Queueing Systems

, Volume 85, Issue 3–4, pp 361–381 | Cite as

A general workload conservation law with applications to queueing systems

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Abstract

In the spirit of Little’s law \(L=\lambda W\) and its extension \(H=\lambda G\) we use sample-path analysis to give a general conservation law. For queueing models the law relates the asymptotic average workload in the system to the conditional asymptotic average sojourn time and service times distribution function. This law generalizes previously obtained conservation laws for both single- and multi-server systems, and anticipating and non-anticipating scheduling disciplines. Applications to single- and multi-class queueing and other systems that illustrate the versatility of this law are given. In particular, we show that, for anticipative and non-anticipative scheduling rules, the unconditional delay in a queue is related to the covariance of service times and queueing delays.

Keywords

Conservation law Workload invariance Multi-server queues Sample-path analysis Scheduling 

Mathematics Subject Classification

Primary 60K25 Secondary 68M20 90B36 

Notes

Acknowledgements

The author wishes to thank the reviewers for their helpful comments.

References

  1. 1.
    Ayesta, U.: A unifying conservation law for single-server queues. J. Appl. Probab. 44, 1078–1087 (2007)CrossRefGoogle Scholar
  2. 2.
    Baccelli, F., Brémaud, P.: Elements of Queueing Theory: Palm Martingale Calculus and Stochastic Recurrences, Applications of Mathematics, vol. 26. Springer, New York (1994)Google Scholar
  3. 3.
    Bartsch, B., Bolch, G.: Conservation law for G/G/m queueing systems. Acta Inform. 810, 105–109 (1978)Google Scholar
  4. 4.
    Dacre, M., Glazebrook, K., Niño-Mora, J.: The achievable region approach to the optimal control of stochastic systems. J. R. Stat. Soc. B 61, 747–791 (1999)CrossRefGoogle Scholar
  5. 5.
    El-Taha, M.: Asymptotic time averages and frequency distributions. Int. J. Stochastic Anal. 2016, Article ID 2741214. doi: 10.1155/2016/2741214 (2016)
  6. 6.
    El-Taha, M.: Invariance of workload in queueing systems. Queueing Syst. 83(1–2), 181–192 (2016)CrossRefGoogle Scholar
  7. 7.
    El-Taha, M., Stidham Jr., S.: Sample-Path Analysis of Queueing Systems. Kluwer Academic Publishing, Boston (1999)CrossRefGoogle Scholar
  8. 8.
    Federgruen, A., Groenevelt, H.: M/G/c queueing systems with multiple customer classes: characterization and control of achievable performance under nonpreemptive priority rules. Manag. Sci. 34, 1121–1138 (1988)CrossRefGoogle Scholar
  9. 9.
    Gelenbe, E., Mitrani, I.: Analysis and Synthesis of Computer Systems. Academic Press, London (1980)Google Scholar
  10. 10.
    Green, T., Stidham Jr., S.: Sample-path conservation laws, with applications to scheduling queues and fluid systems. Queueing Syst. 36, 175–199 (2000)CrossRefGoogle Scholar
  11. 11.
    Gross, D., Shortle, J.F., Thompson, J.M., Harris, C.: Fundamentals of Queueing Theory, 4th edn. Wiley, Hoboken (2008)CrossRefGoogle Scholar
  12. 12.
    Heyman, D., Sobel, M.: Stochastic Models in Operations Research, vol. I. McGraw-Hill, New York (1982)Google Scholar
  13. 13.
    Heyman, D.P., Stidham Jr., S.: The relation between customer and time averages in queues. Oper. Res. 28, 983–994 (1980)CrossRefGoogle Scholar
  14. 14.
    Kleinrock, L.: Queueing Systems, vol. II. Wiley Intersciences, New York (1976)Google Scholar
  15. 15.
    Kleinrock, L., Muntz, R.R., Hsu, J.: Tight bounds on average response time for time-shared computer systems. Proc. IFIP Congr. 1, 124–133 (1971)Google Scholar
  16. 16.
    Niño Mora, J.: Conservation Laws and Related Applications. Wiley Encyclopedia of Operations Research and Management Science, London (2011)Google Scholar
  17. 17.
    O’Donovan, T.M.: Distribution of attained service and residual service in general queueing systems. Oper. Res. 22, 570–574 (1974)CrossRefGoogle Scholar
  18. 18.
    Shanthikumar, J.G., Yao, D.D.: Multiclass queueing systems: polynomial structure and optimal scheduling control. Oper. Res. 40(Suppl 2), S293–S299 (1992)CrossRefGoogle Scholar
  19. 19.
    Tsoucas, P.: The region of achievable performance in a model of klimov. Technical report, Thomas J. Watson IBM Research Center. Research Division., T.J. Watson Research Center, Yorktown Hts., New York, NY 10598 (1991)Google Scholar
  20. 20.
    Whitt, W.: Embedded renewal processes in the \({GI}/{G}/s\) queue. J. Appl. Probab. 9, 650–658 (1972)CrossRefGoogle Scholar
  21. 21.
    Wolff, R.: Stochastic Modeling and the Theory of Queues. Prentice Hall, Englewood Cliffs, NJ (1989)Google Scholar

Copyright information

© Springer Science+Business Media New York 2017

Authors and Affiliations

  1. 1.Department of Mathematics and StatisticsUniversity of Southern MainePortlandUSA

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