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The subject of this paper is open multiclass queueing networks, which are common models of communication networks, and complex manufacturing systems such as wafer fabrication facilities. We provide su cient conditions for the existence of bounds on long-run average moments of the queue lengths at the various stations, and we bound the rate of convergence of the mean queue length to its steady state value. Our work provides a solid foundation for performance analysis either by analytical methods or by simulation. These results are applied to several examples including re-entrant lines, generalized Jackson networks, and a general polling model as found in computer networks applications.

Complex systems like semiconductor wafer fabrication facilities (fabs), networks of data switches, and large-scale call centers all demand efficient resource allocation. Deterministic models like linear programs (LP) have been used for capacity planning at both the design and expansion stages of such systems. LP-based planning is critical in setting a medium range or long-term goal for many systems, but it does not translate into a day-to-day operational policy that must deal with discreteness of jobs and the randomness of the processing environment. A stochastic processing network, advanced by J. Michael Harrison (2000, 2002, 2003), is a system that takes inputs of materials of various kinds and uses various processing resources to produce outputs of materials of various kinds. Such a network provides a powerful abstraction of a wide range of real-world systems. It provides high-fidelity stochastic models in diverse economic sectors including manufacturing, service, and information technology. We propose a family of maximum pressure service policies for dynamically allocating service capacities in a stochastic processing network. Under a mild assumption on network structure, we prove that a network operating under a maximum pressure policy achieves maximum throughput predicted by LPs. These policies are semilocal in the sense that each server makes its decision based on the buffer content in its serviceable buffers and their immediately downstream buffers. In particular, their implementation does not use arrival rate information, which is difficult to collect in many applications. We also identify a class of networks for which the nonpreemptive, non-processor-splitting version of a maximum pressure policy is still throughput optimal. Applications to queueing networks with alternate routes and networks of data switches are presented.

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Reentrant lines can be used to model complex manufacturing systems such as wafer fabrication facilities. As the first step to the optimal or near-optimal scheduling of such lines, we investigate their stability. In light of a recent theorem of Dai (1995) which states that a scheduling policy is stable if the corresponding fluid model is stable, we study the stability and instability of fluid models. To do this we utilize piecewise linear Lyapunov functions. We establish stability of First-Buffer-First-Served (FBFS) and Last-Buffer-First-Served (LBFS) disciplines in all reentrant lines, and of all work-conserving disciplines in any three buffer reentrant lines. For the four buffer network of Lu and Kumar we characterize the stability region of the Lu and Kumar policy, and show that it is also the global stability region for this network. We also study stability and instability of Kelly-type networks. In particular, we show that not all work-conserving policies are stable for such networks; however, all work-conserving policies are stable in a ring network.

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-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 nonsimple convex polyhedrons, our condition is shown to be sufficient. It is an open question as to whether our condition is also necessary in the nonsimple case. From the uniqueness, it follows that an SRBM defines a strong Markov process. Our results are applicable to the study of diffusions arising as heavy traffic limits of multiclass queueing networks and in particular, the nonsimple case is applicable 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. Key words, semimartingale reflecting Brownian motion, diffusion process, nonsimple convex polyhedron, completely-S matrix, martingale problems, multiclass queueing networks, fork and join networks 1. Introduction. This paper is concerned with the existence and uniqueness of a class of semimartingale reflecting Brownian motions which live in a d-dimensional convex polyhedron S (d __> 1). The polyhedron is defined in terms of m (m __> 1) ddimensional unit vectors {hi, E J}, J-= {1,..., m}, and an m-dimensional vector b (bl,... ,bin) , where prime denotes transpose. (Hereafter vectors are taken to be column vectors.) The state space S is defined by (1.1) S :_ {x R d" ni x >= bi for all J}, where ni x nix denotes the inner product of the vectors ni and x. It is assumed that the interior of S is nonempty and that the set ((nl, bl),..., (n,, b,)} is minimal in the sense that no proper subset defines S. That is, for any strict subset K C J, the set (x R d" hi. x __> bi V K} is strictly larger than S. This is equivalent to the assumption that each of the faces (1.2) Fi =_ {x e S: ni.x bi}, e J, has dimension d-1 (cf. [7, Thm. 8.2]). As a consequence, ni is the unit normal to Fi that points into the interior of S.

We consider a class of stochastic processing networks. Assume that the networks satisfy a complete resource pooling condition. We prove that each maximum pressure policy asymptotically minimizes the workload process in a stochastic processing network in heavy traffic. We also show that, under each quadratic holding cost structure, there is a maximum pressure policy that asymptotically minimizes the holding cost. A key to the optimality proofs is to prove a state space collapse result and a heavy traffic limit theorem for the network processes under a maximum pressure policy. We extend a framework of Bramson [Queueing Systems Theory Appl. 30 (1998) 89-148] and Williams [Queueing Systems Theory Appl. 30 (1998b) 5-25] from the multiclass queueing network setting to the stochastic processing network setting to prove the state space collapse result and the heavy traffic limit theorem. The extension can be adapted to other studies of stochastic processing networks. represents a significant advance in finding efficient operational policies for a wide class of networks, and is closely related to Dai and Lin (2005). Readers are referred to Dai and Lin (2005) for an explanation of the major differences of these two works. We note that, contrary to the description in Dai and Lin (2005), Tassiluas and Ephremides (1992, 1993) and Tassiulas (1995) do cover network models, not just one-pass systems. For a recent survey of these policies and their applications to wireless networks, see Georgiadis, Neely and Tassiulas (2006).The remainder of the paper is organized as follows. In Section 1.1 we collect some of the notation used in this paper. In Section 2 we describe a class of stochastic processing networks, and introduce the maximum pressure service policies. We then define the workload process of a stochastic processing network in Section 3, where we also introduce the complete resource pooling condition. The main results of this paper are stated in Section 4. The proofs of the main theorems are outlined in Section 5. A key to the proofs of these theorems is a state space collapse result of the diffusion-scaled network processes under a maximum pressure policy. In Section 6 each fluid model solution under a maximum pressure policy is shown to exhibit a state space collapse. Section 7 applies Bramson's approach [Bramson (1998)] to prove the state space collapse of the diffusion-scaled network processes. The state space collapse result is converted into a heavy traffic limit theorem in Section 8. The limit theorem is used in Section 5 to complete the proofs of the main theorems. In Section 9 we discuss the ε-optimality of maximum pressure policies. A number of technical lemmas as well as Theorem 1 are proved in the Appendix A.