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Stochastic Properties of Waiting Lines

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  • Philip M. Morse

    (Massachusetts Institute of Technology, Cambridge, Massachusetts)

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 - (rho)) -2 as the utilization factor (rho) approaches unity, whereas the mean length L increases as (1 - (rho)) -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. Operations Research , ISSN 0030-364X, was published as Journal of the Operations Research Society of America from 1952 to 1955 under ISSN 0096-3984.

Suggested Citation

  • Philip M. Morse, 1955. "Stochastic Properties of Waiting Lines," Operations Research, INFORMS, vol. 3(3), pages 255-261, August.
    • Handle: RePEc:inm:oropre:v:3:y:1955:i:3:p:255-261
    DOI: 10.1287/opre.3.3.255
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    Cited by:

    1. O'Connell, Neil & Yor, Marc, 2001. "Brownian analogues of Burke's theorem," Stochastic Processes and their Applications, Elsevier, vol. 96(2), pages 285-304, December.
    2. John D. C. Little, 2002. "Philip M. Morse and the Beginnings," Operations Research, INFORMS, vol. 50(1), pages 146-148, February.
    3. Duket, Steven D. & Pritsker, A.Alan B., 1978. "Examination of simulation output using spectral methods," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 20(1), pages 53-60.
    4. Morgan, Lucy E. & Barton, Russell R., 2022. "Fourier trajectory analysis for system discrimination," European Journal of Operational Research, Elsevier, vol. 296(1), pages 203-217.
    5. Shiliang Cui & Xuanming Su & Senthil Veeraraghavan, 2019. "A Model of Rational Retrials in Queues," Operations Research, INFORMS, vol. 67(6), pages 1699-1718, November.

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