ANALYSIS OF PARETO/M/s/c QUEUES USING UNIFORMIZATION

Abstract– Uniformization is a powerful numerical technique for the analysis of continuous time Markov chains that possesses a probabilistic interpretation. In this paper, we propose the combination of the embedded Markov chain approach for queues with uniformization to analyze GI/M/s/c queues. The main steps of the procedure are: (1) To compute the mixed-Poisson probabilities associated to the number of arrival epochs of the uniformizing Poisson process in-between two consecutive customer arrivals to the queueing system; and (2) To compute the conditional embedded uniformized transition probabilities of the number of customers in the queueing system in-between two consecutive customer arrivals to the system. To show the power of the approach, we analyze Pareto/M/s/c queues using a stable recursion for the mixed-Poisson probabilities with a computation time that is linear on the number of computed coefficients. As a result, fast numerical computation of loss probabilities in moderate size queues with Pareto renewal arrival process and exponential servers is achievable. The results for Pareto/M/s/c queues are compared with those obtained for queues with other renewal arrival processes, including Gamma/M/s/c, M/M/s/c and D/M/s/c. The results show that much higher loss probabilities and mean waiting times in queue may be obtained for Pareto renewal arrival processes than for the other renewal arrival processes considered, especially for small customer arrival rates.