Markov approximations to D-BMAPs: Information theoretic bounds on queueing performance

Abstract

We consider the problem of approximating a discrete-time batch Markovian arrival process (D-BMAP) by a simple process such that certain corresponding buffer-related performance measures are close approximations of the true performance measures. The motivation comes from network applications where the D-BMAPs are used to represent bursty traffic streams in high-speed networks. We propose an approach for approximating a D-BMAP by a matched Markov process of finite memory obtained by information-theoretic techniques. The performance measures considered are the probability of cell loss due to buffer overflow and the average cell delay in the buffer. We confirm analytically that the approximating performance measures become increasingly accurate with the memory of the matched Markov process. When the parameters of the D-BMAP are unknown, we estimate instead the parameters of a suitable Markov approximation from samples of an observed cell stream. This affords a key advantage as estimation of the parameters of a Markov approximation is much simpler than that of the D-BMAP. We show that the estimated Markov process, with a fixed memory, comes closer with increasing sample size to the D-BMAP, as do the corresponding performance measures, in accordance with the law of iterated logarithm.