Nigel G Bean and David A Green,
Department of Applied Mathematics,
University of Adelaide,
5005, Australia.
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 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 Bean, Green and Taylor, which leaves the above question of whether the departure process of an MMPP/PH/1 queue is a MAP or not, 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 to determine from its generator when a stationary MAP is a Poisson process. This is not discussed in Olivier and Walrand, nor does it appear to have been discussed elsewhere in the literature. This deficiency is remedied using ideas from non-linear filtering theory, to give a characterisation as to when a stationary MAP is a Poisson process.