Asymptotics for Transient and StationaryProbabilities for Finite and In nite

Consider a discrete time queue with i.i.d. arrivals (see the generalisation below) and a single server with a buuer length m. Let Tm be the rst time an overrow occurs. We obtain asymptotic rate of growth of moments and distributions of Tm as m !1. We also show that under general conditions, the overrow epochs converge to a compound Poisson process. Furthermore, we show that the results for the overrow epochs are qualitatively as well as quantitatively diierent from the excursion process of an innnite buuer queue studied in continuous time in the literature. Asymptotic results for several other characteristics of the loss process are also studied, e.g., exponential decay of the probability of no loss (for a xed buuer length) in time 0; n], as n !1, total number of packets lost in 0; n], maximum run of loss states in 0; n]. We also study tails of stationary distributions. All results extend to multiserver case and most to a Markov modulated arrival process.