When is a MAP poisson?

Abstract

The departure process of a queue is important in the analysis of networks of queues, as it may be the arrival process to another queue in the network. A simple description of the departure process could enable a tractable analysis of a network, saving costly simulation or avoiding the errors of approximation techniques. In a recent paper, Olivier and Walrand [1] conjectured that the departure process of a MAP/PH/1 queue is not a MAP unless the queue is a stationary M/M/1 queue. This conjecture was prompted by their claim that the departure process of an MMPP/M/1 queue is not a MAP unless the queue is a stationary M/M/1 queue. We note that their proof has an algebraic error, see [2], which leaves the above question of whether the departure process of an MMPP/PH/1 queue is a MAP, still open. There is also a more fundamental problem with Olivier and Walrand's proof. In order to identify stationary M/M/1 queues, it is essential to be able determine from its generator when a stationary MAP is a Poisson process. This is not discussed in [1], nor does it appear to have been discussed elsewhere in the literature. This deficiency is remedied using ideas from nonlinear filtering theory, to give a characterisation as to when a stationary MAP is a Poisson process.