defined by the recurrent random sequence w n , n ≥ 0, with an arbitrary initial meaning w 0 ≥ 0 : ( (w n +b n −an E )+ , yn = 1, + (6.6) w n+1 =(b n −an E +yn w n ) = (b n −an E −w n )+ , yn = −1. Here (yn , an, b n ), n ≥ 0, is the sequence of i.i.d. random arrays with independent components, m-dimensional vector E = (1, . . ., 1). The random variables an , yn were defined in a description of the onedimensional model. I.i.d. random vectors b n = (1) (m) (bn , . . . , bn ) with independent and nonnegative components have the common d.f. + (6.3), the vector x + = (x1 , . . ., xm )+ = (x+ 1 , . . . xm ). This model describes m-server queueing system with recurrent input flow 0=t1 ≤ t2 = t1 + a1 ≤ t3 = t2 + a2 ≤ . . . and group arrival. An arriving group n consists of m customers with numbers 1, . . . , m. The customer with the number i of the (i) group n is directed to the server i. It has the service time bn and the waiting time (i) (i) (i) (i) (i) wn : w1 = 0, wn+1 = (bn − an + yn wn )+ , n ≥ 1. Suppose that a customer with (1)

(m)

a minimal waiting time un = min(wn , . . . , wn ) in the group n is a winner in a competition among this group customers. Denote a described system of customers competition by D. Using the corollary 6.2 it is possible to obtain the following statement. Theorem 6.6. If d.f.‘s B (1) (t1 ), . . . , B (m)(tm ) ∈ S then the functions W n (t) = P (wn > t), n ≥ 1, satisfy the formula for t → ∞ lim sup W n (t) ∼ lim inf W n (t) ∼ B(t) n→∞

n→∞

∞ X

pk−1 (1 − p)km .

(6.7)

k=1

Corollary 6.3. If d.f.‘s B (1) (t) = . . . = B (m) (t) = G(t) then in the system D the tail of the customers-winners waiting time limit distribution is lighter than the tail (6.3) in the system C and the tail (6.7) in the system B : for t → ∞ m

lim sup P (un > t) ∼ lim inf P (un > t) ∼ G (t) n→∞

n→∞

∞ X

pk−1 (1 − p)km .

k=1 (i)

Remark 6. The random sequences {wn , n ≥ 1}, 1 ≤ i ≤ m, in the theorem 6.3 are independent but in the theorem 6.6 they are dependent. Nevertheless the formula (6.7) unlike the formula (6.3) is obtained for an arbitrary recurrent input flow.


6.5.

47

Proves of theorems

Proof of Theorem 6.1. Suppose that 0 < b ≤ 1 then there are positive α, m0 so that for 0 < t < 1/m0 the inequality G(t) ≥ αtb is true and so G(t) ≤ exp(−αtb ). Put m > m0 and denote Z ∞ m mG (t)dt = Um + Vm , Jm = 0

Um =

Z

1/m

m

mG (t)dt, Vm =

Z

0

∞

m

mG (t)dt. 1/m

Then Um ≤

Z

1/m

m exp(−αmtb )dt =

Z

0

1/m

m exp(−((αm)1/bt)b ) 0

m = (αm)1/b

m−1

Vm ≤ mG

Z

(αm)1/b /m

m exp(−v )dv ∼ (αm)1/b b

0

(1/m)

Z

= O(m

1−1/b

Z

dt(αm)1/b = (αm)1/b

∞

exp(−v b )dv = 0

), m → ∞,

∞ −b

G(t)dt ≤ m exp(−α(m − 1)m )

1/m

Z

∞

G(t)dt = 0

= o(m1−1/b), m → ∞. So Jm = O(m1−1/b) for m → ∞ and consequently Ω = Jm /J1 = O(m1−1/b), m → ∞. The formula (6.6) is proved. Proof of theorem 6.2. It is simple to prove that P (ζ1 ≤ t) satisfies the same condition of the lemma 6.1 as d.f. G(t). The corollary 6.1, [47, lemma 3] and the inclusion S∗ ⊂ S ⊂ L∗ lead to the equivalence P (ζ1 − ξ1 > t) ∼ P (ζ1 > t), t → ∞. Then the property 2 leads to the inclusion P (ζ1 − ξ1 ≤ t) ∈ S∗ . As {ζk − ξk , k ≥ 0} is the sequence of i.i.d.r. variables and their mathematical expectation bm − Eξ1 < 0 because of the formula (6.2) then the Embrechts–Veraverbeke formula [2], [16], [17] leads to R∞ R∞ P (ζ1 > t)dt x P (ζ1 − ξ1 > t)dt W B (x) ∼ ∼ x , x → ∞. (6.1) Eξ1 − bm Eξ1 − bm Proof of theorem 6.4. Put w0 = 0 and introduce the random sequence vn , n ≥ 0, by the recurrent formulas v0 = 0, vn+1 = I(yn = 1)(bn + vn ) + I(yn = −1)bn , n ≥ 0.

(6.2)

48


Denote by Vn (t) the random variable vn d.f. and by Bn (t) – n-fold convolution of d.f. B. From the formulas (6.1), (6.2) using an induction by n ≥ 0 obtain that Vn (t) = pn Bn (t) +

n X

pk−1 (1 − p)Bk (t), n > 0.

(6.3)

k=1

As P (y0 ≤ 0, b0 − a0 ≤ 0) > 0 then [52, theorem 1] the limit W (t) exists for ∀ w0 and ∀ t. Almost surely wn ≤ vn , n ≥ 0, so W n (t) ≤ V n (t), n ≥ 0.

(6.4)

Put the equality (6.3) into the inequality (6.4) and for n → ∞ obtain W (t) ≤ V (t), V (t) =

∞ X

pk−1 (1 − p)Bk (t).

