A Method for Analyzing Queues in Fuzzy Environment

Introduction

In this chapter a new method for analyzing queues in fuzzy environment is presented. This method combines the Zadeh‘s extension principle, the α-cut approach and the parametric non-linear programing technique. The necessary basic concepts in fuzzy theory are explained first. The crisp case of a queuing model considered is described and the corresponding steady state results are presented next. Some special cases of interest are also considered. The same model in fuzzy environment is described subsequently and the new method of analyzing the fuzzy model is explained. An algorithm is developed for this purpose. A method for evaluating the membership functions of interest for the fuzzy case is described and is illustrated with a numerical study. The new method can be applied to any queuing system in fuzzy environment.

In this method the input parameters like arrival rate, service rate, vacation rate etc., are described by fuzzy numbers of specific type (triangular, trapezoidal, quadratic, Gaussian etc.,) and the system performance measures like average queue suze, average waiting time in the queue, average number of customers in the system are all obtained as fuzzy numbers of the same type, which include the crisp solution.

Queuing models have wide applications in machine repair, toll booths, taxi stands, loading and unloading of ships, scheduling patients in hospitals, computer field with respect to program scheduling, time sharing, system design, telecommunication and service organizations, where in different type of customers are serviced by different type of servers. In the traditional queuing theory, the inter arrival times and service times are assumed to follow certain preassigned probability distributions. However in many practical situations the arrival patterns and service times can be more realistically described by linguistic expressions like fast, moderate, slow, sufficient or inadequate rather than by probability distributions. Thus all the queuing characteristics like arrival rate, service rate and vacation rate can be better specified by fuzzy numbers. This gives the scope of studying queues in the context of fuzzy theory. Fuzzy queues can be effectively applied in the fields like manufacturing system, telecommunication and data processing. Many authors have contributed to the study of Bulk Queues and Vacation Queues in the crisp case. Some notable works are Cooper (1970, 1981), Chaudhry and Templeton (1983), and Yutaka Baba (1986). Teghem (1992) has studied control of service process in queuing system. Doshi (1990) has analyzed single server queues with vacations. Madan and Dayyeh (2012) have studied M/G/1 type bulk queues with modified Bernoulli schedule server vacation with restricted admissibility. Li and Lee (1999) have derived analytical results for two fuzzy queuing systems based on Zadeh’s extension principle (1978). Kalyanaraman et al. (2009) have analyzed a single server queue with fuzzy service time and vacation time distributions and a single server vacation queue with unreliable server based on Zadeh’s extension principle. The same authors have also analyzed a single server vacation queue with unreliable server in fuzzy environment and a fuzzy bulk queue with modified Bernoulli vacation and restricted admissible customers (2010, 2013).