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Stochastic Properties of Waiting Lines
Philip M. Morse
Journal of the Operations Research Society of America
Vol. 3, No. 3 (Aug., 1955), pp. 255-261
Published by: INFORMS
Stable URL: http://www.jstor.org/stable/166559
Page Count: 7
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The stochastic properties of waiting lines may be analyzed by a two-stage process: first solving the time-dependent equations for the state probabilities and then utilising these transient solutions to obtain the auto-correlation function for queue length and the root-mean-square frequency spectrum of its fluctuations from mean length. The procedure is worked out in detail for the one-channel, exponential service facility with Poisson arrivals, and the basic solutions for the m-channel exponential service case are given. The analysis indicates that the transient behavior of the queue length n(t) may be measured by a 'relaxation time,' the mean time any deviation of n(t) away from its mean value L takes to return (1/e) of the way back to L. This relaxation time increases as (1-ρ)-2 as the utilization factor rho approaches unity, whereas the mean length L increases as (1-ρ)-1. In other words, as saturation of the facility is approached, the mean length of line increases; but, what is often more detrimental, the length of time for the line to return to average, once it diverges from average, increases even more markedly.
Journal of the Operations Research Society of America © 1955 INFORMS