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The correlation functions of RBM and M/M/1. (English) Zbl 0712.60104
It is shown that for the regulated or reflecting Brownian motion (RBM) and the M/M/1 continuous-time queue-length process, the correlation function of the stationary process coincides with the complementary stationary-excess cumulative distribution function (cdf) associated with a normalized first-moment cdf. Since on the other hand, for the M/M/1 queue the first-moment cdf is the stationary-excess cdf associated with busy-period cdf, all the moment cdf’s and correlation functions can be expressed in terms of the busy-period cdf. As a consequence, simple hyperexponential approximations of the correlation functions and the moment functions are obtained. Finally, some inequalities and bounds are given.

60K25 Queueing theory (aspects of probability theory)
90B22 Queues and service in operations research
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