On conditional passage time structure of birth-death processes

Abstract

Let N(t) be a birth-death process on N = {0,1,2,· ··} governed by the transition rates λ n > 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 λ n > 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 λ n → λ > 0 and μ η → μ > 0 as n → ∞with ρ = λ /μ < 1, one has as r → ∞where T BP(λ,μ) is the server busy period of an M/M/1 queueing system with arrival rate λand service rate μ.