This paper proposes an algorithm, referred to as BNA/FM (Brownian network analyzer with finite element method), for computing the stationary distribution of a semimartingale reflecting Brownian motion (SRBM) in a hypercube. The SRBM serves as an approximate model of queueing networks with finite buffers. Our BNA/FM algorithm is based on finite element method and an extension of a generic algorithm developed by Dai and Harrison (1991). It uses piecewise polynomials to form an approximate subspace of an infinite dimensional functional space. The BNA/FM algorithm is shown to produce good estimates for stationary probabilities, in addition to stationary moments. This is in contrast to BNA/SM (Brownian network analyzer with spectral method) of Dai and Harrison (1991), where global polynomials are used to form the approximate subspace and it sometime fails to produce meaningful estimates of these stationary probabilities. Extensive computational experiences from our implementation are reported that may be useful for future numerical research on SRBMs. A three-station tandem network with finite buffers are presented to illustrate the effectiveness of the Brownian approximation model and our BNA/FM algorithm.

The diffusion approximation is proved for a class of multiclass queueing networks under FIFO service disciplines. In addition to the usual assumptions for a heavy traffic limit theorem, a key condition that characterizes this class is that a J \Theta J matrix G, known as the workload contents matrix, has a spectral radius less than unity, where J represents the number of service stations. The (j; `)th component of matrix G can be interpreted as the amount of future work for station j that is embodied in per unit of immediate work at station ` at time t. This class includes Rybko-Stolyar network with FIFO service discipline as a special case. The result extends existing diffusion limiting theorems to non-feedforward multiclass queueing networks. In establishing the diffusion limit theorem, a new approach is taken. The traditional approach is based on an oblique reflection mapping, but such a mapping is not well-defined for the network under consideration. Our approach takes two steps: f...

This paper derives the strong approximation for a multiclass queueing network, where jobs after service completion can only move to a downstream service station. Job classes are partitioned into groups. Within a group, jobs are served in the order of arrival, i.e., a rst-in-rst-out (FIFO) discipline is in force, and among groups, jobs are served under a pre-assigned preemptive priority discipline. We obtain the strong approximation for the network, through an inductive application of an input-output analysis for a single station queue. Specically, we show that if the input data (i.e., the arrival and the service processes) satisfy an approximation (such as the functional law-of-iterated logarithm approximation or the strong approximation), then the output data (i.e., the departure processes) and the performance measures (such as the queue length, the workload and the sojourn time processes) satisfy a similar approximation. Based on the strong approximation, some procedures are propo...

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.

This paper introduces the framework of synchronous constrained fluid systems (SCFS) to model

The goal of this chapter is to demonstrate the usefulness of analytical and numerical methods of stochastic control theory in the design, analysis and control of telecommunication networks. The emphasis will be concentrated on the heavy traffic approach for queueing type systems in which there is little idle time and the queue length processes can be approximated by reected diffusion processes under suitable scaling. Three principal problems are considered: the multiplexer system, controlled admission in multiserver systems such as ISDN, and the polling or scheduling problem.