Diffusion Approximations

In this chapter, we shall give an overview of some of the basic applications of the theory of diffusion approximations to operations research. A diffusion approximation is a technique in which a complicated and analytically intractable stochastic process is replaced by an appropriate diffusion process. A diffusion process is a (strong) Markov process having continuous sample paths. Diffusion processes have a great deal of analytical structure and are therefore typically more mathematically tractable than the original process with which one starts. The approach underlying the application of diffusion approximations is therefore comparable to that underlying normal approximation for sums of random variables. In the latter setting, the central limit theorem permits one to replace the analytically intractable sum of random variables by an appropriately chosen normal random variable. In this chapter, we shall describe some of the basic theory of weak convergence that underlies the method of diffusion approximation. We shall then survey various applications of this methodology to the approximation of complex queueing systems. Because we are interested in developing approximations for the distribution of a process (considered as a random function of time), it is necessary for us to describe the basic elements of the theory of weak convergence in a function space. Sections 2 and 3 are therefore devoted to this topic. Section 4 discusses the most basic and easily understood of all diffusion approximations, namely the general principle that sums of random variables (when viewed as stochastic processes) can be approximated by Brownian motion: this result is known, in the literature, as Donsker's theorem. By using the close correspondence between random walk and the single-server queue, Section 5 develops the basic theory of weak convergence for the GI/G/1/oo queue. This forms valuable background for the more complex diffusion approximations that appear in the network setting of Sections 7 and 8. Section 6 gives a brief overview of some of the basic analytical theory of diffusion processes. In particular, we show that a large number of interesting performance measures can be calculated as solutions to certain associated partial differential equations.