An Efficient Method to Compute the Stationary Probabilities of the $G{I}^X\mathrm{/}M\mathrm{/}c\mathrm{/}N$ Model

Consider the batch-arrival $G{I}^X\mathrm{/}M\mathrm{/}c\mathrm{/}N$ 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 $i\mathrm{.}i\mathrm{.}d\mathrm{.}$ and independent of each other. In this article we give a simple efficient algorithm to derive an exact solution for the steady state system size probabilities. The starting point is computing the one-step transition probabilities of the imbedded Markov chain observed at the system arrival epochs of the the corresponding $G\mathrm{/}M\mathrm{/}c$ 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 $GI\mathrm{/}M\mathrm{/}c$ model to batch arrivals with and without a finite buffer, and in the case of finite buffer with partial and full batch rejection. Moreover, we develop an efficient stable algorithm that can accurately solve problems with a larger number of servers than previously known. We give numerical examples to demonstrate the performance of our method.}