A closed-form solution on a level-dependent Markovian arrival process with queueing application

This paper reports a closed-form solution of the arrival events for a particular level-dependent Markovian arrival process (MAP). We apply the Baker–Hausdorff Lemma to the matrix expression of the number of arrival events in (0, t]. The successful derivation depends on the fact that the matrices representing the MAP have a specific structure. We report the results of numerical experiments indicating that the closed-form solution is less time-consuming than the uniformization technique for large values of t. As an application, we consider a finite-capacity, multi-server queueing model with impatient customers for possible use in automatic call distribution (ACD) systems. Our primary interest lies in performance measures related to customer waiting time, and we demonstrate how the closed-form solution is applicable to performance analysis.