We consider the problem of existence and uniqueness of semimartingale reflecting Brownian motions (SRBM's) in convex polyhedrons. Loosely speaking, such a process has a semimartingale decomposition such that in the interior of the polyhedron the process behaves like a Brownian motion with a constant drift and covariance matrix, and at each of the (d \Gamma 1)-dimensional faces that form the boundary of the polyhedron, the bounded variation part of the process increases in a given direction (constant for any particular face), so as to confine the process to the polyhedron. For historical reasons, this "pushing " at the boundary is called instantaneous reflection. For simple convex polyhedrons, we give a necessary and sufficient condition on the geometric data for the existence and uniqueness of an SRBM. For non-simple convex polyhedrons, our condition is shown to be sufficient. It is an open question as to whether our condition is also necessary in the non-simple case. From the uniqueness, it follows that an SRBM defines a strong Markov process. Our results have application to the study of diffusions arising as heavy traffic limits of multiclass queueing networks and in particular, the non-simple case has application to multiclass fork and join networks. Our proof of weak existence uses a patchwork martingale problem introduced by T. G. Kurtz, whereas uniqueness hinges on an ergodic argument similar to that used by L. M. Taylor and R. J. Williams to prove uniqueness for SRBM's in an orthant.

We consider a series of n single-server queues, each with unlimited waiting space and the first-in first-out service discipline. Initially, the system is empty; then k customers are placed in the first queue. The service times of all the customers at all the queues are i.i.d. with a general distribution. We are interested in the time D(k, n) required for all k customers to complete service from all n queues. In particular, we investigate the limiting behavior of D(k, n) as n and/or k . There is a duality implying that D(k, n) is distributed the same as D(n , k) so that results for large n are equivalent to results for large k. A previous heavy-traffic limit theorem implies that D(k, n) satisfies an invariance principle as n , converging after normalization to a functional of k-dimensional Brownian motion. We use the subadditive ergodic theorem and a strong approximation to describe the limiting behavior of D(k n , n) where k n as n . The case of k n = xn corresponds to a hydrodyna...

Abstract. Part II continues the development of policy synthesis techniques for multiclass queueing networks based upon a linear fluid model. The following are shown: (i) A relaxation of the fluid model based on workload leads to an optimization problem of lower dimension. An analogous workload-relaxation is introduced for the stochastic model. These relaxed control problems admit pointwise optimal solutions in many instances. (ii) A translation to the original fluid model is almost optimal, with vanishing relative error as the networkload ρ approaches one. It is pointwise optimal after a short transient period, provided a pointwise optimal solution exists for the relaxed control problem. (iii) A translation of the optimal policy for the fluid model provides a policy for the stochastic networkmodel that is almost optimal in heavy traffic, over all solutions to the relaxed stochastic model, again with vanishing relative error. The regret is of order | log(1 − ρ)|.

This paper is concerned with a class of multidimensional diffusion processes, variously known as reflected Brownian motions, regulated Brownian motions, or just RBM's, that arise as approximate models of queueing networks. We develop an algorithm for numerical analysis of a semimartingale RBM with state space S = R d + (the non-negative orthant of d-dimensional Euclidean space). This algorithm lies at the heart of the QNET method [13] for approximate two-moment analysis of open queueing networks. KEY WORDS: Brownian system model, reflected Brownian motion, stationary distribution, numerical analysis, open queueing networks, performance analysis Contents 1. Introduction 1 2. Definitions and Preliminaries 3 3. The Basic Adjoint Relationship 6 4. An Algorithm 9 5. Choosing a Reference Density and fHng 12 6. Numerical Comparisons 16 7. Analysis of an Illustrative Queueing Network 22 AMS 1980 Subject Classification: primary 60J70, 60K30, 65U05; secondary 65P05, 68M20. Abbreviated titl...

Multidimensional reflected Brownian motions, also called regulated Brownian motions or simply RBM's, arise as approximate models of queueing networks. Thus the stationary distributions of these diffusion processes are of interest for steady-state analysis of the corresponding queueing systems. This paper considers two-dimensional semimartingale RBM's with rectangular state space, which include the RBM's that serve as approximate models of finite queues in tandem. The stationary distribution of such an RBM is uniquely characterized by a certain basic adjoint relationship, and an algorithm is proposed for numerical solution of that relationship. We cannot offer a general proof of convergence, but the algorithm has been coded and applied to special cases where the stationary distribution can be determined by other means; the computed solutions agree closely with previously known results, and convergence is reasonably fast. Our current computer code is specific to two-dimensional rectangle...

