IDEAS home Printed from https://ideas.repec.org/a/inm/oropre/v16y1968i3p651-668.html
   My bibliography  Save this article

Some Inequalities in Queuing

Author

Listed:
  • K. T. Marshall

    (Bell Telephone Laboratories, Inc., Holmdel, New Jersey)

Abstract

Bounds are found for various measures of performance in certain classes of the GI / /G 1 queue. First, the mean wait in queue is found in terms of the mean and variance of the interarrival, service, and idle distributions. Bounds on the idle time moments lead to bounds on the mean wait and number in queue. The interarrival time distribution is then assumed to have mean residual life bounded above by 1/λ (λ = arrival rate); i.e., given a time t since the last arrival, the expected time to the next arrival is no more than 1/λ. With this assumption the mean number in queue (and hence system) is bounded to within (1 + p )/2 customers. Both upper and lower bounds are tight. The stronger assumption that, given time t since the last arrival, the probability an arrival occurs in the next ▵ t is nondecreasing in t , leads to bounds on the mean queue length to within ( c 2 a + p )/2, where c a is the coefficient of variation of the arrival distribution. Again the bounds are tight. Specializing to the D / G /1 queue the mean queue length is found to within p /2

Suggested Citation

  • K. T. Marshall, 1968. "Some Inequalities in Queuing," Operations Research, INFORMS, vol. 16(3), pages 651-668, June.
    • Handle: RePEc:inm:oropre:v:16:y:1968:i:3:p:651-668
    DOI: 10.1287/opre.16.3.651
    as

    Download full text from publisher

    File URL: http://dx.doi.org/10.1287/opre.16.3.651
    Download Restriction: no
    File URL: https://libkey.io/10.1287/opre.16.3.651?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
    ---><---

    Citations

    Citations are extracted by the CitEc Project, subscribe to its RSS feed for this item.
    as


    Cited by:

    1. Jinzhi Bu & Xiting Gong & Dacheng Yao, 2019. "Technical Note—Constant-Order Policies for Lost-Sales Inventory Models with Random Supply Functions: Asymptotics and Heuristic," Operations Research, INFORMS, vol. 68(4), pages 1063-1073, July.
    2. Dequan Yue & Jinhua Cao, 2001. "The NBUL class of life distribution and replacement policy comparisons," Naval Research Logistics (NRL), John Wiley & Sons, vol. 48(7), pages 578-591, October.
    3. Shenghai Zhou & Yichuan Ding & Woonghee Tim Huh & Guohua Wan, 2021. "Constant Job‐Allowance Policies for Appointment Scheduling: Performance Bounds and Numerical Analysis," Production and Operations Management, Production and Operations Management Society, vol. 30(7), pages 2211-2231, July.
    4. Dewan, Isha & Khaledi, Baha-Eldin, 2014. "On stochastic comparisons of residual life time at random time," Statistics & Probability Letters, Elsevier, vol. 88(C), pages 73-79.
    5. Hui-Yu Zhang & Qing-Xin Chen & James MacGregor Smith & Ning Mao & Ai-Lin Yu & Zhan-Tao Li, 2017. "Performance analysis of open general queuing networks with blocking and feedback," International Journal of Production Research, Taylor & Francis Journals, vol. 55(19), pages 5760-5781, October.
    6. Hirotaka Sakasegawa, 1977. "An approximation formulaL q ≃α·ρ β /(1-ρ)," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 29(1), pages 67-75, December.
    7. Xiang Zhong & Peter Hoonakker & Philip A. Bain & Albert J. Musa & Jingshan Li, 2018. "The impact of e-visits on patient access to primary care," Health Care Management Science, Springer, vol. 21(4), pages 475-491, December.
    8. Hirotaka Sakasegawa & Genji Yamazaki, 1977. "Inequalities and an approximation formula for the mean delay time in tandem queueing systems," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 29(1), pages 445-466, December.
    9. Yan Chen & Ward Whitt, 2020. "Algorithms for the upper bound mean waiting time in the GI/GI/1 queue," Queueing Systems: Theory and Applications, Springer, vol. 94(3), pages 327-356, April.
    10. Dai, Melody D. M. & Schonfeld, Paul, 1998. "Metamodels for estimating waterway delays through series of queues," Transportation Research Part B: Methodological, Elsevier, vol. 32(1), pages 1-19, January.
    11. Arijit Patra & Chanchal Kundu, 2019. "On generalized orderings and ageing classes for residual life and inactivity time at random time," Metrika: International Journal for Theoretical and Applied Statistics, Springer, vol. 82(6), pages 691-704, August.

    More about this item

    Statistics

    Access and download statistics

    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:oropre:v:16:y:1968:i:3:p:651-668. 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.

    We have no bibliographic references for this item. You can help adding them by using 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.