(6.5)

k=1

As d.f. B(t) is subexponential then using the Embrechts–Veraverbeke formula obtain V (t) ∼

B(t) , t → ∞. 1−p

(6.6)

Construct now a low bound of the function W (t). Define recurrently the sequence Un (t), n ≥ 0, of d.f.‘s by the equalities: U0 (t) = Θ(t) (where Θ(t) is the function of the single jump in the point zero), Un+1 (t) = pD ∗ Un (t) + (1 − p)F (t), n ≥ 0.

(6.7)

Here P (bn − an ≤ t) = D(t), P (bn − an − v ≤ t) = F (t) and the random variable v does not depend on the random sequence (an , bn ), n ≥ 1, and P (v ≤ t) = V (t). Denote by Dk (t), Fk (t) k-fold convolutions of d.f.‘s D(t), F (t) relatively. From the formulas (6.1), (6.4), (6.5) obtain W n (t) ≥ U n (t), n

Un (t) = p Dn (t) +

n X

(6.8)

pk−1 (1 − p)Dk−1 ∗ F (t) + (1 − p)F (t), n ≥ 1.

k=2

If n → ∞ then W (t) ≥ U (t) ≥ Φ(t), U (t) = (1 − p)F (t) +

∞ X

pk−1 (1 − p)Dk−1 ∗ F (t),

k=2

Φ(t) =

∞ X k=1

pk−1 (1 − p)Fk (t).

(6.9)


49

From [47, lemma 3] obtain that F (t) ∼ B(t), t → ∞. As d.f. B(t) is subexponential then d.f. F (t) is subexponential also. Consequently from the Embrechts– Veraverbeke formula obtain Φ(t) ∼

F (t) B(t) ∼ , t → ∞. 1−p 1−p

(6.10)

The inequalities (6.5), (6.9) lead to the fromula V (t) ≥ W (t) ≥ Φ(t). If t → ∞ then from (6.6), (6.10) obtain (6.2). Proof of theorem 6.5. As d.f.‘s B (1) (t1), . . . , B (m) (tm ) ∈ S then for all k ∈ Q B (t) = K(k ). lim k t →∞ B(t)

(6.11)

From [2, chapter IX, Lemma 1.8] obtain that for any ε > 0 there is L > 0 so that for all k ∈ Q, t > 0 B k (t) ≤ L(1 + ε)H(k ) B(t ). (6.12) From the function V (t ) definition and from the formula (6.12) obtain that for all N >0 X Π(k )B k (t ) ≤ V (t) ≤ k : H(k ) k : H(k ) T 1 B k (t) B k (t + N T (δ)) > (1 − ε), > (1 − ε), 1 ≤ k ≤ N. m k B(t ) B k (t)

(6.19)

Unite the inequalities (6.16) – (6.19) and obtain lim inf W n (t ) ≥ (1 − ε)3 n→∞

∞ X

pk−1 (1 − p)km B(t)

k=1

then lim inf W n (t ) lim inf t →∞

n→∞

B(t )

∞ X

k−1

p

≤ 1. (1 − p)k

m

k=1

The formulas (6.14), (6.20) lead to the formula (6.7).

(6.20)

§ 7. Cooperative Effects in Queueing Systems with Rejection In this paragraph a problem of a maximization of an output flow intensity in aggregated queueing systems with a rejection is considered. For a oneserver system with a rejection the maximization of an output flow intensity is accompanied by a tendence of a rejection probability to one. If an aggregation of oneserver queueing systems with a competition between servers (or without it) takes place then a maximization may be realized practically without losses of input flow customers. If a number of aggregated systems tends to infinity and there is a competition between servers then a maximal output intensity tends to a service intensity. But if a competition is absent then a maximal output intensity tends to an input flow intensity.

7.1.

Preliminaries

Consider oneserver queueing system with a rejection and with Poisson input flow of λ intensity. Denote this system by B1 = M |G|1|0. Divide a time halfaxis t ≥ 0 into cycles which consist of an idle interval and a service time interval. Each cycle length coincides with a sum of two independent random variables: first of them has exponential distribution wuith the parameter λ and second of them coincides with a service time. The mean cycle length equals 1/λ + b1 where b1 is the mean service time. From the integral renewal theorem [7, chapter 9, § 4] obtain that the stationary output flow intensity satisfies the formula I(λ) =

1 + b1 λ

−1

.

(7.1)

Then the stationary rejection probability equals one minus the limit by a probability of a ratio between a number of departed customers to a number of arrival customers at the interval [0, T ], T → ∞. This limit from the law of large numbers for a renewal process [7, chapter 9, § 5] is I(λ) . (7.2) P (λ) = 1 − λ For the system A1 = M |M |1|0 with an exponential service time distribution with the parameter µ, the stationary rejection probability and the stationary output flow intensity equal µλ λ , I(λ) = P (λ) = λ+µ λ+µ where the function P (λ), I(λ) are monotonically increasing. Then the maximal output flow intensity (MOFI) equals J = sup(I(λ) : λ > 0) = lim I(λ) = µ λ→∞

53

54


and lim P (λ) = 1.

(7.3)

λ→∞

So an approach to MOFI in the system A1 is accompanied by a tendence of a stationary rejection probability to one. But an aggregation of A1 systems allows to reach MOFI without losses of input flow customers described by the formula (7.3). This problem is solved by a construction and a consideration of the system Bm = M |G|m|0 with the input flow intensity mλ and with a competition of servers.

7.2.

