Piecewise-Linear Diffusion Processes

Diffusion processes are often regarded as among the more abstruse stochastic processes, but diffusion processes are actually relatively elementary, and thus are natural first candidates to consider in queueing applications. To help demonstrate the advantages of diffusion processes, we show that there is a large class of one-dimensional diffusion processes for which it is possible to give convenient explicit expressions for the steady-state distribution, without writing down any partial differential equations or performing any numerical integration. We call these tractable diffusion processes piecewise linear; the drift function is piecewise linear, while the diffusion coefficient is piecewise constant. The explicit expressions for steady-state distributions in turn yield explicit expressions for long-run average costs in optimization problems, which can be analyzed with the aid of symbolic mathematics packages. Since diffusion processes have continuous sample paths, approximation is required when they are used to model discrete-valued processes. We also discuss strategies for performing this approximation, and we investigate when this approximation is good for the steady-state distribution of birth-and-death processes. We show that the diffusion approximation tends to be good when the differences between the birth and death rates are small compared to the death rates.