On the Analysis of Queues with Heavy Tails: A Non-Extensive Maximum Entropy Formalism and a Generalisation of the Zipf-Mandelbrot Distribution

A critique of a non-extensive maximum entropy (NME) formalism is undertaken in conjunction with its application into the analysis of queues with heavy tails that are often observed in performance evaluation studies of heterogeneous networks exhibiting traffic burstiness, self-similarity and/or long range dependence (LRD). The credibility of the NME formalism, as a method of inductive inference, for the study of non-extensive systems with long-range interactions is explored in terms of four consistency axioms of extensive systems with shortrange interactions. Focusing on a a general physical system and, as a special case, a single server queue with finite capacity, it is shown that the NME state probability is characterised by a generalisation of the Zipf-Mandelbrot (Z-M) type distribution depicting heavy tails and asymptotic power law behaviour. Typical numerical experiments are employed to illustrate the adverse combined impact of traffic burstiness and self-similarity on the behaviour of the queue. A reference to open issues relating to the NME formalism and open queueing networks is included.