Aproximação para sistemas de filas M/M/c com servidores heterogêneos

Abstract

This work is specifically aplied to the case of C heterogenous servers, exponentially distributed, which assist a single FCFS line formed for just one type of customer class. A mathematical formulation is presented to obtain an upper bound for the performance measures. Basically, surch formulation is obtained through an expansion of the state space resulting from the heterogeneity of the severs and afterwards through a reduction of that state space. That reduction is feasible because only the possibilities of larger probality are considered. With that, it is possible to find the worst case for the average waiting time in queue and for the average number of people in the queue, as well as for the average waiting time in the system and the average number of people in the system. In most of the cases that formulation becomes attractive as it is capable to approximate the real behavior of those systems of heterogenous servers with an error that is smaller than if it was calculated using the traditional MMc. Results from simulations, which were run in GPSS, are showed with the intention of validating the created formulation and to compare the relative error resulted from it with the error from other approaches. The Gini's Index is used to make the comparison among systems possible as it makes viable the classification of the systems according to their heterogeneity. As a result of that it is possible to evaluate which is the resultant effect on the performance measures of those queueing systems, In addition, some analyses on the influence which different allocation polices have over the results are also presented.