Approximate mean value analysis algorithms for queuing networks

Abstract

This paper is concerned with the properties of nonlinear equations associated with the Scheweitzer-Bard (S-B) approximate mean value analysis (MVA) heuristic for closed product-form queuing networks. Three forms of nonlinear S-B approximate MVA equations in multiclass networks are distinguished: Schweitzer, minimal, and the nearly decoupled forms. The approximate MVA equations have enabled us to: (a) derive bounds on the approximate throughput; (b) prove the existence and uniqueness of the S-B throughput solution, and the convergence of the S-B approximation algorithm for a wide class of monotonic, single-class networks; (c) establish the existence of the S-B solution for multiclass, monotonic networks; and (d) prove the asymptotic (i.e., as the number of customers of each class tends to m) uniqueness of the S-B throughput solution, and (e) the convergence of the gradient projection and the primal-dual algorithms to solve the asymptotic versions of the minimal, the Schweitzer, and the nearly decoupled forms of MVA equations for multiclass networks with single server and infinite server nodes. The convergence is established by showing that the approximate MVA equations are the gradient vector of a convex function, and by using results from convex programming and the convex duality theory.

Product Form Queueing Networks

“…The Bard-Schweitzer Proportional Estimation [56] is a popular and widely applied approximate algorithm [50]. The average queue length N j (K − 1) of a service center j for a network with K − 1 customers is approximated as follows:…”

“…Linearizer is a quite accurate algorithm and further improvements have been developed [50]. Special extensions of these algorithms have been defined for networks with load dependent service centers.…”

The general form linearizer algorithms: A new family of approximate mean value analysis algorithms

“…Among them, the Bard-Schweitzer Proportional Estimation (PE) algorithm [31] and the Chandy-Neuse Linearizer algorithm [7] have gained wide acceptance among performance analysts. Both the PE and Linearizer algorithms involve iteratively solving a system of nonlinear equations using general purpose numerical techniques such as successive substitution or some variant of Newton's method [26,41]. The computational requirements of these two algorithms are substantially smaller than those of the exact MVA algorithm.…”

“…Most AMVA algorithms involve iteratively solving a set of nonlinear equations using numerical techniques such as successive substitution or some variant of Newton's method [26,41]. The final solution of an iterative AMVA algorithm is independent of the numerical technique chosen if the solution exists and is unique.…”

Approximate Queueing Network Analysis of Patient Treatment Times

“…For certain classes of queueing networks, Mean Value Analysis (MVA) [20,19] and a plethora of related techniques (e.g. [1,16,21,17,22]) provide an efficient and elegant route to mean values of measures of interest (such mean waiting time and throughput), but not higher moments. For closed…”