A LINEAR PROGRAMMING APPROACH TO THE STEADY-STATE ANALYSIS OF REFLECTED BROWNIAN MOTION

We investigate a computational method proposed by Bertsimas, Paschalidis, and Tsitsiklis [[1]] and by Kumar, Kumar, and Meyn 2-4 for characterizing the steady-state behavior of Markov processes by means of bounds on the moments of their stationary distributions. In particular, we apply the method to the reflected Brownian motions (RBMs) used to model queueing networks in heavy traffic. In the RBM setting, the bounds are derived from the basic adjoint relationship (BAR)—an assertion that in stationarity, the expected value of the differential generator of the RBM applied to a “good” test function is zero. Coupled with appropriate choices of test functions, the BAR characterizes the stationary distribution of the RBM as the solution of a countable set of linear equations. Using polynomials as test functions, we derive a series of linear programs involving the moments of the stationary distribution as “decision variables”. We discuss the convergence of this series of linear programs and present numerical results for three sample networks.

[1]  Stefan Rolewicz,et al.  On a problem of moments , 1968 .

[2]  S. Varadhan,et al.  Diffusion processes with boundary conditions , 1971 .

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

[4]  D. Dawson.  Qualitative behavior of geostochastic systems , 1980 .

[5]  J. Harrison,et al.  On the distribution of m ultidimen-sional reflected Brownian motion , 1981 .

[6]  J. Harrison,et al.  Reflected Brownian Motion on an Orthant , 1981 .

[7]  G. Pólya,et al.  Problems and theorems in analysis , 1983 .

[8]  J. Harrison,et al.  A tandem storage system and its diffusion limit , 1984 .

[9]  G. Cassier,et al.  Problème des moments sur un compact de Rn et décomposition de polynômes a plusieurs variables , 1984 .

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

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

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

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

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

[15]  Awi Federgruen,et al.  Characterization and Optimization of Achievable Performance in General Queueing Systems , 1988, Oper. Res..

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

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

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

[19]  S. Meyn,et al.  Stability of Markovian processes I: criteria for discrete-time Chains , 1992, Advances in Applied Probability.

[20]  J. Harrison,et al.  Reflected Brownian Motion in an Orthant: Numerical Methods for Steady-State Analysis , 1992 .

[21]  John N. Tsitsiklis,et al.  Optimization of multiclass queuing networks: polyhedral and nonlinear characterizations of achievable performance , 1994 .

[22]  David D. Yao,et al.  Multiclass Queueing Systems: Polymatroidal Structure and Optimal Scheduling Control , 1992, Oper. Res..

[23]  J. M. Harrison,et al.  Brownian Models of Feedforward Queueing Networks: Quasireversibility and Product Form Solutions , 1992 .

[24]  J. Michael Harrison,et al.  Brownian models of multiclass queueing networks: Current status and open problems , 1993, Queueing Syst. Theory Appl..

[25]  S. Meyn,et al.  Stability of Markovian processes III: Foster–Lyapunov criteria for continuous-time processes , 1993, Advances in Applied Probability.

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

[27]  J. Harrison,et al.  The QNET Method for Two-Moment Analysis of Closed Manufacturing Systems , 1993 .

[28]  S. Meyn,et al.  Stability of Markovian processes II: continuous-time processes and sampled chains , 1993, Advances in Applied Probability.

[29]  Ruth J. Williams,et al.  Lyapunov Functions for Semimartingale Reflecting Brownian Motions , 1994 .

[30]  P. R. Kumar,et al.  Performance bounds for queueing networks and scheduling policies , 1994, IEEE Trans. Autom. Control..

[31]  Viên Nguyen,et al.  Sequential Bottleneck Decomposition: An Approximation Method for Generalized Jackson Networks , 1994, Oper. Res..

[32]  Jan A. Van Mieghem,et al.  Brownian Models of Closed Queueing Networks: Explicit Solutions for Balanced Three-Station Systems , 1994 .

[33]  Sean P. Meyn,et al.  Duality and linear programs for stability and performance analysis of queueing networks and scheduling policies , 1994, Proceedings of 1994 33rd IEEE Conference on Decision and Control.

[34]  Sean P. Meyn,et al.  Stability of queueing networks and scheduling policies , 1995, IEEE Trans. Autom. Control..

[35]  Elizabeth Schwerer.  A linear programming approach to the steady-state analysis of Markov processes , 1996 .

[36]  Ruth J. Williams,et al.  Existence and Uniqueness of Semimartingale Reflecting Brownian Motions in Convex Polyhedrons , 1996 .