On limiting behavior of ordinary and conditional first-passage times for a class of birth-death processes

Abstract

LetN(t) be a birth-death process on 𝒩= {0, 1, 2, ·· ·} governed by the transition ratesλn> 0 (n≧ 0) andμn> 0 (n≧ 1) whereλn→λ> 0 andμn→μ> 0 asn→ ∞ andρ=λ/μ. LetTmnbe the first-passage time ofN(t) frommtonand defineIt is shown that, whenconverges in distribution toTBP(μ,λ)asn → ∞whereTΒΡ (μ,λ)is the server busy period of anM/M/1 queueing system with arrival rateμand service rateλ. CorrespondinglyT0n/E[T0n] converges to 1 with probability 1 asn→∞. Of related interest is the conditional first-passage timemTrnofN(t) from r tongiven no visit tomwherem < r < n.As we shall see, the conditional first-passage time ofN(t) can be viewed as an ordinary first-passage time of a modified birth-death processM(t) governed bywhereare generated fromλnandμn. Furthermore it is shown that forandwhile forandThis enables one to establish the relation between the limiting behavior of the ordinary first-passage times and that of the conditional first-passage times.