ASYMPTOTIC APPROXIMATIONS AND BOTTLENECK ANALYSIS IN PRODUCT FORM QUEUING-NETWORKS WITH LARGE POPULATIONS

Asymptotic approximations are constructed for the performance measures of product form queueing networks consisting of single server, fixed rate nodes with large populations. The approximations are constructed by applying singular perturbation methods to the recursion equations o f Mean Value Analysis. Networks with a single job class are studied fi rst to illustrate the use of perturbation techniques. The leading term in the approximation is related to bottleneck analysis, but fails to be accurate if there is more than one bottleneck node. A uniform appro ximation is constructed which is valid for networks with many bottlene ck nodes. The accuracy of the uniform approximation is demonstrated fo r both small and large population sizes. Next, multiclass networks are considered. The leading term in the asymptotic approximation is again related to bottleneck analysis but fails to be valid across ''switchi ng surfaces''. Across these the bottleneck nodes of the network change as a function of the fraction of jobs in the different job classes. A boundary layer correction is constructed near the switching surfaces which provides an asymptotic connection across the switching surfaces. Numerical examples are presented to demonstrate the accuracy of the r esults. We illustrate the asymptotic approach on some simple networks and indicate how to treat more complicated problems. (C) 1998 Elsevier Science B.V. All rights reserved.