Transient Behavior of the M/G/1 Workload Process

Joseph Abate and Ward Whitt
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Joseph Abate: Bell Communications Research, Piscataway, New Jersey
Ward Whitt: AT&T Bell Laboratories, Murray Hill, New Jersey

Operations Research, 1994, vol. 42, issue 4, 750-764

Abstract: In this paper we describe the time-dependent moments of the workload process in the M / G /1 queue. The k th moment as a function of time can be characterized in terms of a differential equation involving lower moment functions and the time-dependent server-occupation probability. For general initial conditions, we show that the first two moment functions can be represented as the difference of two nondecreasing functions, one of which is the moment function starting at zero. The two nondecreasing components can be regarded as probability cumulative distribution function (cdf's) after appropriate normalization. The normalized moment functions starting empty are called moment cdf's; the other normalized components are called moment-difference cdf's. We establish relations among these cdf's using stationary-excess relations. We apply these relations to calculate moments and derivatives at the origin of these cdf's. We also obtain results for the covariance function of the stationary workload process. It is interesting that these various time-dependent characteristics can be described directly in terms of the steady-state workload distribution.

Keywords: queues; transient results: M/G/1 workload process; queues; busy period analysis: M/G/l queue (search for similar items in EconPapers)
Date: 1994
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