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First-passage-time densities and avoiding probabilities for birth-and-death processes with symmetric sample paths

Part of: Markov processes Special processes

Published online by Cambridge University Press:  14 July 2016

Antonio Di Crescenzo
Antonio Di Crescenzo*
Affiliation:
Università di Napoli ‘Federico II’
*
∗Postal address: Dipartimento di Matematica e Applicazioni ‘R. Caccioppoli’, Università di Napoli ‘Federico II’, via Cintia, 80126, Napoli, Italy. E-mail address: [email protected]
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Abstract

For truncated birth-and-death processes with two absorbing or two reflecting boundaries, necessary and sufficient conditions on the transition rates are given such that the transition probabilities satisfy a suitable spatial symmetry relation. This allows one to obtain simple expressions for first-passage-time densities and for certain avoiding transition probabilities. An application to an M/M/1 queueing system with two finite sequential queueing rooms of equal sizes is finally provided.

Keywords

Truncated processestransition probabilitiesspatial symmetryabsorptionreflectionfirst-passage timeavoiding transition probabilitiesM/M/1 queueing systembusy period

MSC classification

Primary: 60J80: Branching processes (Galton-Watson, birth-and-death, etc.)
Secondary: 60J85: Applications of branching processes 60K30: Applications (congestion, allocation, storage, traffic, etc.)
Type
Research Papers
Information
Journal of Applied Probability , Volume 35 , Issue 2 , June 1998 , pp. 383 - 394
Copyright
Copyright © Applied Probability Trust 1998 

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Footnotes

Work supported in part by Italian National Research Council (CNR) under Contracts Nos. 95.742.01, 95.1090.01, 96.180.01, 96.3859.01 and by MURST (40% funds).

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