IDEAS home Printed from https://ideas.repec.org/a/inm/ormnsc/v44y1998i11-part-2ps137-s144.html
   My bibliography  Save this article

A Note on Approximating Peak Congestion in Mt/G/\infty Queues with Sinusoidal Arrivals

Author

Listed:
  • Linda V. Green

    (Columbia Business School, Columbia University, New York, New York 10027-6902)

  • Peter J. Kolesar

    (Columbia Business School, Columbia University, New York, New York 10027-6902)

Abstract

We study the M t /G/\infty queue where customers arrive according to a sinusoidal function \lambda t = \lambda + A sin(2\pi t/T) and the service rate is \mu . We show that the expected number of customers in the system during peak congestion can be closely approximated by (\lambda + A)/\mu for service distributions with coefficient of variation between 0 and 1. Motivated by a result derived by Eick, Massey, and Whitt that the time lag of the peak congestion from the peak of the customer arrivals is 1/2\mu for models with deterministic service times, we show that the time lag for exponential service times is closely approximated by 1/\mu . Based on a cycle length of 24 hours and regardless of the values of other system parameters, these approximations are of the order of 1% accuracy for \mu = 1, and the accuracy increases rapidly with increasing \mu .

Suggested Citation

  • Linda V. Green & Peter J. Kolesar, 1998. "A Note on Approximating Peak Congestion in Mt/G/\infty Queues with Sinusoidal Arrivals," Management Science, INFORMS, vol. 44(11-Part-2), pages 137-144, November.
    • Handle: RePEc:inm:ormnsc:v:44:y:1998:i:11-part-2:p:s137-s144
    DOI: 10.1287/mnsc.44.11.S137
    as

    Download full text from publisher

    File URL: http://dx.doi.org/10.1287/mnsc.44.11.S137
    Download Restriction: no
    File URL: https://libkey.io/10.1287/mnsc.44.11.S137?utm_source=ideas
    LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
    ---><---

    References listed on IDEAS

    as
    1. Bernard O. Koopman, 1972. "Air-Terminal Queues under Time-Dependent Conditions," Operations Research, INFORMS, vol. 20(6), pages 1089-1114, December.
    2. Linda Green & Peter Kolesar, 1991. "The Pointwise Stationary Approximation for Queues with Nonstationary Arrivals," Management Science, INFORMS, vol. 37(1), pages 84-97, January.
    3. Robert M. Oliver & Aryeh H. Samuel, 1962. "Reducing Letter Delays in Post Offices," Operations Research, INFORMS, vol. 10(6), pages 839-892, December.
    4. Arie Harel, 1988. "Sharp Bounds and Simple Approximations for the Erlang Delay and Loss Formulas," Management Science, INFORMS, vol. 34(8), pages 959-972, August.
    5. Arie Harel, 1988. "Erratum to: Sharp Bounds and Simple Approximations for the Erlang Delay and Loss Formulas," Management Science, INFORMS, vol. 34(10), pages 1277-1277, October.
    6. M. Segal, 1974. "The Operator-Scheduling Problem: A Network-Flow Approach," Operations Research, INFORMS, vol. 22(4), pages 808-823, August.
    7. Ward Whitt, 1991. "The Pointwise Stationary Approximation for Mt/Mt/s Queues Is Asymptotically Correct As the Rates Increase," Management Science, INFORMS, vol. 37(3), pages 307-314, March.
    8. Leslie C. Edie, 1954. "Traffic Delays at Toll Booths," Operations Research, INFORMS, vol. 2(2), pages 107-138, May.
    9. Linda Green & Peter Kolesar, 1989. "Testing the Validity of a Queueing Model of Police Patrol," Management Science, INFORMS, vol. 35(2), pages 127-148, February.
    10. Linda V. Green & Peter J. Kolesar, 1995. "On the Accuracy of the Simple Peak Hour Approximation for Markovian Queues," Management Science, INFORMS, vol. 41(8), pages 1353-1370, August.
    11. Linda Green & Peter Kolesar, 1984. "The Feasibility of One-Officer Patrol in New York City," Management Science, INFORMS, vol. 30(8), pages 964-981, August.
    12. Stephen G. Eick & William A. Massey & Ward Whitt, 1993. "Mt/G/\infty Queues with Sinusoidal Arrival Rates," Management Science, INFORMS, vol. 39(2), pages 241-252, February.
    13. Peter J. Kolesar & Kenneth L. Rider & Thomas B. Crabill & Warren E. Walker, 1975. "A Queuing-Linear Programming Approach to Scheduling Police Patrol Cars," Operations Research, INFORMS, vol. 23(6), pages 1045-1062, December.
    14. Bruce H. Andrews & Henry L. Parsons, 1989. "L. L. Bean Chooses a Telephone Agent Scheduling System," Interfaces, INFORMS, vol. 19(6), pages 1-9, December.
    Full references (including those not matched with items on IDEAS)

