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Stochastic Monotonicity and Queueing Applications of Birth-Death Processes PDF

124 Pages·1981·2.036 MB·English
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Lecture Notes in Statistics Edited by D. Brillinger, S. Fienberg, J. Gani, J. Hartigan, J. Kiefer, and K. Krickeberg 4 Erik van Doorn Stochastic Monotonicity and Queueing Applications of Birth-Death Processes Springer-Verlag New York Heidelberg Berlin E. A. van Doorn Netherlands Postal and Telecommunications Services Dr. Neher - Laboratories Post Office Box 421 2260 AK Leldschendam The Netherlands This monograph is a polished version of the author's dissertation entitled: "Stochastic Monotonicity of the Birth-Death Processes," which was written while he was affiliated with the Department of Applied Mathematics, Twente University of Technology, Enschede. AMS Subject Classifications (1980): primary 6OJ80; secondary 6OK25, 92A15 Library of Congress Cataloging In Publication Data Doorn, Erik van. Stochastic monotonicity and queueing applica tions of birth-death processes. (Lecture notes in statistics; 4) Based on the author's thesis, Twente University of Technology, Enschede. Bibliography: p. Includes Indexes. 1. Birth and death processes (Stochastic processes) 2. Monotone operators. 3. Queueing theory. I. rltle. II. Series. QA274.76.D66 519.2'34 80-25183 ISBN-13: 978-0-387-90547-1 e-ISBN-13: 978-1-4612-5883-4 001: 10.1007/978-1-4612-5883-4 All rights reserved. No part of this book may be translated or reproduced In any form without written permission from Springer-Verlag. © 1981 by Springer-Verlag New York Inc. 9 8 7 8 5 4 3 2 1 PREFACE A stochastic process {X(t): 0 S t < =} with discrete state space S c ~ is said to be stochastically increasing (decreasing) on an interval T if the probabilities Pr{X(t) > i}, i E S, are increasing (decreasing) with t on T. Stochastic monotonicity is a basic structural property for process behaviour. It gives rise to meaningful bounds for various quantities such as the moments of the process, and provides the mathematical groundwork for approximation algorithms. Obviously, stochastic monotonicity becomes a more tractable subject for analysis if the processes under consideration are such that stochastic mono tonicity on an inter val 0 < t < E implies stochastic monotonicity on the entire time axis. DALEY (1968) was the first to discuss a similar property in the context of discrete time Markov chains. Unfortunately, he called this property "stochastic monotonicity", it is more appropriate, however, to speak of processes with monotone transition operators. KEILSON and KESTER (1977) have demonstrated the prevalence of this phenomenon in discrete and continuous time Markov processes. They (and others) have also given a necessary and sufficient condition for a (temporally homogeneous) Markov process to have monotone transition operators. Whether or not such processes will be stochas tically monotone as defined above, now depends on the initial state distribution. Conditions on this distribution for stochastic mono tonicity on the entire time axis to prevail were given too by KEILSON and KESTER (1977). It is very well conceivable that a process with monotone transition operators is not stochastically monotone on the entire positive time axis but on an interval of the form (t 1 ,co), with tl > O. Clearly, i"t is of some interest to know under which cir cumstances this phenomenon occurs. The study of these circumstances is the main sub ject of this monograph. The analysis is restricted to birth-death processes, which form the most important class of temporally homogeneous Markov processes in conti nuous time with monotone transition operators. In some proofs