Journal of Computing Research and Innovation (Mar 2024)
Expression of the Pollaczeck-Khintchine fuzzy formulas for a fuzzy retrial queuing system FM/FG/1-FR
Abstract
The Pollaczek-Khintchine formulas are one of the best and most widely used strategies in the analysis of non-markovian standard or retrial queuing systems with a single server and a general service law. This is particularly the case for classical M/G/1 or M/G/1-R queuing systems because these formulas establish a direct link between the mean number of customers in the system and two first moments of the general service law. The Pollaczek-Khintchine formulas generally allow to evaluate any performance measure Ψ of classical M/G/1-R queuing system by a formula such as: , where are respectively the operating parameters of the system and the two first moments mentioned above. In a fuzzy environment, the literature shows that researchers simply resort to Zadeh's extension principle to obtain fuzzy formula from the classical version above. Instead of doing this to evaluate the performance measures of a non-Markovian fuzzy queuing system denoted FM/FG/1-FR, we have shown in this text that it is possible to derive fuzzy formulas of the kind: , which are an emanation of the fuzzy generating functions of stationary distributions of the number of customers in orbit and in the system; and in which the fuzzy moments of order 1 and 2 follow directly from the fuzzy distribution function of the general service law. This is the originality of this paper and its contribution is to show how Pollakzek-Khintchine fuzzy formulas can be constructed from these two generating functions. The formulas thus obtained are the same as those obtained from the classical versions by extension according to Zadeh's extension principle. So, they can be validly applied in the evaluation of performance measures of the fuzzy retrial queuing system FM/FG/1-FR.
