Mt/G/{infty} Queues with Sinusoidal Arrival Rates

• Published in: Management science Vol. 39; no. 2; pp. 241 - 252
• Main Authors: Eick, Stephen G, Massey, William A, Whitt, Ward
• Format: Journal Article
• Language: English
• Published: Linthicum, MD INFORMS 01.02.1993; Institute of Management Sciences; Institute for Operations Research and the Management Sciences
• Subjects: Applied sciences; Approximation; approximations; Determinism; Exact sciences and technology; Fourier series; infinite-server queues; Linear systems; Management research; Mathematical functions; nonstationary queues; Operational research and scientific management; Operational research. Management science; Perceptron convergence procedure; pointwise stationary approximation; Polynomials; queues; queues with time-dependent arrival rates; Queuing theory. Traffic theory; Sine function; Trade gains; Transaction costs; Time dependence; Service time; Approximation; Polynomial approximation; Arrival process; Non stationary process; Fourier series; Queue
• ISSN: 0025-1909; 1526-5501
• DOI: 10.1287/mnsc.39.2.241

In this paper we describe the mean number of busy servers as a function of time in an M t / G / queue (having a nonhomogeneous Poisson arrival process) with a sinusoidal arrival rate function. For an M t / G / 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 / model and related M t / G / s / r models where some customers are lost or delayed.