explicit use is made of specific properties of birth-death processes, so that it is probably not possible to extend the results to other classes of Markov processes. The main results of this monograph are obtained in chapter 5 where necessary and sufficient conditions are given for a birth-death process to be stochastically mono tone in the long run when the state space is a semi-infinite lattice of integers and the initial state distribution is supported by finitely many points. The theory needed to arrive at these results is developed in the chapters 3 and 4. It appears that the concept of dual processes, which is only touched upon in the existing literature, is very fruitful and of intrinsic interest. In chapter 1 and chapter 2 many known facts about birth-death processes are collect ed. Also in chapter 2 some preliminary analysis is done with a view to the chapters 6, 7 and 8, where the results are applied to specific processes. To do this one needs iv at least partial knowledge of the so-called spectral representation of the transition probabilities. As for the linear growth, birth-death processes of chapter 8 (including the M/M/m queue length process) this knowledge is available and application of the results of chapter 5 to these processes is straightforward. This is not the case, however, with the MIMls queue length process of chapter 6 and the queue length process of chapter 7 which models a system where potential customers are discouraged by queue length. A substantial part of this monograph, in fact the main part of the chapters 6 and 7 is therefore concerned with obtaining these representations, which have an interest of their own. Our findings in this respect extend the results previously obtained by KARLIN and McGREGOR (1958a) and NATVIG (1974), respectively. In chapter 9 various aspects of the first moment of birth-death processes are dis cussed. It appears to behave very regularly in a number of important cases. Finally, birth-death processes with a finite state space are considered in chapter 10. Although the analysis of the phenomenon of stochastic monotonicity may be performed through the concept of dual processes as in the infinite case, an entirely different approach is chosen. I take pleasure in closing this preface by acknowledging the support of Professor Jos H.A. de Smit of Twente University of Technology who provided the key references. and by thanking Miss Bea Bhola of the Dr. Neher - Laboratories for the careful typing of the manuscript. Leidschendam, August 1980 Erik van Doorn TABLE OF CONTENTS Chapter I : PRELIMINARIES 1.1 Markov processes I 1.2 Stochastic monotonicity 3 1.3 Birth-death processes 6 1.4 Some notation and terminology 8 Chapter 2: NATURAL BIRTH-DEATH PROCESSES 2. I Some basic properties II 2.2 The spectral representation 12 2.3 Exponential ergodicity 17 2.4 The moment problem and related topics 18 Chapter 3: DUAL BIRTH-DEATH PROCESSES 3. I Introduction • 22 3.2 Duality relations 23 3.3 Ergodic properties 26 Chapter 4: STOCHASTIC KlNOTONICITY: GENERAL RESULTS 4. I The case llO - 0 28 4.2 The case llO > 0 32 4.3 Properties of E(t) 35 Chapter 5 : STOCHASTIC KlNOTONICITY: DEPENDENCE ON THE INITIAL STATE DISTRIBUTION 5. I Introduction to the case of a fixed initial state 38 5.2 The transient and null recurrent process • • • 40 5.3 The positive recurrent process • • • • • • • • 41 5.4 The case of an initial state distribution with finite support 41 Chapter 6 : THE MIMls QUEUE LENGTH PROCESS . 6. I Introduction 44 6.2 The spectral function 46 6.3 Stochastic mono tonicity 60 6.4 Exponential ergodicity • 65 Chapter 7: A QUEUEING MODEL WHERE POTENTIAL CUSTOMERS ARE DISCOURAGED BY QUEUE LENGTH 7. I Introduction 66 7.2 The spectral representation 67 vi 7.3 Stochastic monotonicity and exponential ergodicity • • • • • . . 