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Consider the batch-arrival $GI^X/M/c/N$ model with $c$ servers, generalinter-arrival batch times, finite buffer, and exponential service times.Inter-arrival batch times, batch sizes, and service times are $i.i.d.$ andindependent of each other. In this article we give a simple efficient algorithmto derive an exact solution for the steady state system size probabilities. Thestarting point is computing the one-step transition probabilities of theimbedded Markov chain observed at the system arrival epochs of the thecorresponding $G/M/c$ model. The one-step transition probabilities are computedexactly by converting a numerical integration problem into a finite sum.Another key contribution is generating the transition probabilities of thebatch-arrival model by using a simple and intuitive method to extend theresults of the standard $GI/M/c$ model to batch arrivals with and without afinite buffer, and in the case of finite buffer with partial and full batchrejection. Moreover, we develop an efficient stable algorithm that canaccurately solve problems with a larger number of servers than previouslyknown. We give numerical examples to demonstrate the performance of ourmethod.}

An Efficient Method to Compute the Stationary Probabilities of the $GI^X/M/c/N$ Model