On finite capacity queues with time dependent arrival rates

We consider the finite capacity M / M / 1 − K queue with a time dependent arrival rate λ ( t ) . Assuming that the capacity K is large and that the arrival rate varies slowly with time (as t / K ), we construct asymptotic approximations to the probability of finding n customers in the system at time t , as well as the mean number. We consider various time ranges, where the system is nearly empty, nearly full, or is filled to a fraction of its capacity. Extensive numerical studies are used to back up the asymptotic analysis.