71 Chapter 8: LINEAR GROWTH BIRTH-DEATH PROCESSES 8. I Introduction • • . • • • 72 8.2 Stochastic monotonicity 74 Chapter 9: THE MEAN OF BIRTH-DEATH PROCESSES 9. I Introduction • • 76 9.2 Representations 76 9.3 Sufficient conditions for finiteness 80 9.4 Behaviour of the mean in special cases 82 Chapter 10 : THE TRUNCATED BIRTH-DEATH PROCESS 10.1 Introduction........ 87 10.2 Preliminaries • . . • • • • 88 10.3 The sign structure of P'(t) 93 10.4 Stochastic monotonicity 97 Appendix I: PROOF OF THE SIGN VARIATION DIMINISHING PROPERTY OF STRICTLY TOTALLY POSITIVE MATRICES . • • • • • • • • • • • • • • • • • • • • • • . • 100 Appendix 2: ON PRODUCTS OF INFINITE MATRICES ••.•••••••••••..• 103 Appendix 3: ON THE SIGN OF CERTAIN QUANTITIES • • • • • • . • • • • •• . . •• 104 Appendix 4: PROOF OF THEOREM 10.2.8 •••••••.•••••.•••••••• 107 REFERENCES • • • • • • • • • • • • • • • • • • • • • • • • . • • • • • • • • • • I I I NOTATION INDEX • • • • • • • . • . • • • • . • • • . • • • . • • • • • . • . . . I 13 AUTHOR INDEX . • • • • . • • • • • • • • • • • • • • • • • • • . • • • • • . • • I 16 SUBJECT INDEX . • • . • • • • • • • • • • • • • • . • • • • • • • • . . • • • • I I 7 I • PRELIMINARIES 1.1 tfarkov processes (CHUNG (1967). FREEDMAN (1971). REUTER (1957» By a Map/'Ov pPOae88 we shall understand a continuous time stochastic process {X(t): 0 S t < co} which has a denumerable state space S and which possesses the M=kov pPOpepty. i.e .• for every n;" 2. 0 s tl < ..... < tn and any i l ...... in in S one has (I. I. I) pdX(tn )-i n I X(tl)-il •••••• X(tP I)-iP I}- Pr{X(tn ) - i n I X(t n-I) - i n-I}' The process is supposed to be temporaLLy homogeneous. i.e •• for every i. j in S the conditional probability Pr{X(t+s) - j I Xes) - i} does not depend on s. In this case we may put (I. I. 2) p l.J. (t) - Pr{X(t+s) a j IX(s) - i} t ;" O. The function p .. (.) is the transition ppobabiLity jUnation from the 8tate i to lJ the 8tate j. The ab80Lute di8tPibution at time t;" 0 is defined to be {Pi (t): i € S}; where (1.1.3) Pi (t) - pr{X(t) - i}. l.: p. (t)· I. i 1 The absolute distribUtion at time t -0 is called the initiaL di8tPibution. We have the obvious relation (1.1.4) t ;" O. (.». The denumerable array of functions (p l.J. i. j € S. is the tronsition mat1'i:x: of the Markov process. It satisfies for every i. j and s. t the conditions (1.1.5) p .. (t);"O lJ (1.1.6) l.: p ..( t) - I j lJ (1.1.7) ~ Pik (s) Pkj (t) a Pij (t+s) • Conversely. any array of functions (p .. (.» satisfying (1.1.5) - (1.1.7) and lJ the initial condition p .. (0) - cs •• (cs •• is Kroneckers's delta) for every i.j € S lJ lJ lJ and s. t;"O. is the transition matrix of a Markov process {X(t): OSt<oo} 2 for which (I. 1.2) holds. A transition matrix is called standard iff (I.1.8) H!!! p •• (t) • p .• (O) ;; 6 ••• t+o 1J 1J 1J A standard transition matrix has the property that each Pij('} is uniformly continuous. Furthermore, the right-hand derivatives (I.1.9) q 1•J• • p!1 J• (O) • lUimO (P1'J' (t) - 61·J·}/t (p! . (t) will denote the right-hand derivative when t. 0, and the two-sided 1J derivative when t > O) exist, and they are finite except possibly when i· j. Always (I. L JO) i I' j, (I.I.II) 1: q .. s -q .. s ...., jl'i 1J 11 and inmost cases of practical interest (I. I .12) q 1••1 > -... i € S, (I.1.13) 1: q ••• 0 i € S. j 1J The matrix (q .. ) • (p! .(O}) is the q-matrire of the transition matrix (p .. (.}); 1J 1J 1J it is stable iff (1.1.12) holds and conservative iff it is stable and (1.1.13) holds. The elements p .. (.) of a stable transition matrix, i.e., a transition 1J matrix having a stable q-matrix, have continuous derivatives. The functions p .. (.) satisfy the bacTaJard equa-tione 1J (I.1.14) iff the q-matrix is conservative. Given a conservative q-matrix, i.e., a matrix (qij) with non-negative elements off the main diagonal and satisfying both (1.1.12) and (1.1.13), then there exists at least one standard transition matrix (p •• (.