Modelling the flow of coalescing data streams through a processor

Abstract

In a data processing network, two data streams A and B arrive at a node independently at the same Poisson rate λ. Service at exponential rate µ can take place iff there is at least one of each of A and B present. The output is the combined processed data AB. We consider models of this situation with finite buffers, with infinite buffers and with finite buffers for the excess of each input type over the other. We apply the filtering theory for point process functionals of a Markov chain to study whether the output flow is Poisson in equilibrium. The motivation is to examine, if the output is to subsequently be processed by a queueing system, whether it can be treated as an independent Poisson input to that system. A result of independent interest is that a subset of transitions of a countable-state Markov process does not yield a Poisson process when counted, if the rate matrix of counted transitions is nilpotent, and we prove a generalization of Pakes' lemma for countable-state Markov chains.