Static Optimization of Queueing Systems

Abstract

This paper discusses some recent developments in the static: optimization of queueing systems. Special attention is given to three problem classes: (i) the optimal allocation of servers, or service capacity, to queues in a network; (ii) the optimal allocation of the visits of a single server to several queues (a polling system); (iii) the optimal allocation of a single arrival stream to several single server queues. l. INTRODUCTIONWhen several users compete for the use of a common resource, the limited capacity of the resource can give rise to congestion. This situation occurs in a plethora of everyday situations: people queue at a counter in a bank or supermarket, congestion occurs in road traffic, products encounter delays at machines during their production process, messages wait for access to a common transmission channel and computer jobs for the use of a set of processors. Queueing occurs even when the service capacity of the resource strongly exceeds the demand. This is due to the fact that the interarrival times of the users, and their required service times, are generally not fixed. A mathematical model of congestion phenomena therefore usually represents interarrival and service times of users by random variables. The resources are called service facilities, with a single server or multiple servers, and the users are called customers. Customers often visit a number of service facilities, encountering several queues during their stay in the system. Queueing theory is devoted to the description, analysis and optimization of such queueing systems. It concentrates on a few key performance measures, like queue lengths and waiting times. Due to the stochastic nature of the arrival and service processes, and of the routing process of customers through a network of queues, the main performance measures are also random variables (or moments thereof). Generally, costs are in a natural way associated with these performance measures. The ultimate goal of performance analysis is optimization -and that is the subject of this paper. While an enormous amount of literature has been devoted to the probabilistic analysis of queueing systems, their optimization is somewhat lagging behind. This is partly due

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