Approximations for Superposition Arrival Processes in Queues

S. L. Albin has described extensive simulations of queue behavior for a system with a single server and an arrival process that is a superposition of n renewal processes. The simulations show, among other things, that as n increases for a fixed traffic intensity \rho, the queue behavior approaches that of the M/M/1 system. The rate of convergence, however, becomes much slower as the traffic intensity \rho comes closer to 1. Several qualitative effects shown in the simulations are explained here. In particular it is shown that the approach to the M/M/1 system requires that n(1 - \rho)² \gg 1.