Consider a job-shop or batch-flow manufacturing system in which new jobs are introduced only as old ones depart, either because of physical constraints or as a matter of management policy. Assuming that there is never a shortage of new work to be done, the number of active jobs remains constant over time, and the system can be modeled as a kind of closed queueing network. With manufacturing applications in mind, we formulate a general closed network model and develop a mathematical method to estimate its steady-state performance characteristics. A restrictive feature of our network model is that all the job classes that are served at any given node or station share a common service time distribution. Our analytical method, which is based on an algorithm for computing the stationary distribution of an approximating Brownian model, is motivated by heavy traffic theory; it is precisely analogous to a method developed earlier for analysis of open queueing networks. The required inputs include not only first-moment information, such as average product mix and average processing rates, but also second-moment data that serve as quantitative measures of variability in the processing environment. We present numerical examples that show that system performance is very much affected by changes in second-moment data. In these few numerical examples, our estimates of average throughput rates and average throughput times for different product families are generally accurate when compared against simulation results.
"The QNET Method for Two-Moment Analysis of Closed Manufacturing Systems." Ann. Appl. Probab. 3 (4) 968 - 1012, November, 1993. https://doi.org/10.1214/aoap/1177005269