We consider a single class open queueing network, also known as a generalized Jackson network (GJN). A classical result in heavytraffic theory asserts that the sequence of normalized queue length processes of the GJN converge weakly to a reflected Brownian motion (RBM) in the orthant, as the traffic intensity approaches unity. However, barring simple instances, it is still not known whether the stationary distribution of RBM provides a valid approximation for the steady-state of the original network. In this paper we resolve this open problem by proving that the re-scaled stationary distribution of the GJN converges to the stationary distribution of the RBM, thus validating a so-called “interchange-of-limits” for this class of networks. Our method of proof involves a combination of Lyapunov function techniques, strong approximations and tail probability bounds that yield tightness of the sequence of stationary distributions of the GJN.

In recent years a new approach has emerged for analyzing the stability properties of constrained stochastic processes. In this approach, one associates with the stochastic model a deterministic model (or a family of deterministic models), and, under appropriate conditions, stability of the stochastic model follows if all solutions of the deterministic model are attracted to the origin. In the present work we show that a rather sharp characterization for the stability of the deterministic model is possible when it can be represented in terms of what we call a "regular" Skorokhod Map. Let G be a convex polyhedral cone with vertex at the origin given as the intersection of half spaces {G i , i = 1, ..., N}, where n i and d i denote the inward normal and direction of constraint associated with G i . Suppose that the Skorokhod Problem defined by the data {(n i , d i ), i = 1, ..., N} is regular. Under these conditions, the deterministic model mentioned above will correspond to a law of...

We consider a very general type of d-station open queueing network, with multiple customer classes and a more or less arbitrary service discipline at each station, but restricted by the requirement that customers always flow from lowered numbered stations to higher numbered ones. To approximate the behavior of such a queueing network under heavy traffic conditions, a corresponding Brownian network model is proposed, and it is shown that the approximating Brownian model reduces to a d-dimensional reflected Brownian motion W whose state space is the non-negative orthant. A necessary and sufficient condition for W to have a product form stationary distribution (that is, a stationary distribution with independent components), and a probabilistic interpretation for that condition, are given. Our interpretation involves a notion of quasireversibility analogous to that introduced by F. P. Kelly and elaborated by J. Walrand in their brilliant analysis of product form solutions for conventional...

We consider a connection-level model of Internet congestion control, introduced by Massoulié and Roberts [36], that represents the randomly varying number of flows present in a network. Here bandwidth is shared fairly amongst elastic document transfers according to a weighted α-fair bandwidth sharing policy introduced by Mo and Walrand [37] (α ∈ (0,∞)). Assuming Poisson arrivals and exponentially distributed document sizes, we focus on the heavy traffic regime in which the average load placed on each resource is approximately equal to its capacity. A fluid model (or functional law of large numbers approximation) for this stochastic model was derived and analyzed in a prior work [29] by two of the authors. Here we use the long time behavior of the solutions of this fluid model established in [29] to derive a property called multiplicative state space collapse, which loosely speaking shows that in diffusion scale the flow count process for the stochastic model can be approximately recovered as a continuous lifting of the workload process. Under weighted proportional fair sharing of bandwidth (α = 1) and a mild

This paper is concerned with the estimation of performance measures of two priority disciplines in a d-station re-entrant queueing network. Such networks arise from complex manufacturing systems such as wafer fabrication facilities. The priority disciplines considered are first-buffer-first-served and last-buffer-first-served. An analytical method is developed to estimate the long-run average workload at each station and the mean sojourn time in the network. When the first-buffer-first-served discipline is used, a refined estimate of the mean sojourn time is also developed. The workload estimation has two steps. In the first step, following Harrison and Williams (1992), we use a d-dimensional reflecting Brownian motion (RBM) to model the workload process. We prove that the RBM exists and is unique in distribution, and it has a unique stationary distribution. We then use an algorithm of Dai and Harrison (1992) to compute the stationary distribution of the RBM. Our method uses both the ...