On a Bicriterion Server Allocation Problem for a Multidimensional Erlang Loss System

Abstract

In this work an optimization problem on a classical elementary stochastic system system, modeled as an Erlang-B (M/M/x) loss system, is formulated by using a bicriteria approach. The problem is focused on the allocation of a given total of k servers to a number of groups of servers capable of carrying certain offered traffic processes assumed as Poissonian in nature. Two main objectives are present in this formulation. Firstly a criterion of equity in the grade of service, measured by the call blocking probabilities, entails that the absolute difference between the blocking probabilities experienced by the calls in the different service groups must be as small as possible. Secondly a criterion of system economic performance optimization requires the total traffic carried by the system, to be maximized. Relevant mathematical results characterizing the two objective functions and the set N of the non-dominated solutions, are presented. An algorithm for traveling on N based on the resolution of single criterion convex problems, using a Newton-Raphson method, is also proposed. In each iteration the two first derivatives of the Erlang-B function in the number of circuits (a difficult numerical problem) are calculated using a method earlier proposed. Some computational results are also presented.