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Linda V. Green & Peter J. Kolesar & João Soares, 2001. "Improving the Sipp Approach for Staffing Service Systems That Have Cyclic Demands," Operations Research, INFORMS, vol. 49(4), pages 549-564, August.
    2. Ward Whitt & Wei You, 2019. "Time-Varying Robust Queueing," Operations Research, INFORMS, vol. 67(6), pages 1766-1782, November.
    3. 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.
    4. Ali Yousefi & Reza Pourtaheri & Mohammad Reza Salehi Rad, 2024. "Analysis of the Mt/M/1 Queueing System with Impatient Customers and Single Vacation," Sankhya B: The Indian Journal of Statistics, Springer;Indian Statistical Institute, vol. 86(2), pages 690-712, November.
    5. Defraeye, Mieke & Van Nieuwenhuyse, Inneke, 2016. "Staffing and scheduling under nonstationary demand for service: A literature review," Omega, Elsevier, vol. 58(C), pages 4-25.
    6. Ali Yousefi & Reza Pourtaheri, 2024. "The Mt/M/1 queueing system with impatient customers and multiple vacation," OPSEARCH, Springer;Operational Research Society of India, vol. 61(4), pages 2002-2022, December.
    7. Noah Gans & Ger Koole & Avishai Mandelbaum, 2003. "Telephone Call Centers: Tutorial, Review, and Research Prospects," Manufacturing & Service Operations Management, INFORMS, vol. 5(2), pages 79-141, September.
    8. 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.
    9. Alnowibet, Khalid Abdulaziz & Perros, Harry, 2009. "Nonstationary analysis of the loss queue and of queueing networks of loss queues," European Journal of Operational Research, Elsevier, vol. 196(3), pages 1015-1030, August.
    10. Zohar Feldman & Avishai Mandelbaum & William A. Massey & Ward Whitt, 2008. "Staffing of Time-Varying Queues to Achieve Time-Stable Performance," Management Science, INFORMS, vol. 54(2), pages 324-338, February.
    11. Stolletz, Raik, 2008. "Approximation of the non-stationary M(t)/M(t)/c(t)-queue using stationary queueing models: The stationary backlog-carryover approach," European Journal of Operational Research, Elsevier, vol. 190(2), pages 478-493, October.
    12. Wang, Haiyan & Olsen, Tava Lennon & Liu, Guiqing, 2018. "Service capacity competition with peak arrivals and delay sensitive customers," Omega, Elsevier, vol. 77(C), pages 80-95.
    13. Achal Bassamboo & J. Michael Harrison & Assaf Zeevi, 2006. "Design and Control of a Large Call Center: Asymptotic Analysis of an LP-Based Method," Operations Research, INFORMS, vol. 54(3), pages 419-435, June.
    14. P. Daniel Wright & Matthew J. Liberatore & Robert L. Nydick, 2006. "A Survey of Operations Research Models and Applications in Homeland Security," Interfaces, INFORMS, vol. 36(6), pages 514-529, December.
    15. Aykin, Turgut, 2000. "A comparative evaluation of modeling approaches to the labor shift scheduling problem," European Journal of Operational Research, Elsevier, vol. 125(2), pages 381-397, September.
    16. Dietz, Dennis C., 2011. "Practical scheduling for call center operations," Omega, Elsevier, vol. 39(5), pages 550-557, October.
    17. Daniel Adelman, 2007. "Price-Directed Control of a Closed Logistics Queueing Network," Operations Research, INFORMS, vol. 55(6), pages 1022-1038, December.
    18. Achal Bassamboo & J. Michael Harrison & Assaf Zeevi, 2009. "Pointwise Stationary Fluid Models for Stochastic Processing Networks," Manufacturing & Service Operations Management, INFORMS, vol. 11(1), pages 70-89, August.
    19. Gary M. Thompson, 1997. "Labor staffing and scheduling models for controlling service levels," Naval Research Logistics (NRL), John Wiley & Sons, vol. 44(8), pages 719-740, December.
    20. Banu Sungur & Cemal Özgüven & Yasemin Kariper, 2017. "Shift scheduling with break windows, ideal break periods, and ideal waiting times," Flexible Services and Manufacturing Journal, Springer, vol. 29(2), pages 203-222, June.

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:inm:ormnsc:v:44:y:1998:i:11-part-2:p:s137-s144. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Chris Asher (email available below). General contact details of provider: https://edirc.repec.org/data/inforea.html .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.