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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.

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