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An Empirical Investigation of the Transient Behavior of Stationary Queueing Systems

Author

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  • Amedeo R. Odoni

    (Massachusetts Institute of Technology, Cambridge, Massachusetts)

  • Emily Roth

    (Carnegie-Mellon University, Pittsburgh, Pennsylvania)

Abstract

This paper examines the transient behavior of infinite-capacity, single-server, Markovian queueing systems. It estimates Q ( t ), the expected number of customers in queue at time t , by numerically solving the sets of simultaneous, first-order differential equations that describe these systems. Empirical results have been drawn from these observations. For small values of t , the behavior of Q ( t ) is strongly influenced by the initial state of the queueing system. For systems with deterministic initial conditions, one can roughly predict which of a small set of patterns this behavior will follow. After an initial period of time and independently of initial conditions, Q ( t ) approaches Q (∞) in a manner that can be approximated through a decaying exponential function. On the basis of experimental evidence, we have developed an expression that provides a good approximation to the observed values of the time constant associated with this exponential function. This expression can also be used to determine an upper bound for the amount of time required until Q ( t ) is close to Q (∞).

Suggested Citation

  • Amedeo R. Odoni & Emily Roth, 1983. "An Empirical Investigation of the Transient Behavior of Stationary Queueing Systems," Operations Research, INFORMS, vol. 31(3), pages 432-455, June.
    • Handle: RePEc:inm:oropre:v:31:y:1983:i:3:p:432-455
    DOI: 10.1287/opre.31.3.432
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    Citations

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    Cited by:

    1. Missbauer, Hubert, 2009. "Models of the transient behaviour of production units to optimize the aggregate material flow," International Journal of Production Economics, Elsevier, vol. 118(2), pages 387-397, April.
    2. Xiao Chen & Carolina Osorio & Bruno Filipe Santos, 2019. "Simulation-Based Travel Time Reliable Signal Control," Transportation Science, INFORMS, vol. 53(2), pages 523-544, March.
    3. Britt Mathijsen & Bert Zwart, 2017. "Transient error approximation in a Lévy queue," Queueing Systems: Theory and Applications, Springer, vol. 85(3), pages 269-304, April.
    4. William H. Kaczynski & Lawrence M. Leemis & John H. Drew, 2012. "Transient Queueing Analysis," INFORMS Journal on Computing, INFORMS, vol. 24(1), pages 10-28, February.
    5. Flötteröd, G. & Osorio, C., 2017. "Stochastic network link transmission model," Transportation Research Part B: Methodological, Elsevier, vol. 102(C), pages 180-209.
    6. Nikolic, Nebojsa, 2008. "Statistical integration of Erlang's equations," European Journal of Operational Research, Elsevier, vol. 187(3), pages 1487-1493, June.
    7. Jacquillat, Alexandre & Odoni, Amedeo R., 2018. "A roadmap toward airport demand and capacity management," Transportation Research Part A: Policy and Practice, Elsevier, vol. 114(PA), pages 168-185.
    8. Chaithanya Bandi & Nikolaos Trichakis & Phebe Vayanos, 2019. "Robust Multiclass Queuing Theory for Wait Time Estimation in Resource Allocation Systems," Management Science, INFORMS, vol. 65(1), pages 152-187, January.
    9. J. D. Griffiths & G. M. Leonenko & J. E. Williams, 2008. "Approximation to the Transient Solution of the M/E k /1 Queue," INFORMS Journal on Computing, INFORMS, vol. 20(4), pages 510-515, November.
    10. Shiliang Cui & Xuanming Su & Senthil Veeraraghavan, 2019. "A Model of Rational Retrials in Queues," Operations Research, INFORMS, vol. 67(6), pages 1699-1718, November.
    11. Chaithanya Bandi & Dimitris Bertsimas & Nataly Youssef, 2018. "Robust transient analysis of multi-server queueing systems and feed-forward networks," Queueing Systems: Theory and Applications, Springer, vol. 89(3), pages 351-413, August.
    12. Linsen Chong & Carolina Osorio, 2018. "A Simulation-Based Optimization Algorithm for Dynamic Large-Scale Urban Transportation Problems," Transportation Science, INFORMS, vol. 52(3), pages 637-656, June.
    13. Carolina Osorio & Jana Yamani, 2017. "Analytical and Scalable Analysis of Transient Tandem Markovian Finite Capacity Queueing Networks," Transportation Science, INFORMS, vol. 51(3), pages 823-840, August.
    14. Walid W. Nasr & Michael R. Taaffe, 2013. "Fitting the Ph t / M t / s / c Time-Dependent Departure Process for Use in Tandem Queueing Networks," INFORMS Journal on Computing, INFORMS, vol. 25(4), pages 758-773, November.
    15. Leonenko, G.M., 2009. "A new formula for the transient solution of the Erlang queueing model," Statistics & Probability Letters, Elsevier, vol. 79(3), pages 400-406, February.
    16. Armann Ingolfsson & Elvira Akhmetshina & Susan Budge & Yongyue Li & Xudong Wu, 2007. "A Survey and Experimental Comparison of Service-Level-Approximation Methods for Nonstationary M(t)/M/s(t) Queueing Systems with Exhaustive Discipline," INFORMS Journal on Computing, INFORMS, vol. 19(2), pages 201-214, May.
    17. Hai Wang & Amedeo Odoni, 2016. "Approximating the Performance of a “Last Mile” Transportation System," Transportation Science, INFORMS, vol. 50(2), pages 659-675, May.
    18. Julia Pahl & Stefan Voß & David Woodruff, 2007. "Production planning with load dependent lead times: an update of research," Annals of Operations Research, Springer, vol. 153(1), pages 297-345, September.

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