On finite capacity queues with time dependent arrival rates

We consider the finite capacity M/M/1−KM/M/1−K queue with a time dependent arrival rate λ(t)λ(t). Assuming that the capacity KK is large and that the arrival rate varies slowly with time (as t/Kt/K), we construct asymptotic approximations to the probability of finding nn customers in the system at time tt, 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.