Mathematical Methods of Operations Research

, Volume 87, Issue 3, pp 411–430 | Cite as

Approximate sojourn time distribution of a discriminatory processor sharing queue with impatient customers

Original Article
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Abstract

We consider a two-class processor sharing queueing system scheduled by the discriminatory processor sharing discipline. Poisson arrivals of customers and exponential amounts of service requirements are assumed. At any moment of being served, a customer can leave the system without completion of its service. In the asymptotic regime, where the ratio of the time scales of the two-class customers is infinite, we obtain the conditional sojourn time distribution of each class customers. Numerical experiments show that the time scale decomposition approach developed in this paper gives a good approximation to the conditional sojourn time distribution as well as the expectation of it.

Keywords

Discriminatory processor sharing Impatient customers Sojourn time distribution Time scale decomposition method 

Notes

Acknowledgements

The author would like to thank the associate editor and the anonymous referees for their comments and suggestions on the first draft of this paper. Their suggestions have greatly improve the quality of the paper. This research was supported by the 2014 Research Fund of University of Seoul.

References

  1. Altman E, Jimenez T, Kofman D (2004) DPS queues with stationary ergodic service times and the performance of TCP in overload. In: Proceedings of IEEE Infocom 2004Google Scholar
  2. Altman E, Avrachenkov K, Ayesta U (2006) A survey on discriminatory processor sharing. Queueing Syst 53(1):53–63MathSciNetCrossRefMATHGoogle Scholar
  3. Barry C (2015) invLT: Inversion of Laplace-Transformed Functions. https://CRAN.R-project.org/package=invLT. R package version 0.2.1
  4. Bonald T, Roberts J (2003) Congestion at flow level and the impact of user behaviour. Comput Netw 42:521–536CrossRefMATHGoogle Scholar
  5. Boxma OJ, Hegde N, Nunez-Queija R (2006) Exact and approximate analysis of sojourn times in finite discriminatory processor sharing queues. Int J Electron Commun 60:109–115CrossRefGoogle Scholar
  6. Coffman E, Puhalskii A, Reiman M, Wright P (1994) Processor-shared buffers with reneging. Perform Eval 19:25–46MathSciNetCrossRefMATHGoogle Scholar
  7. Evans G, Chung K (2000) Laplace transform inversions using optimal contours in the complex plane. Int J Comput Math 73(4):531–543MathSciNetCrossRefMATHGoogle Scholar
  8. Fayolle G, Mitrani I, Iasnogorodski R (1980) Sharing a processor among many job classes. J ACM 27(3):519–532MathSciNetCrossRefMATHGoogle Scholar
  9. Guillemin F, Robert P, Zwart B (2004) Tale asymptotics for processor sharing queues. Adv Appl Probab 36:525–543CrossRefMATHGoogle Scholar
  10. Jones WB, Thron W (2009) Continued fractions: analytic theory and applications. Cambridge University Press, reissue editionGoogle Scholar
  11. Karlin S, McGregor J (1957) The differential equations of birth-and-death processes, and the Stieltjes moment problem. Trans Am Math Soc 85:489–546MathSciNetCrossRefMATHGoogle Scholar
  12. Kim S (2014) Approximate queue length distribution of a discriminatory processor sharing queue with impatient customers. J Korean Stat Soc 43:105–118MathSciNetCrossRefMATHGoogle Scholar
  13. Kleinrock L (1967) Time-shared system: a theoretical treatment. J ACM 14:242–261MathSciNetCrossRefMATHGoogle Scholar
  14. Massoulie L, Roberts J (2001) Bandwidth sharing: objectives and algorithms. IEEE/ACM Trans Netw 10:320–328CrossRefGoogle Scholar
  15. Murphy J, O’Donohoe M (1975) Some properties of continued fractions with applications in markov processes. IMA J Appl Math 16:57–71MathSciNetCrossRefMATHGoogle Scholar
  16. O’Donovan T (1974) Direct solutions of M/G/1 processor-sharing models. Oper Res 22(6):1232–1235CrossRefMATHGoogle Scholar
  17. Rege KM, Sengupta B (1994) A decomposition theorem and related results for the discriminatory processor sharing queue. Queueing Syst 18(3):333–351MathSciNetCrossRefMATHGoogle Scholar
  18. Sengupta B, Jagerman D (1985) A conditional response time of the M/M/1 processor-sharing queue. AT T Tech J 64(2):409–421MathSciNetCrossRefMATHGoogle Scholar
  19. van Kessel G, Nunez-Queija R, Borst S (2005) Differentiated bandwidth sharing with disparate flow sizes. In: Proceedings of Infocom 2005Google Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2017

Authors and Affiliations

  1. 1.Department of StatisticsUniversity of SeoulSeoulRepublic of Korea

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