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.

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 building block for analyzing much more complicated systems, but also provides insights to the understanding of basic structure of general systems with correlated demand. This model is motivated by several typical applications in production systems and scheduling. In such a system, it is very common that one resource is demanded by multiple demand processes, similarly, one class of customer demand could contain a combination of different products. The correlation induced by this commonality poses difficulty even for the simplest version of the model. In this paper, after briefly introducing the mathematical model and stochastic processes that characterize the basic underlying relationship, we provide an diffusion approximation to the stochastic processes in the form of a reflected two-dimensional Brownian motion, which is just an abstraction of previous known results in various forms. Then our main effort will be concentrated on the analysis of some key quantities of this diffusion process. More specifically, we are interested at the joint probability density of the Brownian motion and its coordinate running maximum. To produce the computational result, we relate its computation to finding a proper solution to a classic elliptic partial differential equation, the extended Helmhotz equation.

The rest of the paper will be organized as follows, in Section

Suppose that there are two types of products, type

Diffusion approximation can be obtained for this system, see, for example [

From now on, we will focus on the calculation of the probability density function calculation. More specifically, let

In the case of one-dimensional Brownian motion, the density of the Brownian motion and its running maximum is a classical result obtained through reflection principle. In two dimension, the dependence between the two coordinates creates a barrier for generalizing the result easily. In this paper, first, we convert the calculation of the quantity (

There are many different approaches to the classic problem of the joint distribution of a standard Brownian motion and its running maximum, see, for example, [

For the two-dimensional Brownian motion, applying the same argument, calculating

To calculate the stationary distribution of the reflected Brownian motion

Functions

Notice that the boundary in the problem we study consists of the two axis. So the surface measure is basically the Lebesgue measure on

The main idea in this paper is to find a function

Now plug

Assume that

The Laplace transform of the stationary

Recall that our goal is to get an expression for

The laplace transform of

We studied a simple but representative operations research model for performance analysis and capacity planning. Through diffusion approximation, we identify the problem of characterizing the joint distribution of a two-dimensional Brownian motion and its coordinate running maximum, which is of independent interests in the theory of probability, as well as other applications. Using some probabilistic techniques, we are able to reduce the problem to the solution of an extended Helmholtz equation, and a careful exam of the solution to this well-known partial differential equation leads to the calculation of the Laplace transform of the desired joint probability distribution.

Standard inversion methods exist for estimating the probabilities from the Laplace transform, in the ongoing research, we are exploring the special structure of our solution, and tailor the inversion method to our special needs, thus provide a very efficient and accurate way of estimating these fundamentally important quantities for both probability theory and operations research.