Congestion in a system of N service centers with interference.

Abstract

We have a system of N sendox machines (service centers) which can transmit or receive (but not simultaneously) photocopies to and from each other. The stochastic process {X(M)(t),t ≥ 0}, where X(M)(t) is the number of messages undergoing transmission in the system at epoch t, is selected to be the measure of system congestion. Under the assumptions of exponential inter-arrival and transmission times and that messages not transmitted immediately are lost, X(M)(t) is a birth and death process. We call the model under these assumptions the "loss" model. It is shown that X(M)(t) can be approximated by a diffusion process. The steady state results of the diffusion model can be calculated with ease and are in close agreement with those of the birth and death model and the simulation "loss" model. A more realistic "no loss" model is also developed where messages that are not transmitted immediately are allowed to queue up. It is shown, however, that this "no loss" model can be quite accurately approximated by an appropriate "loss" model.