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

TL;DR

This paper introduces an efficient algorithm for calculating the steady state probabilities of the $GI^X/M/c/N$ queueing model, handling batch arrivals, finite buffers, and multiple servers with improved accuracy and scalability.

Contribution

The authors develop a simple, exact method converting numerical integration into finite sums, extending standard models to batch arrivals and large server systems.

Findings

Exact steady state probabilities computed efficiently.

Method handles large number of servers with high accuracy.

Numerical examples demonstrate improved performance.

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 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 $G/M/c$ 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. 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.}