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.

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This work is concerned with the existence and uniqueness of a strong Markov process that has continuous (i) The state space is an infinite two-dimensional wedge, and the process behaves in the interior of (ii) The process reflects instantaneously at the boundary of the wedge, the angle of reflection being (iii) The amount of time that the process spends at the comer of the wedge is zero (i.e., the set of Hereafter, let .$ be the angle of the wedge (0 < .$ < 2 4 , let 8, and 8, be the angles of reflection on the two sides of the wedge, measured from the inward normals, the positive angles being toward the comer (-4-< e,, 8, < h), and set a = (8, + 8~/ . $ .The question of existence and uniqueness is recast as a submartingale problem in the style used by Stroock and Varadhan (Difusion processes with boundary conditions, Comm. Pure Appl. Math. 24, 1971, pp. 147-225), for diffusions on smooth domains with smooth boundary conditions. It is shown that no solution exists if a 2 2. In this case, there is a unique continuous strong Markov process satisfying (i) and (ii) above; it reaches the comer of the wedge almost surely and it remains there. If a < 2, however, then there is a unique continuous strong Markov process satisfying (i)-(iii). It is shown that starting away from the comer this process does not reach the comer of the wedge if a 5 0, and does reach the comer if 0 < a < 2.The general theory of multi-dimensional diffusions does not apply to the above problem because in general the boundary of the state space is not smooth and there is a discontinuity in the direction of reflection at the comer. For some values of a, the process arises from diffusion approximations to storage systems and queueing networks. sample paths and the following additional properties: the wedge like an ordinary Brownian motion. constant along each side. times for which the process is at the comer has Lebesgue measure zero).

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Overloaded enzymatic processes are shown to create indirect coupling between upstream components in cellular networks. This has important implications for the design of synthetic biology devices and for our understanding of currently inexplicable links within endogenous biological systems.

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This reprint differs from the original in pagination and typographic detail. 1 2 KANG, KELLY, LEE AND WILLIAMS approximation for the flow count process. In this case, the workload diffusion behaves like Brownian motion in the interior of a polyhedral cone and is confined to the cone by reflection at the boundary, where the direction of reflection is constant on any given boundary face. When all of the weights are equal (proportional fair sharing), this diffusion has a product form invariant measure. If the latter is integrable, it yields the unique stationary distribution for the diffusion which has a strikingly simple interpretation in terms of independent dual random variables, one for each of the resources of the network.We are able to extend this product form result to the case where document sizes are distributed as finite mixtures of exponentials and to models that include multi-path routing. We indicate some difficulties related to extending the diffusion approximation result to values of α = 1. We illustrate our approximation results for a few simple networks. In particular, for a two-resource linear network, the diffusion lives in a wedge that is a strict subset of the positive quadrant. This geometrically illustrates the entrainment of resources, whereby congestion at one resource may prevent another resource from working at full capacity. For a four-resource network with multi-path routing, the product form result under proportional fair sharing is expressed in terms of independent dual random variables, one for each of a set of generalized cut constraints.

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