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On conditional passage time structure of birth-death processes
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- Journal:
- Journal of Applied Probability / Volume 21 / Issue 1 / March 1984
- Published online by Cambridge University Press:
- 14 July 2016, pp. 10-21
- Print publication:
- March 1984
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- Article
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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.
, for 
is considerably different from that of the ordinary first-passage time 
where, under certain conditions, exponentiality sets in as 
as 