Customer Equilibrium and Social Optimization in a Markovian Queue with Fuzzy Parameters

Abstract

This paper analyzes the customers’ equilibrium strategy and optimal social benefit in a Markovian queueing system, in which the arrival rate, service rate of customers, as well as the reward and holding cost are all fuzzy numbers. Based on Zadeh’s extension principle, we investigate the membership functions of the optimal and equilibrium strategies in both observable and unobservable cases. Furthermore, by applying the \(\alpha \)-cut approach, the family of crisp strategy is described by formulating a pair of parametric nonlinear programs, through which the membership functions of the strategy can be derived. Finally, numerical examples are solved successfully to illustrate the validity of the proposed approach and to show the relationship of these strategies and social benefits. Our main contribution is showing that the value of equilibrium and optimal strategies have no deterministic relationship, which are different from the results in the corresponding crisp queues. Moreover, the successful extension of queue game to fuzzy environments can provide more precise information to the system managers.