Stationary characteristics of an aggregated system

Consider the queueing system Am = M |M |m|0 with the input flow intensity mλ and the service intensities µ at all m servers. The system Am is an aggregation of m systems A1 . A number of customers in the system Am is described by the death and birth process xm (t) with the following death and birth intensities λm (k) = mλ, 0 ≤ k < m, µm (k) = kµ, 0 < k ≤ m. Denote by P (m) (λ) the stationary rejection probability in the system Am and by I (m)(λ) the stationary intensity of output flow. Then MOFI equals J (m) = sup(I (m)(λ) : λ > 0). Theorem 7.1. The following formulas are true: J (m) ∼ µ, m → ∞, P (m) (λ) → 1 −

µ , λ > µ, P (m) (λ) → 0, λ ≥ µ, m → ∞. λ

Proof. Denote pm (k) = lim P (xm (t)=k), 0 ≤ k ≤ m. Then t→∞

pm (m − 1) = pm (m)

µ 2 m(m − 1) µ m · , pm (m − 2) = pm (m) · ,... λ m λ m2

From the integral renewal theorem and the law of large numbers for the renewal process [7, chapter 9, § 4, 5] the rejection probability P (m) satisfies the equality 

P (m) (λ) = pm (m) = 1 +

m k−1 X µ k Y k=1

λ

j=0

 −1 j  . 1− m

(7.4)

So the stationary output flow intensity in the aggregated system Am (for a single server) equals (7.5) I (m)(λ) = λ(1 − P (m) (λ)), λ > 0.

Cooperative Effects in Queueing Systems with Rejection

55

From the formulas (7.4), (7.5) obtain that I (m)(µ) ≤ J (m) ≤ µ.

(7.6)

Fix ε, 0, 5 < ε < 1, and denote c=−

ln(1 − ε) 1−ε

then 1 − x ≥ e−cx , 0 < x < 1 − ε, and so X

(P (m) (µ))−1 ≥ 1 +

 k−1 2 X X ck j  > 1+ ≥ exp −c exp − m 2m 

j=0

1≤k≤m(1−ε)

≥

Z

m(1−ε)+1 0

1≤k≤m(1−ε)

r cx2 πm dx ∼ , m → ∞. exp − 2m 2c

(7.7)

Consequently from (7.5), (7.6), (7.7) obtain the formulas: J (m) ∼ µ, P (m) (µ) → 0, m → ∞. The formulas (7.4), (7.8) lead to P (m) (λ) → 0, m → ∞, λ ≤ µ. From the formula P (m) (λ) >

m X µ k k=0

λ

!−1

=

1 − µ/λ 1 − (µ/λ)m+1

and from the formula (7.7) for any 0 < γ < 1/2 

P (m) (λ)< 

 −1 p X µ k exp(cm−γ )(1 − µ/λ) ck2  exp − ≤ 1/2−γ λ 2m 1 − (µ/λ)m 1/2−γ

0≤k≤m

obtain P (m) (λ) → 1 − The theorem 7.1 is proved.

µ , m → ∞, λ > µ. λ

(7.8)

56

7.3.


An aggregated system with a competition of servers

Consider the queueing system Bm = M |G|m|0 with the Poisson input flow with the intensity mλ and a competition of servers. This system is an aggregation of m systems B1 . Suppose that a customer i arrives in the system Bm at the time moment ti . If the system is empty then the customer receives an information about (1) (m) its possible service times ηi , . . ., ηi at the system servers and chooses a server (1) (m) with the minimal service time ζi = min(ηi , . . . , ηi ), i ≥ 1. If the system is not empty then the customer is rejected. During a customer service at some server all (j) other servers do not work. Random variables ηi , i≥1, 1 ≤ j ≤ m, are i.i.d. with the common d.f. G(t). The system Bm may be interpreted as the oneserver queueing system M |G|1|∞ with the Poisson input flow 0 = t1 ≤ t2 = t1 + ξ1 ≤ t3 = t2 + ξ2 ≤ . . . with the intensity mλ and with i.i.d. service times ζ1, ζ2, . . . which have the mean Z ∞ (G(t))m dt > 0. bm = 0

From the formulas (7.1), (7.2) obtain that in the system Bm the stationary output flow intensity (for each server) I (m)(λ) and the stationary rejection probability P (m) (λ) satisfy the equalities I

(m)

(λ)=

1 + Km λ

−1 I (m)(λ) , Km = mbm . , P (m) (λ) = 1− λ

(7.9)

It is proved in [50] that if for some a, b > 0 G(x) ∼ axb , x → 0, then for Km for m → ∞ the following formulas are true: Km = O(m1−1/b), 0 < b ≤ 1, 1/Km = O(1/m1−1/b), 1 ≤ b. So if 0 < b < 1 then from (7.9), (7.10) for any λ > 0 obtain I (m) (λ) → λ, P (m) (λ) → 0, m → ∞.

(7.10)

§ 8. Asymptotic Analysis of Logical Systems with Unreliable Elements In this paragraph models of networks with unreliable arcs are investigated. Asymptotic formulas for probabilities of the networks work or failure and the networks lifetime distributions are obtained. Direct calculations of these characteristics in general case [40], [44] demand sufficiently large volumes of arithmetical operations. Main parameters of the asymptotic formulas are minimal way length and minimal section ability to handle. A series of new algorithms and formulas to calculate parameters of asymptotic formulas are developed.

8.1.

Main characteristics

Define oriented graph Γ with finite number of nodes U and the set W of arcs (u, v). In this graph there is single node u∗ without input arcs and single node u∗ without output arcs, the graph has not arcs (u, u). Suppose that n(s) is a number of arcs of a subgraph s, s⊆W . For S⊆{s : s⊆W } put X Y c(u, v), n(S) = min n(s), D(S) = s∈S

s: n(s)=n(S) (u,v)∈s

C(S) = min C(s), C(s) = s∈S

X

c(u, v),

(u,v)∈s

C1 (S) = min C1 (s), C1 (s) = max c(u, v), s∈S

Th (S) =

(u,v)∈S

X

Y

exp(−h−c(u,v) ),

s: C1 (s)=C1 (S) (u,v)∈s

c(u, v) is positive and integer function. Designate by N (S), N1(S), N∗(S) numbers of s ∈ S : C(s) = C(S), C1 (s) = C1 (S), n(s) = n(S), correspondingly. Put R the set of all ways R from u∗ to u∗ without selfintersections. Consider the sets / A}, L=L(A)={(u, v) : u ∈ A, v ∈ / A} A={A ⊂ U, u∗ ∈ A, u∗ ∈ and L = {L(A), A ∈ A} is the set of all sections in the graph Γ.

