The Conditional Distribution of Waiting Time Given Queue Length

In this paper we propose a conditional distribution of the waiting time given the queue length. We assume that the number of customers in a queue, that is the queue length, follows a geometric distribution and the waiting time of a newly arrived customer in a queue follows an exponential distribution. We model the conditional distribution of the waiting time given the queue length using Gumbel's bivariate exponential copula. Parameters are estimated using the likelihood approach with a two stage estimation procedure. A simulation study indicates the performance of the estimators. Asymptotic properties of the estimators of the parameters are studied. We apply the model to a real data set to illustrate the dependence between the waiting time and the queue length.