Optimal selection of the service rate for a finite input source fuzzy queuing system

摘要

Finite input source queuing models have a wide range of applications in many practical situations and are frequently found in machine maintenance and repair operations. Owing to uncontrollable factors, finite input source queuing problems parameters may contain uncertainty. That is why to begin with, in this paper it is proposed the analysis, development and design of a fuzzy queuing model with finite input source in which the arrival pattern as well as the service pattern follow an exponential distribution under uncertain parameter. In order to design the system, the established optimization criterion consists in determining the optimal number of servers so as to minimize the expected total cost function per unit time. To choose among uncertain alternatives, three techniques to make decisions under uncertainty are proposed. This type of systems has a finite waiting line. As a result, the system of differential equations that governs the model's behaviour consists of a finite number of equations. In the second place, this feature of the finite input source queuing systems enables us to bring forward in this paper two new contributions: (ⅰ) in the first one the following doubt has been raised and solved: if the convergence of the probabilities p_i (t) obtained from the fuzzy differential equations that govern the behaviour of the finite source system is worked out; does the result match the steady-state fuzzy probabilities obtained by applying Zadeh's Extension Principle to the classic queuing model?; and (ⅱ) it has been performed a fuzzy simulation experiment on the differential equations using Bontempi's Qua. Si. HI algorithm. This enables us to know the time evolution of the probabilities P_i(t) and its convergence on the one hand. On the other we can compare the results of the simulation to those analytically obtained by applying Zadeh's Extension Principle. This way the validity of the proposed procedure for the simulation is verified by means of the theoretical results. The extension of queuing decision models to fuzzy environments enables the decision maker to obtain more informing results and wider knowledge on the behaviour of the system, since the results obtained in the fuzzy queuing model are fuzzy subsets containing the whole initial information, that is why the finite input source queuing models with uncertain data can be of more use and can have a broader range of applications.