Asymptotic properties of first passage times for tandem Jackson networks I: Buildup of large queue lengths

The authors consider d M/M/1 queues in series. Asymptotic expansions are constructed for the mean time until the total customer population in the network reaches a large number K. The present approach employs formal asymptotic techniques such as boundary layer theory and matched asymptotic expansions. The authors give formulas which show that the mean time behaves as αK’βexp(γK) for K⇒•, where α,β,γ are explicitly given in terms of the arrival and service rates.