8.2.

Graphs with unreliable arcs

For each the graph Γ define arc define the number α(u, v) = I(the arc (u, v) works) 57

58


where I(G) is an indicator function of the event G. It is not difficult to confirm that ^ _ _ ^ α(u, v) = α(u, v). (8.1) R∈R (u,v)∈R

L∈L (u,v)∈L

Denote α(Γ) the quantity of both sides of the equality (8.1) which characterizes the graph Γ work. Suppose that α(u, v), (u, v)∈W, are independent random variables, P (α(u, v)=1)= pu,v (h), qu,v (h) = 1 − pu,v (h) where h is small parameter: h → 0. Then the following asymptotic formulas are true for h → 0. 1. If pu,v (h) ∼ c(u, v)h, then P (α(Γ) = 1) ∼ hn(R) D(R). 2. If pu,v (h) ∼ hc(u,v) , then P (α(Γ) = 1) ∼ N (R)hC(R). 3. If pu,v (h) ∼ exp(−h−c(u,v) ), then P (α(Γ) = 1) ∼ Th (R). 4. If qu,v (h) ∼ c(u, v)h, then P (α(Γ) = 0) ∼ hn(L) D(L). 5. If qu,v (h) ∼ hc(u,v) , then P (α(Γ) = 0) ∼ N (L)hC(L) . 6. If qu,v (h) ∼ exp(−h−c(u,v) ), then P (α(Γ) = 0) ∼ Th (L).

8.3.

Applications to lifetime models

Suppose that τ (u, v) independent random variables are arcs (u, v)∈W lifetimes. Denote P (τ (u, v) > t) = pu,v (h) and put the graph Γ lifetime τ (Γ) = min max τ (u, v). R∈R (u,v)∈R

Suppose that h = h(t) is monotonically decreasing and continuous function and h → 0, t → ∞, then asymptotic formulas 1, 2, 3 are true if P (α(Γ) = 1) is replaced by P (τ (Γ) > t). Suppose that h is monotonically increasing and continuous function and h → 0, t → 0, then the formulas 4, 5, 6 are true if P (α(Γ) = 0) is replased by P (τ (Γ) ≤ t).

8.4.

Calculation of graph characteristics

For A ∈ A define Q(A) = {v 6∈ A : ∃u ∈ A, (u, v) ∈ W } and construct the sets A1 = Q(A0) = {u∗}, Ak+1 = Ak ∪ Q(Ak ), k = 1, 2, . . . Denote n = n(R) = min(k : u∗ ∈ Ak ). Designate by ϕ(u, v), (u, v) ∈ W, integer and nonnegative function: X X ϕ(u, v) = ϕ(v, u), ϕ(v, u) ≤ c(u, v), (u, v) ∈ W, (u,v)∈W

(v,u)∈W

and call it a flow. A quantity of the flow is the sum

X

ϕ(u∗, v).

(u∗ ,v)∈W

Denote by Γ1 k Γ2 the graph constructed from the graphs Γ1 , Γ2 by a connection of their initial and final nodes correspondingly and by Γ1 → Γ2 - the graph constructed by a connection of the graph Γ1 final node with the graph Γ2 initial

Asymptotic Analysis of Logical Systems with Unreliable Elements

59

node. Consider the sets R1, L1 , R2, L2 for the graphs Γ1 , Γ2 in the same sense as the sets R, L for the graph Γ. Suppose that further ui ∈ Q(Ai−1 ), i = 1, . . . , n. Calculation of D(R) : D(u1 ) = 1, u1 ∈ A1, X D(uk )c(uk , uk+1 ), 1 ≤ k < n, D(R) = D(u∗). D(uk+1 ) = uk ∈Q(Ak−1)

Calculation of N∗(R) : N∗(un−1 ) = 1, un−1 ∈ Q(An−2 ), X N∗ (un−k )I((un−k−1 , un−k ) ∈ W ), 1 ≤ k < n − 1, N∗(un−k−1 ) = un−k ∈Q(An−k−1)

N∗(R) = N∗(u∗). Calculation of C(R), N (R): each arc (u, v) of the graph G is divided into arcs with initial lengths (because the function c(u, v) is integer). Then the graph Γ is transformed into the graph Γ1 with single lengths arcs. Applying the n = n(R), N∗(R) calculation procedures to the graph Γ1 obtain C(R), N (R) for the graph Γ. Calculation of C(L), n(L) : using the theorem [5] of coincidence of maximal flow value and minimal section ability to handle C(L) and Ford-Falkerson algorithm define C(L). Then n(L) equals to C(L) for c(u, v) ≡ 1. Suppose that W = {(uk , uk+1 ), ui ∈ Q(Ai−1 ), i = 1, . . . , n} in next five points. Calculation of C1 (R) : C1 (u1) = 0, u1 ∈ A1 , C1 (uk+1 )= min max(C1 (uk ), c(uk , uk+1 )), 1≤k xj+1 ) = 1 or if the inequalities

62


chain 1 = x1 ≥ x2 ≥ . . . ≥ xn = 0 becomes incorrect. So the equality C(L(A)) = C(L) (and consequently the equality X = 1) leads that for any k, l satisfying the conditions k < l the following formulas are true: {k, k + 1, . . ., l} ∩ M = ∅ or xk+1 = xk+2 = . . . = xl = 1, or xk = xk+1 = . . . = xl = 0. So the equality C(L(A)) = C(L) takes place only if for some k ∈ M nj = Nj , 1 ≤ j ≤ k, nj = 0, k < j ≤ n. So the decisive formulas for C(L), N (L) are proved.

