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The M / M /∞ Queue with Varying Arrival and Departure Rates

Author

Listed:
  • T. Collings

    (University of Strathclyde, Glasgow, Scotland)

  • C. Stoneman

    (University of Strathclyde, Glasgow, Scotland)

Abstract

This paper derives the queue size distribution for infinite server queues with Poisson arrivals and exponential service times when the parameters of both distributions are allowed to vary with time. Both continuous and discrete variation are examined. Infinite server queues realistically describe those queues in which sufficient service capacity exists to prevent virtually any customer waiting time. Such queues are not uncommon in the health services, when a delay in service can sometimes mean death. Specifically, we discuss the queuing problem of an intensive care unit and show that it is unlikely that the hourly variation in the arrival rate of patients to the unit will significantly affect the number of beds occupied.

Suggested Citation

  • T. Collings & C. Stoneman, 1976. "The M / M /∞ Queue with Varying Arrival and Departure Rates," Operations Research, INFORMS, vol. 24(4), pages 760-773, August.
    • Handle: RePEc:inm:oropre:v:24:y:1976:i:4:p:760-773
    DOI: 10.1287/opre.24.4.760
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    Cited by:

    1. Anatoly Nazarov & Alexander Dudin & Alexander Moiseev, 2022. "Pseudo Steady-State Period in Non-Stationary Infinite-Server Queue with State Dependent Arrival Intensity," Mathematics, MDPI, vol. 10(15), pages 1-12, July.
    2. Giorno, Virginia & Nobile, Amelia G., 2022. "On some integral equations for the evaluation of first-passage-time densities of time-inhomogeneous birth-death processes," Applied Mathematics and Computation, Elsevier, vol. 422(C).
    3. Tan, Xiaoqian & Knessl, Charles & Yang, Yongzhi (Peter), 2013. "On finite capacity queues with time dependent arrival rates," Stochastic Processes and their Applications, Elsevier, vol. 123(6), pages 2175-2227.
    4. Schwarz, Justus Arne & Selinka, Gregor & Stolletz, Raik, 2016. "Performance analysis of time-dependent queueing systems: Survey and classification," Omega, Elsevier, vol. 63(C), pages 170-189.
    5. Amir Elalouf & Guy Wachtel, 2022. "Queueing Problems in Emergency Departments: A Review of Practical Approaches and Research Methodologies," SN Operations Research Forum, Springer, vol. 3(1), pages 1-46, March.
    6. Harris, K.R. & Trappeniers, N.J., 1980. "The density dependence of the self-diffusion coefficient of liquid methane," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 104(1), pages 262-280.
    7. Michels, J.P.J. & Trappeniers, N.J., 1980. "Molecular dynamical calculations on the transport properties of a square-well fluid," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 104(1), pages 243-254.

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