Stochastic Monotonicity and Queueing Applications of Birth-Death Processes

by Erik van Doorn

Overview

A shastic process {X(t): 0 S t < =} with discrete state space S c ~ is said to be shastically increasing (decreasing) on an interval T if the probabilities Pr{X(t) > i}, i E S, are increasing (decreasing) with t on T. Shastic 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, shastic monotonicity becomes a more tractable subject for analysis if the processes under consideration are such that shastic mono tonicity on an inter­ val 0 < t < E implies shastic 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 "shastic 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 shas­ tically monotone as defined above, now depends on the initial state distribution. Conditions on this distribution for shastic mono tonicity on the entire time axis to prevail were given too by KEILSON and KESTER (1977).

Product Details

ISBN-13:	9780387905471
Publisher:	Springer New York
Publication date:	02/20/1981
Series:	Lecture Notes in Statistics , #4
Edition description:	Softcover reprint of the original 1st ed. 1981
Pages:	118
Product dimensions:	6.10(w) x 9.25(h) x 0.01(d)

Table of Contents

1 : Preliminaries.- 1.1 Markov processes.- 1.2 Shastic monotonicity.- 1.3 Birth-death processes.- 1.4 Some notation and terminology.- 2 : Natural Birth-Death Processes.- 2.1 Some basic properties.- 2.2 The spectral representation.- 2.3 Exponential ergodicity.- 2.4 The moment problem and related topics.- 3 : Dual Birth-Death Processes.- 3.1 Introduction.- 3.2 Duality relations.- 3.3 Ergodic properties.- 4 : Shastic Monotonicity: General Results.- 4.1 The case—0 = 0.- 4.2 The case—0 > 0.- 4.3 Properties of E(t).- 5 : Shastic Monotonicity: Dependence on the Initial State Distribution.- 5.1 Introduction to the case of a fixed initial state.- 5.2 The transient and null recurrent process.- 5.3 The positive recurrent process.- 5.4 The case of an initial state distribution with finite support.- 6 : The M/M/S Queue Length Process.- 6.1 Introduction.- 6.2 The spectral function.- 6.3 Shastic monotonicity.- 6.4 Exponential ergodicity.- 7 : A Queueing Model Where Potential Customers are Discouraged by Queue Length.- 7.1 Introduction.- 7.2 The spectral representation.- 7.3 Shastic monotonicity and exponential ergodicity.- 8 : Linear Growth Birth-Death Processes.- 8.1 Introduction.- 8.2 Shastic monotonicity.- 9 : The Mean of Birth-Death Processes.- 9.1 Introduction.- 9.2 Representations.- 9.3 Sufficient conditions for finiteness.- 9.4 Behaviour of the mean in special cases.- 10 : The Truncated Birth-Death Process.- 10.1 Introduction.- 10.2 Preliminaries.- 10.3 The sign structure of P’(t).- 10.4 Shastic monotonicity.- Appendix 1 : Proof of the Sign Variation Diminishing Property of Strictly Totally Positive Matrices.- Appendix 2: On Products of Infinite Matrices.- Appendix 3: On the Sign of Certain Quantities.- Appendix 4: Proof of Theorem 10.2.8.-References.- Notation Index.- Author Index.