}) for which p! .(O) • q .. , but in 1J 1J 1J general this transition matrix is not unique. When it is unique the conservative q-matrix will be called noPmal. For the elements p .. (.) of a stable transition \ll8trix to satisfy the fOThJal'd 1J equations (I.I.IS) it is sufficient (but not necessary) that the q-matrix is normal. 3 Given a transition matrix (p •• (.». i.j € S. the state i is called absorbing iff 1J Pii(t) • I for all t > O. A necessary and sufficient condition for this is (I. I. 16) The state i is .r.e OU1'1'ent iff .... (1.1.17) !p .. (t) o u otherwise it is called transient. A recurrent state i is called positive or null according as Pii > 0 or Pii • O. where (1.1.18) p ••• lim p •• (t). 1J t-+oo 1J The latter limit exists for every i and j. A Markov process with transition matrix (Pij('» is called transient (null PeOU1'1'ent. positive PeOU1'1'ent) iff every state i € S is transient (null recurrent. positive recurrent). The process is said to be i~duaible on S' c S when p •• (t) > 0 for all i.j € S' and t> O. It was shown by 1J KENDALL (1959) that a Markov process with normal q-matrix has p •. (t) > 0 for all 1J t> 0 iff there exists a finite sequence (i.kl ••••• kr.j) with r > 0 and satisfying (1.1.19) 1.2 StOChastic monotonicity We define R as the set of probability distribution vectors on E _ {O.I.2 •••• }.i.e •• (1.2.1) r .• J} 1 (superscript T will denote transpose). DEFINITION 1.2.1. Let ~(I) .~(2) € R. Then r(l) dominates r(2) (~(I) o~ ~(2» iff for i • 1.2 .... (I) (2) (1.2.2) .r..r. ~ .r..r. Jl!1 J Jl!l. J The vector ~(I) stPiotly dominates ~(2) (~(I) 0> ~(2» iff strict inequality holds in (l.i.2) for i • 1.2 ••••• DEFINITION 1.2.2. An operator L mapping R into R is monotone iff for every pair ~(I) .~(2) € R with ~(I) 0> ~(2) (1.2.3) L(~(I» 0> L(~(2». 4 Now let {X(t): O:s; t < oo} be II Markov process with state space = S = E {O, I, }, say. The transition matrix (Pij('» defines a set (in fact, a semigroup) of operators P t' t ~ 0, mapping R into R by means of T (\.2.4) (Pt(-r»J. - Ei r.1.p 1...J (t), -r - (rO' rl, •••• ) € R. {Pt: O:S;t<oo} is the set of transition operutors of {X(t)}. If one denotes the probability distribution vector of {X(t)} at time t ~ 0 by (1.2.5) the relation (1.1.4) may be written as (1.2.6) More generally one has for s, t ~ 0 (1.2.7) as can easily be verified. In view of definition 1.2.1 it is natural to introduce the concept of stochastic monotonicity in terms of {X(t)} as follows. DEFINITION 1.2.3. The process {X(t)} :i.sstoahaatiooZZy inareasing (deareaaing) on the interval (tl, t2) iff for every pair TI ' T2 with o:s; tl:S; TI < T2 < t2 :s; .. (1.2.8) The process is striatZy stoahastiaaZZy inareasing (deareasing) iff strict domination prevails throughout. It appears from the next theorem that in the context of stochastic monotonicity, .monotone transition operators are of particular interest. THEOREM 1.2.4 (STOYAN (1977), Satz 4.2.4b). The Markov proaess {X(t)} is stoahastiaaZZy inareasing (deareasing) on the intervaZ (tl, .. ) if the transition operutors P t' t ~ 0, of {X(t)} are monotone and there e:x:ists a number T > 0 suah that {X(t)} is stoahastiaaZZy inareasing (deareasing) on the intervaZ (tl, tl + T). Several authors have given necessary and suffieient conditions for the transition operator P t to be monotone for all t ~ 0 (KEItsON and KESTER (\977), KIRSTEIN (\976), STOYAN (1977». The following is STOYAN's result.

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