§ 9. Cooperation and Competition in Risk Models In this paper different models of a cooperation and a competition of insurance companies are constructed and investigated. For this aim the following risk models are considered: classical risk model, risk model under constant interest force and risk model under stochastic interest force with discrete time. A cooperation of insurance companies means an aggregation of their input flows, initial capitals and risks and a segregation of reserve companies. A competition of insurance companies means that aggregated companies compete for each customer. An insurance of forward contracts are considered in a suggestion that the company insures both sides of the forward contract. In this paper well known heavy-tailed technique gives convenient mathematical instrument to obtain sufficiently simple asymptotic estimates of cooperation and competition effects.

9.1.

Preliminaries

1. Classical risk model. The classical risk model has Poisson arrival process with the intensity λ and constant premium rate c. Suppose that the sequence of claim sizes {Zk , k ≥ 1} consists of independent and identically distributed random variables (i.i.d.r.v.‘s) with EZk = b and the distribution function (d.f.) B(t), t ≥ 0, so that B(t) = 1 − B(t) > 0, t ≥ 0. If u is an initial capital then this model is characterized by the parameters (u, λ, c, B(t)). Denote Z 1 z B(t) dt, z ≥ 0, (9.1) F (z) = b 0 and put ρ = λb. Suppose that d.f. F (z), z ∈ [0, ∞), belongs to the class of subexponential distributions S, that is for n ≥ 2 : lim

z→∞

F ∗n (z) = n. F (z)

Here F ∗n (z) is n-fold conjuncture of d.f. F (z). If ρ < c then [17] Embrechts– Veraverbeke asymptotic formula is true ψ0 (u) ∼

ρ F (u), c−ρ

u → ∞,

(9.2)

where ψ0(u) is infinite-time ruin probability of an insurance company as a function of an initial capital u. 2. Risk model under constant interest force. In [45] a constant interest force r > 0 is introduced into the classical risk model. For F (z) ∈ S the formula Z λ ∞ dz (9.3) B(z) , u → ∞, ψr (u) ∼ r u z 63

64


is proved where ψr (u) is ruin probability in risk model under constant interest force r. This model has the parameters (u, λ, c, B(t), r). 3. Risk model under stochastic interest force. Consider a risk model with discrete time n = 1, 2, . . . and denote Xn the insurer‘s net loss (the total claim amount minus the total incoming premium) within period n and Yn the discount factor from time n to time n − 1. Here Xn is called insurance risk and Yn is called financial risk. These random variables are independent with d.f.‘s B1 (t)=B(t+1), G(t) relatively. Denote Ψ(u) infinite-time ruin probability of this model with initial capitalnu. If B 1 (t) =ol(t)t−α , 0 < α < ∞, where l(t) is slowly varying function and E max Y1α−δ , Y1α+δ < 1 for some 0 < δ < α then [46] Ψ(u) ∼

EY1α EY1α B (u) ∼ B(u), 1 1 − EY1α 1 − EY1α

u → ∞.

(9.4)

Denote parameters of this model by (u, B1 (t), G(t)).

9.2.

Cooperative effects

Consider an aggregation of m identical insurance companies of the same type and analyze their common ruin probability if B(t) = l(t)t−α , 1 < α < ∞, where l(t) is slowly varying function (and consequently B(t) ∈ S). Further only these distributions B(t) will be considered. Model 1. The aggregation of m models described by the parameters (u, λ, c, B(t)) is characterized by the parameters (mu, mλ, mc, B(t)).

(9.5)

(m)

The ruin probability of the aggregated model ψ0 (u) from (9.2), (9.5) satisfies the formula (m) ψ0 (u) ∼ m−α+1 , u → ∞. (9.6) ψ0(u) Model 2. The aggregation of m models with the parameters (u, λ, c, B(t), r) is characterized by the parameters (mu, mλ, mc, B(t), r). (m)

(9.7)

The ruin probability of the aggregated model ψr (u) from (9.3), (9.7) satisfies the formula (m) ψr (u) ∼ m−α+1 , u → ∞. (9.8) ψr (u)

Cooperation and Competition in Risk Models

65

Model 3. The aggregation of m models with the parameters (u, B1 (t), G(t)) is characterized by the parameters (mu, B1∗m (t), G(t))

(9.9)

where B1∗m (t) is m-fold conjuncture of d.f. B1 (t). The ruin probability of the aggregated model Ψ(m) (u) from (9.4), (9.9) satisfies the formula Ψ(m) (u) ∼ m−α+1 , Ψ(u)

9.3.

u → ∞.

(9.10)

Individual and group risks

In this section consider another structure of an aggregation of insurance companies if B(t) = l(t)t−α , 1 < α < ∞. Suppose that [mγ], 0 < γ < 1, (reserve) companies of m aggregated companies put their initial capitals into reserve and direct their claims into a group of other (working) m − [mγ] companies. (Here [a] is integer part of a real a). Suppose that a choice of [mγ] reserve companies among m companies is with equal probabilities. Then an initial capital of each insurance company may be put into a reserve with the probability [mγ]/m. So the ruin probability of all m companies (group risk) equals 0 and the ruin probability of each company (individual risk) equals the ruin probability of working companies multiplied by (m − [mγ])/m. (m) Model 1. Denote ψb0 (u) the individual risk in so aggregated model then (m) (m − [mγ])−α+2 ψb0 (u) ∼ , ψ0 (u) m

u → ∞.

