# Stochastic Monotonicity and Queueing Applications of Birth-Death Processes

• E. A. van Doorn
Book

Part of the Lecture Notes in Statistics book series (LNS, volume 4)

1. Front Matter
Pages i-vi
2. E. A. van Doorn
Pages 1-10
3. E. A. van Doorn
Pages 11-21
4. E. A. van Doorn
Pages 22-27
5. E. A. van Doorn
Pages 28-37
6. E. A. van Doorn
Pages 38-43
7. E. A. van Doorn
Pages 44-65
8. E. A. van Doorn
Pages 66-71
9. E. A. van Doorn
Pages 72-75
10. E. A. van Doorn
Pages 76-86
11. E. A. van Doorn
Pages 87-99
12. Back Matter
Pages 100-118

### Introduction

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).

### Keywords

Geburts- und Todesprozess (Statistik) Markov chain Markov process Monotoner Operator Warteschlange birth-death process ergodicity stochastic process

#### Authors and affiliations

• E. A. van Doorn
• 1
1. 1.Dr. Neher — LaboratoriesNetherlands Postal and Telecommunications ServicesLeidschendamThe Netherlands

### Bibliographic information

• DOI https://doi.org/10.1007/978-1-4612-5883-4
• Copyright Information Springer-Verlag New York 1981
• Publisher Name Springer, New York, NY
• eBook Packages
• Print ISBN 978-0-387-90547-1
• Online ISBN 978-1-4612-5883-4
• Series Print ISSN 0930-0325
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