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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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Stochastic Properties of Waiting Lines
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Abstract

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.

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