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On conditional passage time structure of birth-death processes

Published online by Cambridge University Press:  14 July 2016

Ushio Sumita
Ushio Sumita*
Affiliation:
The University of Rochester
*
∗ Postal address: The Graduate School of Management, The University of Rochester, Rochester, NY 14627, U.S.A.
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Abstract

Let N(t) be a birth-death process on N = {0,1,2,· ··} governed by the transition rates λn > 0 (n ≧ 0) and μ η > 0 (n ≧ 1). Let mTm be the conditional first-passage time from r to n, given no visit to m where m <r < n. The downward conditional first-passage time nTm is defined similarly. It will be shown that , for any λn > 0 and μ η > 0. The limiting behavior of is considerably different from that of the ordinary first-passage time where, under certain conditions, exponentiality sets in as n →∞. We will prove that, when λn → λ > 0 and μ η → μ > 0 as n → ∞with ρ = λ /μ < 1, one has as r → ∞where TBP(λ,μ) is the server busy period of an M/M/1 queueing system with arrival rate λand service rate μ.

Keywords

FIRST-PASSAGE TIMESCONDITIONAL FIRST-PASSAGE TIMES
Type
Research Papers
Information
Journal of Applied Probability , Volume 21 , Issue 1 , March 1984 , pp. 10 - 21
Copyright
Copyright © Applied Probability Trust 1984 

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Footnotes

This research was done while the author was at Syracuse University and was published as Working Paper No. 82-001.

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