On the tail behavior of queues with discrete autoregressive arrivals

Abstract

Autoregressive arrival models are described by a few parameters and provide a simple mean to obtain analytical models for matching first and second order statistics of the measured data. We consider a discrete time queueing system where one customer (cell) per slot is transmitted, and the arrival process is governed by DAR(1) (Discrete AutoregRessive process of order 1) characterized by an arbitrary stationary batch size distribution and a correlation coefficient. The tail behavior of the queue length and waiting time distributions are examined. In particular, it is shown that unlike the classical queueing models with Markovian arrival processes, the correlation in the DAR(1) model results in a non-geometric tail behavior of the queue occupancy (and the waiting time) if the stationary distribution of the DAR(1) has an infinite support. A complete characterization of the geometric tail behavior of the queue occupancy (and the waiting time) is also presented showing the impact of the stationary distribution and the correlation coefficient when the stationary distribution of the DAR(1) has a finite support.