(9.11)

b (m) (u) the individual risk in so aggregated model then Model 2. Denote Ψ (m) ψbr (u) ∼ (m − [mγ])−α+1, ψr (u)

u → ∞.

(9.12)

b (m) (u) the individual risk in so aggregated model then Model 3. Denote Ψ b (m) (u) Ψ ∼ (m − [mγ])−α+1, Ψ(u)

u → ∞.

(9.13)

As a result a zeroise of the group risk is accompanied by sufficiently small increase (for large m) of the individual risk.

66

9.4.


Competition and cooperation

Consider an aggregation of m insurance companies and suppose that there is a competition between [mβ ] (randomly choused) companies, 0 < β ≤ 1, for each claim. The claim arrives into the company which suggests maximal payout. Model 2. The aggregation of m insurance models with the parameters (u, λ, c, B(t), r) and with the competition gives the model with the parameters β

(mu, mλ, mc, B [m ] (t), r).

(9.14) (m,β)

Denote the ruin probability of so aggregated model by ψr (9.14) obtain (m,β) (u) ψr ∼ m−α+1 [mβ ], u → ∞. ψr (u)

(u). Then from (9.3), (9.15)

Model 3. The aggregation of m insurance models with the parameters (mu, B1∗m (t), G(t)) and with the competition gives the model with the parameters [m]β ∗m (t), G(t)). (9.16) (mu, B1 Denote the ruin probability of so aggregated model by Ψ(m,β)(u). Then from (9.4), (9.16) obtain Ψ(m,β) (u) ∼ m−α+1 [mβ ], u → ∞. (9.17) Ψ(u)

9.5.

Insurance of forward contracts

Consider a financial management model of forward contracts insurance suggested in [1]. Suppose that there are two insurance companies insuring both participants of some forward contract and working in discrete time with the claims ξ + η, ξ − η. Here ξ is common nonnegative summand for claims of these two companies. Then η is individual claim of the first participant of the contract and −η is individual claim of the second participant. The claim η (the claim −η) may be considered as a premium for η < 0 (for η > 0). Suppose that ξ is much smaller than η in some sense. Now consider this suggestion in the following probabilistic sense: P (ξ > x) = o(P (η > x)), P (ξ > x) = o(P (−η > x)), x → ∞, and assume that ξ, η are independent r.v.‘s and ξ, η, −η are subexponential r.v.‘s. Denote one-step ruin probabilities of the companies insuring the first and the second participants of the contract by A1(x) = P (ξ + η > x), A2 (x) = P (ξ − η > x)

Cooperation and Competition in Risk Models

67

and the ruin probability of both companies by A(x) = P (ξ + η > x, ξ − η > x) = P (ξ − |η| > x). Here A1 (x), A2 (x) characterize individual risks of the insurance companies and A(x) characterizes their group risk. Compare A1 (x), A2 (x), A(x) with one-step ruin probability of the aggregation of these two companies C(x) = P (ξ + η + ξ − η > x + x) = P (ξ > x). Here C(x) characterizes as individual so group risks. Using well known properties of sums of subexponential distributions (see [11], [12], [15]) obtain for x → ∞ : A1 (x) ∼ P (η > x), A2 (x) ∼ P (−η > x). Analogously [47, lemma 3] leads to the formula: A(x) ∼ C(x) ∼ P (ξ > x) = o(Ai (x)), i = 1, 2. So the aggregation of these two companies allows to decrease individual risks to small P (ξ > x) and to conserve the group risk at sufficiently small level P (ξ > x).

Bibliography [1] Afanasiev A.A. Commercial Banks at Market of Derivative Financial Instruments. Vladivostok, Far Eastern State Academy of Economics and Control, 2002, 308 p. [2] Asmussen S. Ruin Probabilities. Singapore: World Scientific, 2000, 388 p. [3] Asmussen S. Subexponential asymptotics for stochastic processes: extremal behaviour, stationary distributions and first passage probabilities. The Annals of Applied Probability, 8 (1998), pp. 354–374. [4] Basharin G.P., Tolmachev A.L. The theory of queuing networks and its applications to the analysis of information and computation systems. The totals of science and engineering, the probability theory. M. VINITI, 1983, pp. 3–119. [5] Belov V.V., Vorobiev E.M., Shatalov V.E. Belov V.V., Vorobiev E.M., Shatalov V.E. Graph theory. Education guidance for technical universities. Moscow: High School, 1976 (in Russian). [6] Berezko M.P., Vishnevskiy V.M., Levner E.V., Fedotov E.V. Mathematical models of routing algorithms investigations in data transmission networks. Information Processes, 1 (2001), No. 2, pp. 103–125. [7] Borovkov A.A. Probability Theory. M.: Science, 1986, 432 p. (in Russian). [8] Boxma O.J., Vlasiou M. On queues with service and interarrival times depending on waiting times. Queueing Systems, to appear. [9] Chang C.S., Thomas J.A., Kiang S.H. On the stability of open networks: a unified approach by stochastic dominance. Queueing Systems, 15 (1994), pp. 239–260. [10] Chistyakov V. A theorem on sums of independent random variables and its applications in branching processes. Theory of Probab. Appl., 9 (1964), pp. 640–648. [11] Cline D.B.H. Convolution tails, product tails and domains of attraction. Probab. Theory Relat. Fields, 72 (1986), No. 4, pp. 529–557. 69

