Estimating the stationary distribution in a GI/M/1-queue

Abstract

We consider nonparametric estimation of the stationary distribution of the number of customers in a GI/M/1-queueing system just before arrival times, in the situation where the service time distribution is assumed to be exponential with known mean and the interarrival time distribution is unknown. It is supposed that a random sample of interarrival times is available. The proposed estimator is the stationary distribution for the GI/M/1-queue with the same service time distribution as for the original queueing system and with interarrival time distribution given by the empirical distribution based on the sample of interarrival times. The stationary distribution is defined in terms of the solution to an equation involving the Laplace transform of the interarrival time distribution; this solution is estimated by the solution of the corresponding equation in terms of the empirical Laplace transform. We derive strong consistency and asymptotic normality of this estimated solution and use these results to obtain limiting results for the law of the L 1 and the L 2 distance of the estimated stationary distribution from the unknown true distribution.