Performance Analysis of Production Systems with Correlated Demand via Diffusion Approximations

We investigate the performance of a production system with correlated demand through diffusion approximation. The key performance metric under consideration is the extreme points that this system can reach. This problem is mapped to a problem of characterizing the joint probability density of a two-dimensional Brownian motion and its coordinate running maximum. To achieve this goal, we obtain the stationary distribution of a reflected Brownian motion within the positive quarter-plane, which is of independent interest, through investigating a solution of an extended Helmhotz equation. 1. Introduction There are extensive studies on some classic one-dimensional models in the field of operations research, most notably, the work on the behavior and scheduling of single-server queuing system, as well as the understanding of performance and management of single-item inventory system. Probabilistic tools and techniques such as random walks and integral transformations are traditionally used in analyzing them. More recently, concepts and methodologies in dynamical systems and diffusion processes are brought in through fluid and diffusion approximations, and they extend our understanding and capability of analyzing and control of these systems substantially. While highly desirable, extending these studies to their multidimensional counterparts is a rather difficult problem. Extensions of classic probabilistic methods and techniques, such as random walk and integral transform, introduce a new level of complication that requires deeper understanding in algebraic and complex geometry for their general treatment. For the approximations methods, their multidimensional counterparts, multidimensional dynamical systems, and multidimensional diffusion processes, also pose significant barriers for either qualitative understanding and quantitative computation. While a general and sophisticated methodology is lacking, some very interesting results have been obtained in establishing convergence results that lay the foundation to both fluid and diffusion approximation schemes. Meanwhile, the quantitative aspect of the problem seriously lagged behind, and results in approximation and control of multidimensional systems are very limited. This calls for more focus be put on computational efforts on such system, so that new tools and techniques can be added to the arsenal of attacking these problem. In this paper, we aim at extend some theoretical and computational understanding to a simple but representative two-dimensional operations research model. Not only can it serve as a