Flows in Queueing Networks: A Martingale Approach

This paper applies the filtering theory for point process observations to the analysis of flows in networks. An abstract model for the study of Markovian queueing systems is developed and used to derive the filtering equations satisfied by the conditional probability distribution of the state given the past of an observed set of flows. These filtering equations are used to obtain a necessary and sufficient condition for that observation to be independent of the value of the state of the system at any time. This approach also yields a probabilistic interpretation of a necessary condition for a given flow in a multi-class Jackson network to be Poisson.

These results are combined to give a proof of the “loop criterion” for single-class Jackson network and to show that this criterion does not apply to multi-class networks. Examples of loop-free Jackson networks are given for which some of the flows are not Poisson.