Queues with dependency between interarrival and service times using mixtures of bivariates

Abstract

We analyze queueing models where the joint density of the interarrival time and the service time is described by a mixture of joint densities. These models occur naturally in multiclass populations serviced by a single server through a single queue. Other motivations for this model are to model the dependency between the interarrival and service times and consider queue control models. Performance models with component heavy tailed distributions that arise in communication networks are difficult to analyze. However, long tailed distributions can be approximated using a finite mixture of exponentials. Thus, the models analyzed here provide a tool for the study of performance models with heavy tailed distributions. The joint density of A and X , the interarrival and service times respectively, f ( a , x ), will be of the form f(a, x) = Sigma(i=1)(M) p(i)f(i)(a,x) where p(i) > 0 and Sigma(i=1)(M) p(i)=1. We derive the Laplace Stieltjes Transform of the waiting time distribution. We also present and discuss some numerical examples to describe the effect of the various parameters of the model.