We use cookies to distinguish you from other users and to provide you with a better experience on our websites. Close this message to accept cookies or find out how to manage your cookie settings.
We use cookies to distinguish you from other users and to provide you with a better experience on our websites. Close this message to accept cookies or find out how to manage your cookie settings.
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.
