Brownian models of open queueing networks with homogeneous customer populations^∗

Abstract

We consider a family of multidimensional diffusion processes that arise as heavy traffic approximations for open queueing networks. More precisely, the diffusion processes considered here arise as approximate models of open queueing networks with homogeneous customer populations, which means that customers occupying any given node or station of the network are essentially indistinguishable from one another. The classical queueing network model of J. R. Jackson fits this description, as do other more general types of systems, but multiclass network models do not.The objectives of this paper are (a) to explain in concrete terms how one approximates a conventional queueing model or a real physical system by a corresponding Brownian model, and (b) to state and prove some new results 77 78 J. M. HARRISON AND R. J. WILLIAMS regarding stationary distributions of such Brownian models. The part of the paper aimed at objective (a) is largely a recapitulation of previous work on weak convergence theorems, with the emphasis placed on modeling intuition. With respect to objective (b), several important foundational issues are resolved here and under certain conditions we are able to express the stationary distribution and related performance measures in explicit formulas. More specifically, it is shown that the stationary distribution of the Brownian model has a separable (product form) density if and only if its data satisfy a certain condition, in which case the stationary density is exponential, and all relevant performance measures can be written out in explicit formulas.KEY WORDS: Brownian system models, heavy traffic, performance analysis, product form solutions, queueing networks, stationary distributions.

A phase transition behavior for Brownian motions interacting through their ranks

“…Arratia in [5] (see also [4]) observes that instead of Brownian motion, if one considers the exclusion process associated with the nearest neighbor random walk on Z, then the corresponding stationary system of spacings between particles can be interpreted as a finite or infinite series of queues. We direct the reader to the seminal articles by Harrison and his coauthors for a background on systems of Brownian queues: [16], [17], [18], [19], and [20]. Baryshnikov [7] establishes the connections between Brownian queues and GUE random matrices.…”

An Ergodic Control Problem for Constrained Diffusion Processes: Existence of Optimal Markov Control

“…if b ∈ C then the corresponding constrained diffusion is transient [5]. In the context of the constant drift case, this necessary and sufficient condition for positive recurrence, for constrained diffusions which correspond to single class networks, was first proved in [14].…”

Heavy Traffic Analysis of a Simple Closed-Loop Supply Chain

“…Static planning problems are formulated by constructing a system where the fluid-scaled processes are replaced by their long-run averages (or fluid limits) and solving a suitable optimization problem involving those averages (see [19,25,28,29]). In the fluid limit, we formulate a deterministic problem to choose ρ andμ that maximize the profit rate subject to stability conditions on both queues.…”