Mt/G/\infty Queues with Sinusoidal Arrival Rates

Stephen G. Eick, William A. Massey and Ward Whitt
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Stephen G. Eick: AT&T Bell Laboratories, Murray Hill, New Jersey 07974-0636
William A. Massey: AT&T Bell Laboratories, Murray Hill, New Jersey 07974-0636
Ward Whitt: AT&T Bell Laboratories, Murray Hill, New Jersey 07974-0636

Management Science, 1993, vol. 39, issue 2, 241-252

Abstract: In this paper we describe the mean number of busy servers as a function of time in an M t /G/\infty queue (having a nonhomogeneous Poisson arrival process) with a sinusoidal arrival rate function. For an M t /G/\infty model with appropriate initial conditions, it is known that the number of busy servers at time t has a Poisson distribution for each t, so that the full distribution is characterized by its mean. Our formulas show how the peak congestion lags behind the peak arrival rate and how much less is the range of congestion than the range of offered load. The simple formulas can also be regarded as consequences of linear system theory, because the mean function can be regarded as the image of a linear operator applied to the arrival rate function. We also investigate the quality of various approximations for the mean number of busy servers such as the pointwise stationary approximation and several polynomial approximations. Finally, we apply the results for sinusoidal arrival rate functions to treat general periodic arrival rate functions using Fourier series. These results are intended to provide a better understanding of the behavior of the M t /G/\infty model and related M t /G/s/r models where some customers are lost or delayed.

Keywords: queues; nonstationary queues; infinite-server queues; queues with time-dependent arrival rates; approximations; pointwise stationary approximation (search for similar items in EconPapers)
Date: 1993
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