Fuzzy Computational Procedure for System Length and Waiting Time Distribution in an $M^{X}/D/c/N$ Queue under Parameter Uncertainty

Abstract

This paper introduces a fuzzy computational method for analyzing a finite buffer multiple server $M^{X}/D/c/N$ queuing system with uncertain variables. Unlike common models that assume precise knowledge of arrival, service, and batch size distributions, this model handles these variables as triangular fuzzy numbers. This reflects parameter uncertainty caused by limited data and operational changes. The fuzzy characteristic equation is created, and using the $\alpha$-cut method, the problem changes into a set of interval polynomial equations. We show that fuzzy stationary probabilities do exist for the system. Using interval arithmetic and the Laplace transform with Pad\'e approximation, we find solutions for fuzzy mean queue length, arrival rate, and mean waiting time. We then use a practical example based on public services to show how the method works. A sensitivity study measures how uncertainty spreads into congestion levels. The results show that parameter uncertainty grows when systems approach saturation, requiring strong capacity planning. This approach maintains the analytical style of the root-based method while giving specific performance limits, making it a mathematically sound and understandable extension of multi-server queuing theory under uncertainty.