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 ∞) 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.

This paper presents new formulations of the approximate mean value analysis (MVA) algorithms for the performance evaluation of closed product-form queueing networks. The key to the development of the algorithms is the derivation of vector nonlinear equations for the approximate network throughput. We solve this set of throughput equations using a nonlinear Gauss-Seidel type distributed algorithms, coupled with a quadratically convergent Newton's method for scalar nonlinear equations. The throughput equations have enabled us to: (a) derive bounds on the approximate throughput; (b) prove the existence, uniqueness, and convergence of the Schweitzer-Bard (S-B) approximation algorithm for a wide class of monotone, single class networks, (c) establish the existence of the S-B solution for multi-class, monotone networks, and (d) prove the asymptotic (i.e., as the number of customers of each class tends to ∞) uniqueness of the S-B throughput solution, and the asymptotic convergence of the various versions of the distributed algorithms in multi-class networks with single server and infinite server nodes. The asymptotic convergence is established using results from convex programming and convex duality theory. Extension of our algorithms to mixed networks is straighfoward. Only multi-class results are presented in this paper.