The correlation functions of RBM and M/M/1.

This memorandum is the fifth in a series investigating the time-dependent behavior of queues and related stochastic models. This memorandum 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 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.