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Some Inequalities in Queuing

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  • 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
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    Cited by:

    1. 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.
    2. 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.
    3. 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.
    4. 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.
    5. 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.
    6. 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.
    7. 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.
    8. 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.
    9. 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.
    10. 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.
    11. 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.

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