Sachin R. Gurnule
Classical queueing models usually assume that arrival and service parameters are known precisely; however, in many practical service systems these parameters are uncertain due to demand variation, service fluctuations, incomplete information, and operational ambiguity. This paper presents the performance evaluation of a finite-capacity M/M/1 Jackson-type network queue under fuzzy and intuitionistic fuzzy environments. The suggested network is composed of two service nodes and one Central Server (Head Office). Customers are coming from outside the system at nodes 1, 2 and the Head Office and if a customer needs more services at nodes 1 and 2, he may be referred to the Head Office. The transient behaviour of the system is formulated through Kolmogorov forward equations and represented using an infinitesimal generator matrix. Since closed-form solutions are difficult for the considered finite-capacity network with routing and impatience, the system is solved numerically using the fourth-order Runge–Kutta method in MATLAB. Triangular and trapezoidal fuzzy numbers are used to represent uncertain arrival and service rates, and α-cut analysis is applied to obtain interval bounds of performance measures. The model is further extended to intuitionistic fuzzy environments by applying the α-cut and the β-cut techniques which include membership, non-membership and hesitation. The numerical and graphical results indicate that the uncertainty decreases with the increase of the α-cut value and the intuitionistic fuzzy model with the β-cut gives a more compact and stable performance interval. It is observed that as per the node-wise comparison the Head Office has the highest expected system length and mean waiting time and the same can be interpreted that the Head Office is the main bottleneck of the network.