Markov-Modulated Flow Model for the Output Queues of a Packet Switch

In this paper, we study the outuut queues of an M x N packet switch using a Markov-modulated flow model. The switching element is a central server which sequentially routes packets from the inputs to the outputs. We focus on systems where the server speed is such that the bulk of the queueing takes place in the output queues. For such systems, accurate sizing of the output buffers is an important design issue and requires a correct characterization of the arrival processes to the output buffers. The conventional point process approach neglects the impact of switching and transmission time. We attempt to account for these finite system speeds by using a Markov-modulated continuous flow to approximate the arrival process to an output queue. This model captures the dependency between arrivals at different outputs and reflects the fact that packet arrivals and departures are not instantaneous. The output queue content distribution is obtained, for both infinite and finite buffer systems, from the spectral expansion of the solution of a system of differential equations. Numerical examples and comparisons with the results of an M/M/1 approximation are presented.