Asymptotics for transient and stationary probabilities for finite and infinite buffer discrete time queues

Abstract

Consider a discrete time queue with i.i.d. arrivals (see the generalisation below) and a single server with a buffer length m. Let {T}_{m} be the first time an overflow occurs. We obtain asymptotic rate of growth of moments and distributions of {T}_{m} as m \to \infty. We also show that under general conditions, the overflow epochs converge to a compound Poisson process. Furthermore, we show that the results for the overflow epochs are qualitatively as well as quantitatively different from the excursion process of an infinite buffer 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 fixed buffer length) in time [0, n], as n \to \infty, 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 the multiserver case and most to a Markov modulated arrival process.