This paper concerns a dynamic scheduling problem for a queueing system that has two streams of arrivals to infinite capacity buffers and two (non-identical) servers working in parallel. One server can only process jobs from one buffer, whereas the other server can process jobs from either buffer. The service time distribution may depend on the buffer being served and the server providing the service. The system manager dynamically schedules waiting jobs onto available servers. We consider a parameter regime in which the system satisfies both a heavy traffic condition and a resource pooling condition. Our cost function is a mean cumulative discounted cost of holding jobs in the system, where the (undiscounted) cost per unit time is a linear function of normalized (with heavy traffic scaling) queue length. We first review the analytic solution of the Brownian control problem (formal heavy traffic approximation) for this system. We "interpret" this solution by proposing a threshold contro...

scientific non-commercial use only for individuals, with permission from the authors. We consider a parallel server queueing system consisting of a bank of buffers for holding incoming jobs and a bank of flexible servers for processing these jobs. Incoming jobs are classified into one of several different classes (or buffers). Jobs within a class are processed on a first-in-first-out basis, where the processing of a given job may be performed by any server from a given (class-dependent) subset of the bank of servers. The random service time of a job may depend on both its class and the server providing the service. Each job departs the system after receiving service from one server. The system manager seeks to minimize holding costs by dynamically scheduling waiting jobs to available servers. We consider a parameter regime in which the system satisfies both a heavy traffic and a complete resource pooling condition. Our cost function is an expected cumulative discounted cost of holding jobs in the system, where the (undiscounted) cost per unit time is a linear function of normalized (with heavy traffic scaling) queue length. In a prior work [40], the second author proposed a continuous review threshold control policy for use in such a parallel server system. This policy was advanced as an “interpretation ” of the analytic solution to an associated Brownian control problem (formal heavy

Using a slight modification of the framework in Bramson [7] and Williams [52], we prove heavy traffic limit theorems for six families of multiclass queueing networks. The first three families are single station systems operating under first-in first-out (FIFO), generalized head-of-the-line proportional processor sharing (GHLPPS) and static buffer priority (SBP) service disciplines. The next two families are re-entrant lines operating under first-buffer-first-serve (FBFS) and last-buffer-first-serve (LBFS) service disciplines; the last family consists of certain 2-station, 5-class networks operating under an SBP service discipline. Some of these heavy traffic limits have appeared earlier in the literature; our new proofs demonstrate the significant simplifications that can be achieved in the present setting.

We consider a class of open stochastic processing networks, with feedback routing and overlapping server capabilities, in heavy traffic.

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

Abstract. The paper studies a closed queueing network containing two types of node. The first type (server station) is an infinite server queueing system, and the second type (client station) is a single server queueing system with autonomous service, i.e. every client station serves customers (units) only at random instants generated by strictly stationary and ergodic sequence of random variables. It is assumed that there are r server stations. At the initial time moment all units are distributed in the server stations, and the ith server station contains Ni units, i = 1,2,..., r, where all the values Ni are large numbers of the same order. The total number of client stations is equal to k. The expected times between departures in the client stations are small values of the order O(N −1) (N = N1 +N2 +...+Nr). After service completion in the ith server station a unit is transmitted to the jth client station with probability pi,j (j = 1, 2,..., k), and being served in the jth client station the unit returns to the ith server station. Under the assumption that only one of the client stations is a bottleneck node, i.e. the expected number of arrivals per time unit to the node is greater than the expected number of departures from that node, the paper derives the representation for non-stationary queue-length distributions in non-bottleneck client stations.

Abstract — In this paper we consider packet networks loaded by admissible traffic patterns, i.e. by traffic patterns that, if optimally routed, do not overload network resources. In these conditions, we study the combined behavior of distributed dynamic routing and scheduling algorithms based upon link state information, with no knowledge of the average traffic pattern, and we prove that simple schemes can achieve the same network throughput as optimal centralized routing and scheduling algorithms with complete information on the traffic pattern. Our study is based on a flow-level abstract model of the network, and considers elastic traffic, i.e., we assume that flows can adapt their transmission rates to network conditions. As a result, our model captures some of the main features of Internet traffic and of quality-of-service routing approaches being

In this paper, we extend the work of Chen and Zhang (2000b) and establish a new sufficient condition for the existence of the (conventional) diffusion approximation for multiclass queueing networks under priority service disciplines. This sufficient condition relates to the weak stability of the fluid networks and the stability of the high priority classes of the fluid networks that correspond to the queueing networks under consideration. Using this sufficient condition, we prove the existence of the diffusion approximation for the last-bufferfirst-served reentrant lines. We also study a three-station network example, and observe that the diffusion approximation may not exist, even if the “proposed” limiting semimartingale reflected Brownian motion (SRBM) exists.

Semimartingale Reflecting Brownian Motions (SRBMs) living in the closures of domains with piecewise smooth boundaries are of interest in applied probability because of their role as heavy traffic approximations for some stochastic networks. In this paper, assuming certain conditions on the domains and directions of reflection, a perturbation result, or invariance principle, for SRBMs is proved. This provides sufficient conditions for a process that satisfies the definition of an SRBM, except for small random perturbations in the defining conditions, to be close in distribution to an SRBM. A crucial ingredient in the proof of this result is an oscillation inequality for solutions of a perturbed Skorokhod problem. We use the invariance principle to show weak existence of SRBMs under mild conditions. We also use the invariance principle, in conjunction with known uniqueness results for SRBMs, to give some sufficient conditions for validating approximations involving (i) SRBMs in convex polyhedrons with a constant reflection vector field on each face of the polyhedron, and (ii) SRBMs in bounded domains with piecewise smooth boundaries and possibly non-constant reflection vector fields on the boundary surfaces. Copyright c○W. Kang and R. J. Williams, 2006. All print and electronic rights reserved.

We consider a queueing network of d single server stations. Each station has a finite capacity waiting buffer, and all customers served at a station are homogeneous in terms of service requirements and routing. The routing is assumed to be deterministic and hence feedforward. A server stops working when the downstream buffer is full. We show that a properly normalized d-dimensional queue length process converges in distribution to a d-dimensional semimartingale reflecting Brownian motion (RBM) in a d-dimensional box under a heavy traffic condition. The conventional continuous mapping approach does not apply here because the solution to our Skorohod problem may not be unique. Our proof relies heavily on a uniform oscillation result for solutions to a family of Skorohod problems. The oscillation result is proved in a general form that may be of independent interest. It has the potential to be used as an important ingredient in establishing heavy traffic limit theorems for general finite buffer networks. Key words and phrases: Finite capacity network, blocking probabilities, loss network, semimartingale reflecting Brownian motion, RBM, heavy traffic, limit theorems, oscillation estimates.