On Poisson Approximations for Superposition Arrival Processes in Queues

We report on simulations of \Sigma _iGI_i/M/1 queues; the arrival process is the superposition (sum) of up to 1024 i.i.d. renewal processes and there is a single exponential server. As one might anticipate, the simulation estimate of the expected number of customers in a \Sigma _iGI_i/M/1 queueing system approaches the expected number in an M/M/1 queueing system as the number of arrival processes, n, increases. However, for a given n, the difference between the expected numbers in the M/M/1 and \Sigma _iGI_i/M/1 queueing systems dramatically increases as the traffic intensity increases from \rho = 0.5 to \rho = 0.9. This difference is approximated by a formula which is a function of the traffic intensity, the number of component arrival processes and the squared coefficient of variation of the component interarrival times.