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Invariance of workload in queueing systems

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Abstract

In multi-server systems, invariance of workload is a key step in establishing conservation and strong conservation laws. Although several articles have dealt with the issue, invariance of workload has been either assumed, asserted without proof, or the proof given is unsatisfactory, except when service times are exponential. In this article we consider a multi-server queueing system with a work-conserving, non-preemptive, and non-anticipative scheduling rule and give a proof for the invariance of the workload in queue and in system for all such rules. We also show that invariance of workload holds for a subclass of strict priority rules.

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References

  1. Bartsch, B., Bolch, G.: Conservation law for G/G/m queueing systems. Acta Inform. 810, 105–109 (1978)

    Google Scholar 

  2. Bertsimas, D.: The achievable region method in the optimal control of queueing systems. Queueing Syst. 21, 337–389 (1995)

    Article  Google Scholar 

  3. 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)

    Article  Google Scholar 

  4. El-Taha, M.: Invariance of workload in queueing systems: application to multi-server and batch arrival systems (2014) (Preprint)

  5. El-Taha, M., Stidham Jr, S.: Sample-Path Analysis of Queueing Systems. Kluwer, Boston (1999)

    Book  Google Scholar 

  6. Federgruen, A., Groenevelt, H.: Characterization and optimization of achievable performance in general queueing systems. Oper. Res. 36, 733–741 (1988)

    Article  Google Scholar 

  7. 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)

    Article  Google Scholar 

  8. Foss, S.G.: Comparison of servicing strategies in multichannel queueing systems. Sib. Math. J. 22, 142–147 (1981)

    Article  Google Scholar 

  9. Gelenbe, E., Mitrani, I.: Analysis and Synthesis of Computer Systems. Academic Press, London (1980)

    Google Scholar 

  10. Green, T., Stidham Jr., S.: Sample-path conservation laws, with applications to scheduling queues and fluid systems. Queueing Syst. 36, 175–199 (2000)

  11. Heyman, D., Sobel, M.: Stochastic Models in Operations Research, vol. 1. McGraw-Hill, New York (1982)

    Google Scholar 

  12. Kelton, W.D., Sadowski, R.P., Swets, N.B.: Simulation with Arena, 5th edn. McGraw-Hill, New York (2010)

    Google Scholar 

  13. Shanthikumar, J.G., Yao, D.D.: Multiclass queueing systems: polynomial structure and optimal scheduling control. Oper. Res. 40, S293–S299 (1992). Supplement 2

  14. Stidham Jr., S.: Optimal Design of Queueing Systems. CRC Press, Taylor and Francis Group, Boca Raton (2009)

  15. Wolff, R.: Stochastic Modeling and the Theory of Queues. Prentice Hall, NJ (1989)

    Google Scholar 

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Acknowledgments

The author wishes to thank anonymous reviewers for their careful reading of the manuscript and their helpful comments and suggestions.

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  1. Department of Mathematics and Statistics, University of Southern Maine, 96 Falmouth Street, Portland, ME, 04104-9300, USA

    Muhammad El-Taha

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  1. Muhammad El-Taha
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Correspondence to Muhammad El-Taha.

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El-Taha, M. Invariance of workload in queueing systems. Queueing Syst 83, 181–192 (2016). https://doi.org/10.1007/s11134-016-9474-1

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  • DOI: https://doi.org/10.1007/s11134-016-9474-1

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