Asymptotics for Steady-state Tail Probabilities in Structured Markov Queueing Models

In this paper we establish asymptotics for the basic steady-state distributions in a large class of single-server queues. We consider the waiting time, the workload (virtual waiting time) and the steady-state queue lengths at an arbitrary time, just before an arrival and just after a departure. We start by establishing asymptotics for steady-state distributions of Markov chains of M/GI/1 type. Then we treat steady-state distributions in the BMAP/GI/1 queue, which has a batch Markovian arrival process (BMAP). The BMAP is equivalent to the versatile Markovian point process or Neuts (N) process; it generalizes the Markovian arrival process (MAP) by allowing batch arrivals. The MAP includes the Markov-modulated Poisson process (MMPP), the phasetype renewal process (PH) and independent superpositions of these as special cases. We also establish asymptotics for steady-state distributions in the MAP/MSP/1 queue, which has a Markovian service process (MSP). The MSP is a MAP independent of the arrival process generating service completions during the time the server is busy. In great generality (but not always), the basic steady-state distributions have asymptotically exponential tails in all these models. When they do, the asymptotic parameters of the different distributions are closely related.