Diffusion approximations for complex repair systems

For many stochastic models in applied probability complicated Markov chains arise which are impossible to analyze directly. A classical approach to this problem, dating back to Bachelier, is to show that a sequence of Markov chains with appropriate time and state scales converges at a given time point (or weakly) to a limiting diffusion process. In these instances the limiting diffusion process may hold out the only hope for providing useful approximations to practical problems. When the Markov chains are one-dimensional birth-death processes in either discrete or continuous time, Stone has developed a complete theory for the weak convergence of these Markov chains, to a limiting diffusion. Roughly speaking, Stone’s results require convergence of the infinitesimal mean and variance to those of the limiting diffusion plus convergence of boundary conditions when appropriate. In this article a comparable development in higher dimensions will be applied for a restricted class of limiting diffusions: multivariate Ornstein-Uhlenbeck processes. These results will then be applied to three generalized repairman models.