Efficient analysis of the queue length moments of the MMAP/MAP/1 preemptive priority queue

The analysis of priority queues where both the arrival and the service processes are correlated does not have a long literature. Only a few results are known, that attack the problem with the matrix geometric machinery. Unfortunately, these results have some restrictions that limit their usability significantly, for example they require the calculation of infinite series of matrices and infinite summations, that can be implemented only by truncation.The method presented in this paper calculates only the queue length moments, but without relying on infinite series of matrices and provides procedures to calculate the arising infinite sums accurately in an efficient way by means of linear equations, matrix-quadratic equations and a coupled matrix-quadratic equation.The numerical examples demonstrate that the presented method is several orders of magnitudes faster than the existing ones. From the large number of queue length moments it is possible to obtain lower and upper bounds for the queue length distribution by using existing moment based distribution estimation results.