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On the sojourn time distribution in a finite capacity processor shared queue

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Charles KnesslAuthors Info & Claims

Journal of the ACM (JACM), Volume 40, Issue 5

Pages 1238 - 1301

https://doi.org/10.1145/174147.169736

Published: 01 November 1993 Publication History

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Abstract

We consider a processor shared M/M/1 queue that can accommodate at most a finite number K of customers. Using singular perturbation techniques, we construct asymptotic approximations to the distribution of a customer's sojourn time. We assume that K is large and treat several different cases of the model parameters and also treat different time scales. Extensive numerical comparisons are used to back up our asymptotic formulas.—Author's Abstract

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Cited By

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Zhang XYin B(2017)Performance analysis of CDN-P2P networks based on processer-sharing queues2017 8th IEEE International Conference on Software Engineering and Service Science (ICSESS)10.1109/ICSESS.2017.8342856(24-27)Online publication date: Nov-2017
https://doi.org/10.1109/ICSESS.2017.8342856
Zhang XYin BShi H(2016)Performance analysis of multi-server based on processor-sharing queue2016 18th International Conference on Advanced Communication Technology (ICACT)10.1109/ICACT.2016.7423582(843-848)Online publication date: Jan-2016
https://doi.org/10.1109/ICACT.2016.7423582
Tikhonenko OKempa W(2015)Queueing system with processor sharing and limited memory under control of the AQM MechanismAutomation and Remote Control10.1134/S000511791510006976:10(1784-1796)Online publication date: 1-Oct-2015
https://dl.acm.org/doi/10.1134/S0005117915100069
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Index Terms

On the sojourn time distribution in a finite capacity processor shared queue
1. Mathematics of computing
  1. Probability and statistics
    1. Statistical paradigms
      1. Queueing theory

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Reviews

Reviewer: Robert B. Cooper

The paper begins with a set of birth-and-death differential-difference equations, of the standard variety, that determine a set of state probabilities that describe the behavior of the model described in the abstract. Knessl then provides a long and complicated mathematical analysis that leads to asymptotic approximations for the density function of the response times. The accuracy of the approximations is evaluated by comparing them with corresponding values obtained by numerical integration of the original birth-and-death equations. The essence of the paper is the use of classical techniques of applied mathematics in the analysis of queueing models. From the viewpoint of practical application for performance analysis, one could conclude that, in this case, mathematical analysis is much less attractive than numerical analysis (which the author uses, in passing, to check the accuracy of his results).

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Association for Computing Machinery

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Publication History

Published: 01 November 1993

Published in JACM Volume 40, Issue 5

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Zhang XYin B(2017)Performance analysis of CDN-P2P networks based on processer-sharing queues2017 8th IEEE International Conference on Software Engineering and Service Science (ICSESS)10.1109/ICSESS.2017.8342856(24-27)Online publication date: Nov-2017
https://doi.org/10.1109/ICSESS.2017.8342856
Zhang XYin BShi H(2016)Performance analysis of multi-server based on processor-sharing queue2016 18th International Conference on Advanced Communication Technology (ICACT)10.1109/ICACT.2016.7423582(843-848)Online publication date: Jan-2016
https://doi.org/10.1109/ICACT.2016.7423582
Tikhonenko OKempa W(2015)Queueing system with processor sharing and limited memory under control of the AQM MechanismAutomation and Remote Control10.1134/S000511791510006976:10(1784-1796)Online publication date: 1-Oct-2015
https://dl.acm.org/doi/10.1134/S0005117915100069
Tikhonenko O(2015)Egalitarian Processor Sharing System with Demands of Random Space RequirementComputer Networks10.1007/978-3-319-19419-6_33(347-356)Online publication date: 28-May-2015
https://doi.org/10.1007/978-3-319-19419-6_33
Zhen QKnessl C(2013)Asymptotic Analysis of Spectral Properties of Finite Capacity Processor Shared QueuesStudies in Applied Mathematics10.1111/sapm.12020131:2(179-210)Online publication date: 24-Jun-2013
https://doi.org/10.1111/sapm.12020
Zhen Qvan Leeuwaarden JKnessl C(2010)On a processor sharing queue that models balkingMathematical Methods of Operations Research10.1007/s00186-010-0328-z72:3(453-476)Online publication date: 9-Sep-2010
https://doi.org/10.1007/s00186-010-0328-z
Zhen QKnessl C(2009)On sojourn times in the finite capacity M/M/1 queue with processor sharingOperations Research Letters10.1016/j.orl.2009.09.00737:6(447-450)Online publication date: 1-Nov-2009
https://dl.acm.org/doi/10.1016/j.orl.2009.09.007
Chen NJordan S(2008)Violation Probability in Processor-Sharing QueuesIEEE Transactions on Automatic Control10.1109/TAC.2008.92987553:8(1956-1961)Online publication date: Sep-2008
https://doi.org/10.1109/TAC.2008.929875
Yang YKnessl C(2008)Heavy traffic asymptotics of the queue length in the GI/M/l – K queueEuropean Journal of Applied Mathematics10.1017/S09567925000025397:05Online publication date: 26-Sep-2008
https://doi.org/10.1017/S0956792500002539
Chen NJordan S(2007)Throughput in Processor-Sharing QueuesIEEE Transactions on Automatic Control10.1109/TAC.2006.88790652:2(299-305)Online publication date: Feb-2007
https://doi.org/10.1109/TAC.2006.887906
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Recommendations

Sojourn time distribution in polling systems with processor-sharing policy

The processor-sharing queue with bulk arrivals and phase-type services

Sojourn times in a processor sharing queue with multiple vacations

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