Performance bounds for queueing networks and scheduling policies

Abstract

Except for the class of queueing networks and scheduling policies admitting a product form solution for the steady-state distribution, little is known about the performance of such systems. For example, if the priority of a part depends on its class (e.g., the buffer that the part is located in), then there are no existing results on performance, or even stability. In most applications such as manufacturing systems, however, one has to choose a control or scheduling policy, i.e., a priority discipline, that optimizes a performance objective. In this paper the authors introduce a new technique for obtaining upper and lower bounds on the performance of Markovian queueing networks and scheduling policies. Assuming stability, and examining the consequence of a steady state for general quadratic forms, the authors obtain a set of linear equality constraints on the mean values of certain random variables that determine the performance of the system. Further, the conservation of time and material gives an augmenting set of linear equality and inequality constraints. Together, these allow the authors to bound the performance, either above or below, by solving a linear program. The authors illustrate this technique on several typical problems of interest in manufacturing systems. For an open re-entrant line modeling a semiconductor plant, the authors plot a bound on the mean delay (called cycle-time) as a function of line loading. It is shown that the last buffer first serve policy is almost optimal in light traffic. For another such line, it is shown that it dominates the first buffer first serve policy. For a set of open queueing networks, the authors compare their lower bounds with those obtained by another method of Ou and Wein (1992). For a closed queueing network, the authors bracket the performance of all buffer priority policies, including the suggested priority policy of Harrison and Wein (1990). The authors also study the asymptotic heavy traffic limits of the lower and upper bounds. For a manufacturing system with machine failures, it is shown how the performance changes with failure and repair rates. For systems with finite buffers, the authors show how to bound the throughput. Finally, the authors illustrate the application of their method to GI/GI/1 queues. The authors obtain analytic bounds which improve upon Kingman's bound (1970) for E/sub 2//M/1 queues.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>