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[12] Cline D.B.H. Convolutions of distributions with exponential and subexponential tails. J. Austral. Math. Soc. Ser. A, 43 (1987), pp. 347–365. [13] Cline D.B.H., Resnick S.I. Multivaiate subexponential distributions. Stochastic Processes and their Applications, 42 (1992), pp. 49–72. [14] David H.A. Order Statistics. John Wiley and Sons, New York, 1970. [15] Embrechts P., Goldie C.M. On closure and factorization properties of subexponential and related distributions. J. Austral. Math. Soc. Ser. A, 29 (1980), No. 2, pp. 243–256. [16] Embrechts P., Kluppelberg C., Mikosch T. Modelling Extremal Events for Insurance and Finance. Berlin: Springer, 1997. [17] Embrechts P., Veraverbeke N. Estimates for the probability of ruin with special emphasis on the possibility of large claims. Insurance: Math. and Econom., 1 (1982), pp. 55–72. [18] Fikhtengoltz G.M. Course of differential and integral calculus. M.: Science, 1966, vol. 2, 800 p. (in Russian). [19] Foss S.G. Ergodicity of queueing networks. Siberian Mathematical Journal, 32 (1991), No. 4, pp. 184–203 (in Russian). [20] Foss S., Korshunov D. Heavy tails in multiserver queue. Queueing Systems, 52 (2006), pp. 31–48. [21] Freund J.E. A bivariate extension of the exponential distribution. J. Amer. Statist. Assoc., 56 (1961), No. 296, pp. 971–977. [22] Gabasov R., Kirillova F.M. Methods of linear programming. Minsk: BSU, 1978, part 2, 239 p. (in Russian). [23] Goldie C.M., Kluppelberg C. Subexponential Distributions. Preprint. Johannes Guttenberg – Universitat Mainz, 1996, No. 96–1, 20 p. [24] Gordon K.D., Newell G.F. Closed Queuing Systems with Exponential Servers. Oper. Research, 15 (1967), No. 2, pp. 254–265. [25] Jackson J.R. Networks of Waiting Lines. Oper. Res., 5 (1957), No. 4, pp. 518–521. [26] Ivchenko G.I., Kashtanov V.A., Kovalenko I.N. Queueing theory. M.: High school, 1982, 256 p. (in Russian). [27] Ivnitskiy V.A. Theory of queueing networks. M.: Physmathedit, 2004, 772 p. (in Russian).

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[28] Karlin S., Mac–Gregor J. The differential equations of birth–and–death processes and the Stieltjes moment problem. Trans. Amer. Math. Soc., 85 (1957), pp. 489–546. [29] Karlin S., Mac–Gregor J. The classification of birth and death processes. Trans. Amer. Math. Soc., 86 (1957), pp. 366–400. [30] Kelly F.P. Reversibility and Stochastic Networks. John Wiley & Sons, 1979. [31] Kiefer J., Wolfowitz J. On the theory of queues with many servers. Trans. Amer. Math. Soc., 78 (1955), pp. 147–161. [32] Kovalenko I.N., Kuznetsov N.Yu., Shurenkov V.M. Random processes. Kiev: Naukova dumka, 1983, 366 p. (in Russian). [33] Krylenko A.V. Queueing networks with some types of customers, instant serving and customers bypasses of nodes. Problems of Information Transmission, 33 (1997), No. 3, pp. 91–101 (in Russian). [34] D.V. Lindley. The theory of queues with a single server. Proceedings Cambridge Philosophical Society, 48 (1952), pp. 277–289. [35] Malinkovskiy Yu.V., Nikitenko O.A. Stationary distribution of states with bypasses and negative customers. Automatics and Remote Control, 2000, No. 8, pp. 79–85. (in Russian). [36] Mashunin Yu.K. Methods and models of vector optimization. M.: Science, 1986, 141 p. (in Russian). [37] Osipova M.A. Development and investigation of queueing networks models by decomposition method of special kind. Autoreferat of candidate thesis. Vladivostok: FES, 2003, 19 p. (in Russian). [38] Park B.C., Park J.Y., Foley R.D. Carousel system performance. Journal of Applied Probability, 40 (2003), No. 3, pp. 602–612. [39] Pittel B. Closed exponential networks of queues with saturation: the Jackson– type stationary distribution and its asymptotic analysis. Math. Oper. Res., 4 (1979), No. 4, pp. 357–378. [40] Riabinin I.A. Logic–probability calculus as method of reliability and safety investigation in complex systems with complicated structure. Automatics and Remote Control, 2003, No 7, pp. 178–186 (in Russian). [41] Rozanov Yu.A. Lectures on probability theory. M.: Science, 1986, 120 p. (in Russian).

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[42] Serfozo R. Introduction to Stochastic Networks. Springer Verlag, New York, 1999. [43] Shiriaev A.N. Probability. M.: Science, 1989, 640 p. (in Russian). [44] Solojentsev E.D. Features of logic–probability risk theory with groups of antithetical events. Automatics and Remote Control, 2003, No. 7, pp. 187–203 (in Russian). [45] Sundt B., Teugels J.L. Ruin estimates under interest force. Insurance: Mathematics and Economics, 16 (1995), pp. 7–22. [46] Tang Q., Tsitsiashvili G.Sh. Finite and Infinite Time Ruin Probabilities in the Presence of Stochastic Returns on Investments. Advances in Applied Probability, 36 (2004), No. 4, pp. 1278–1299. [47] Tsitsiashvili G.Sh., Markova N.V. Asymptotic characteristics of output flows inqueueing networks. Far Eastern Mathematical Journal, 4 (2003), No. 1, pp. 36–43 (in Russian). [48] Tsitsiashvili G.Sh., Osipova M.A. New product theorems for queueing networks. Problems of Information Transmission, 41 (2005), No. 2, pp. 111–122 (in Russian). [49] Tsitsiashvili G.Sh., Osipova M.A. Probability distribution in queueing networks with variable structure. Problems of Information Transmission, 42 (2006), No. 2, pp. 101–108 (in Russian). [50] Tsitsiashvili G.Sh. Cooperative Effects in Multi–Server Queueing Systems. Mathematical Scientist, 30 (2005), part 1, pp. 17–24. [51] Vlasiou M., Adan I.J.B.F., Wessels J. A Lindley–type equation arising from a carousel problem. J. Appl. Probab., 41 (2004), No. 4, pp. 1171–1181. [52] Whitt W. Queues with service times and interarrival times depending linearly and randomly upon waiting times. Queuing Systems, 6 (1990), No. 4, pp. 335–351.

