The QNET Method for Re-Entrant Queueing Networks with Priority Disciplines

Abstract

This paper is concerned with the estimation of performance measures of two priority disciplines in a d-station re-entrant queueing network. Such networks arise from complex manufacturing systems such as wafer fabrication facilities. The priority disciplines considered are First-Buffer-First-Served (FBFS) and Last-Buffer-First-Served (LBFS). An analytical method is developed to estimate the long-run average workload at each station and the mean sojourn time in the network. When the first-buffer-first-served discipline is used, a refined estimate of the mean sojourn time is also developed. The workload estimation has two steps. In the first step, following Harrison and Williams (Harrison, J. M., R. J. Williams. 1992. Brownian models of feedforward queueing networks: Quasireversibility and product form solutions. Anns. Appl. Prob. 2 263–293.), we use a d-dimensional reflecting Brownian motion (RBM) to model the workload process. We prove that the RBM exists and is unique in distribution and that it has a unique stationary distribution. We then use an algorithm of Dai and Harrison (Dai, J. G., J. M. Harrison. 1992. Reflected Brownian motion in an orthant: Numerical methods for steady-state analysis. Anns. Appl. Prob. 2 65–86.) to compute the stationary distribution of the RBM. Our method uses both the first and second moment information, and it is rooted in heavy traffic theory. It is closely related to the QNET method of Harrison and Nguyen (Harrison, J. M., V. Nguyen. 1993. Brownian models of multiclass queueing networks: Current status and open problems. Queueing Systems: Theory and Appl. 13 5–40.) for two-moment analysis of First-In-First-Out (FIFO) discipline. Our performance estimates of several example problems are compared to the simulation estimates to illustrate the effectiveness of our method.