Time-dependent queue length distribution in queues fed by K customers in a finite interval

Abstract

Abstract We consider queueing models, where customers arrive according to a continuous-time binomial process on a finite interval. In this arrival process, a total of K customers arrive in the finite time interval [0, T ], where arrival times of those K customers are independent and identically distributed according to an absolutely continuous distribution defined by its probability density function f ( t ) on (0, T ]. To analyze the time-dependent queue length distribution of this model, we introduce an auxiliary model with non-homogeneous Poisson arrivals and show that the time-dependent queue length distribution in the original model is given in terms of the time-dependent joint distribution of the numbers of arrivals and departures in the auxiliary model. Next, we consider a numerical procedure for computing the time-dependent queue length distribution in Markovian models with piecewise constant f ( t ). A particular feature of our computational procedure is that the truncation error bound can be specified as an input parameter. Some numerical examples are also provided.