Reflected Brownian Motion in an Orthant: Numerical Methods for Steady-State Analysis

This paperis concernedwith a classof multidimensionaldiffusion processes, variously known asreflectedBrownianmotions,regulatedBrownianmotions,or just RBM’s, thatarise asapproximatemodelsof queueingnetworks.We developanalgorithmfor numericalanalysis of a semimartingaleRBM with statespaceS Rd (thenon-negativeorthantof d-dimensional Euclideanspace).This algorithmlies at the heartof the QNETmethod[13] for approximate two-momentanalysisof openqueueingnetworks.

[1]  J. Harrison.  The diffusion approximation for tandem queues in heavy traffic , 1978, Advances in Applied Probability.

[2]  Gerard J. Foschini.  Equilibria for diffusion models of pairs of communicating computers - Symmetric case , 1982, IEEE Trans. Inf. Theory.

[3]  W. Whitt,et al.  The Queueing Network Analyzer , 1983, The Bell System Technical Journal.

[4]  W. Whitt,et al.  Performance of the Queueing Network Analyzer , 1983, The Bell System Technical Journal.

[5]  Martin I. Reiman,et al.  Open Queueing Networks in Heavy Traffic , 1984, Math. Oper. Res..

[6]  Analysis of Brownian motion with drift, confined to a quadrant by oblique reflection , 1984 .

[7]  J. Harrison,et al.  The Stationary Distribution of Reflected Brownian Motion in a Planar Region , 1985 .

[8]  Ruth J. Williams,et al.  Conformal mapping solution of Laplace's equation on a polygon with oblique derivative boundary conditions , 1986 .

[9]  Guy Pujolle,et al.  Introduction to queueing networks , 1987 .

[10]  Ruth J. Williams,et al.  Multidimensional Reflected Brownian Motions Having Exponential Stationary Distributions , 1987 .

[11]  Ruth J. Williams.  Reflected Brownian motion with skew symmetric data in a polyhedral domain , 1987 .

[12]  Ruth J. Williams,et al.  Brownian Models of Open Queueing Networks with Homogeneous Customer Populations , 1987 .

[13]  Ruth J. Williams,et al.  A boundary property of semimartingale reflecting Brownian motions , 1988 .

[14]  J. Michael Harrison,et al.  The QNET method for two-moment analysis of open queueing networks , 1990, Queueing Syst. Theory Appl..

[15]  J. Harrison,et al.  Steady-State Analysis of RBM in a Rectangle: Numerical Methods and A Queueing Application , 1991 .

[16]  A. Bernard,et al.  Regulations dÉterminates et stochastiques dans le premier “orthant” de RN , 1991 .

[17]  William P. Peterson,et al.  A Heavy Traffic Limit Theorem for Networks of Queues with Multiple Customer Types , 1991, Math. Oper. Res..

[18]  J. Dai.  Steady-state analysis of reflected Brownian motions: characterization, numerical methods and queueing applications , 1991 .

[19]  R. J. Williams,et al.  Existence and uniqueness of semimartingale reflecting Brownian motions in an orthant , 1993 .