Transition Probabilities of the GI/M/c and GIX/M/c/N Models

Abstract

Consider the batch-arrival GI^X/M/c/N model with c servers, general inter- arrival batch times, finite buffer, and exponential service times. Inter-arrival batch times, batch sizes, and service times are i.i.d. and independent of each other. In this article we give a simple efficient method to derive the one-step transition probabilities of the imbedded Markov chain observed at the system arrival epochs of the corresponding G/M/c model. The one-step transition probabilities are computed exactly by converting a numerical integration problem into a finite sum. Another key contribution is generating the transition probabilities of the batch-arrival model by using a simple and intuitive method to extend the results of the standard GI/M/c model to batch arrivals with and without a finite buffer, and in the case of finite buffer with partial and full batch rejection. We give examples to demonstrate the performance of our method.