The correlation functions of rbm and m/m/1

Abstract

This paper describes the (auto) correlation functions of regulated or reflecting Brownian motion (RBM) and several processes associated with the M/M/1 queue. For RBM and the M/M/1 continuous-time queue-length process, the correlation function of the stationary process coincides with the complementary stationary-excess cdf (cumulative distribution function) associated with a previously studied first-moment cdf. The first-moment cdf is the mean as a function of time given that the process starts at the origin, normalized by dividing by the steady-state limit. The M/M/1 first-moment cdf in turn is the stationary-excess cdf associated with the M/M/1 busy-period cdf. In fact, all the moment cdfs and correlation functions can be expressed directly in terms of the busy-period cdf. This structure provides the basis for simple approximations of the correlation functions and the moments as functions of time by hyperexponentials.