INDEX A D accuracy, 39 aggregation, 3, 21, 53, 54, 56, 63, 64, 65, 66, 67 algorithm, 10, 19, 20, 33, 59 alternative, 17 asymptotics, 69

B behavior, 43 birth, 32, 54, 71 bounds, 39 branching, 69

C calculus, 70, 71 circulation, 1 classes, 3, 39 classification, 71 closure, 70 communication, 5 competition, 2, 39, 40, 43, 44, 46, 53, 54, 56, 63, 66 components, 2, 20, 32, 38, 44, 45, 46 computation, 69 connectivity, 10, 11, 20 construction, 19, 20, 54 control, 3, 13 customers, 1, 2, 1, 2, 3, 6, 7, 8, 9, 13, 14, 15, 16, 17, 18, 19, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 37, 38, 39, 40, 41, 42, 43, 44, 46, 53, 54, 71 cycles, 53

death, 32, 54, 71 decomposition, 71 definition, 10, 11, 31, 49, 60 demand, 20, 57 discipline, 13, 29, 43 displacement, 23, 27, 29 distribution, 1, 2, 4, 5, 6, 7, 8, 9, 10, 13, 14, 15, 16, 17, 18, 19, 22, 23, 24, 25, 26, 27, 28, 29, 30, 37, 39, 40, 42, 43, 44, 46, 53, 63, 70, 71, 72 distribution function, 29, 39, 63 division, 21 dominance, 69

E education, 69 environment, 3, 1, 9 equality, 11, 19, 25, 29, 30, 31, 32, 41, 42, 48, 54, 58, 61, 62

F failure, 14, 15, 16, 17, 26, 57 Far East, 69, 72 Ford, 59

G generalization, 44, 46 grants, 3 graph, 2, 3, 8, 10, 11, 20, 57, 58, 59, 60 groups, 44, 72 guidance, 69

74

Index

I identity, 30 incidence, 10 inclusion, 40, 47, 51 induction, 11, 31, 48 inequality, 10, 31, 41, 42, 45, 47, 48, 50, 60, 61 infinite, 32, 63, 64 insurance, 63, 64, 65, 66, 67 intensity, 1, 13, 14, 15, 16, 18, 19, 21, 22, 25, 26, 27, 29, 30, 32, 35, 38, 53, 54, 56, 63 interaction, 1, 7 interpretation, 3 interval, 41, 53

O one dimension, 3 optimization, 29, 30, 33, 71

P parameter, 13, 53, 58 performance, 71 Poisson arrival process, 63 probability, 1, 27, 29, 30, 45, 46, 53, 54, 56, 63, 64, 65, 66, 67, 69, 70, 71, 72 probability distribution, 45, 46 probability theory, 69, 71

L lifetime, 2, 57, 58 limitation, 29 linear dependence, 39, 44 linear programming, 70

Q queuing networks, 13, 69

R

M management, 66 Markov chain, 29, 41 matrix, 2, 1, 2, 3, 7, 8, 10, 16, 17, 18, 19, 20, 23, 25, 26, 27, 29, 30, 33, 34, 35, 36, 37, 38 memory, 10, 11 models, 2, 3, 1, 13, 14, 21, 39, 57, 58, 63, 64, 65, 66, 69, 71 monograph, 3 Moscow, 69 motion, 16 multidimensional, 46 multivariate, 39, 45

N nanotechnology, 35 network, 1, 3, 1, 2, 3, 5, 6, 7, 8, 9, 10, 13, 14, 15, 16, 17, 18, 19, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 33, 34, 35, 37, 38, 60 New York, 70, 72 nodes, 1, 2, 3, 5, 6, 7, 8, 13, 14, 15, 16, 17, 18, 19, 21, 22, 23, 24, 25, 26, 27, 28, 29, 33, 34, 35, 37, 38, 57, 58, 60, 61, 71

recursion, 10, 20, 44 reduction, 3 rejection, 53, 54, 56 reliability, 3, 13, 71 repair, 13, 14, 15, 16, 17, 18, 25, 26 repetitions, 11 returns, 19 risk, 3, 63, 64, 65, 67, 72 routing, 69

S safety, 71 saturation, 71 school, 70 science, 3, 69 search, 60 segregation, 63 series, 57 Singapore, 69 stability, 69 stationary distributions, 69 stochastic processes, 69 systems, 3, 1, 13, 15, 32, 36, 39, 40, 41, 42, 43, 53, 54, 56, 69, 71

T theory, 3, 10, 13, 69, 70, 71, 72

Index time, 21, 22, 25, 29, 30, 40, 41, 42, 43, 44, 46, 53, 56, 63, 64, 66 timing, 21, 22 transformations, 19, 21 transition(s), 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 13, 14, 15, 16, 17, 18, 19, 21, 22, 23, 24, 25, 26, 27, 28, 29 transmission, 69 transportation, 29, 34

V values, 29 variable(s), 1, 5, 6, 7, 8, 9, 21, 22, 23, 24, 29, 30, 39, 40, 43, 44, 46, 47, 48, 53, 56, 58, 63, 64, 69, 72 vector, 2, 11, 14, 15, 16, 17, 18, 21, 22, 26, 27, 30, 32, 33, 36, 38, 45, 46, 71

W

U universities, 69

75

Weibull distribution, 42

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