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On the three-queue priority polling system with threshold service policy

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Abstract

In this paper, a priority polling system consisting of three M / M / 1 queues, served by a single server is investigated. Queue 1 has the Head-of-Line (HoL) priority and Queue 2 has a higher priority over Queue 3 with threshold N. All the switches are instantaneous and preempting. Using the Kernel method we derive the probability of generating functions of the stationary joint queue-length distributions, which yields the mean queue lengths and the mean sojourn times. Furthermore, we consider the limit behaviors in the light-traffic and heavy-traffic scenarios. And an interpolation approximation for the sojourn times utilizing the light and heavy traffic limits are illustrated. To test the validity, we also undertake some simulation works.

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References

  1. Adan, I.J.B.F., Van Leeuwaarden, J.S.H., Winands, E.M.M.: On the application of Rouché’s theorem in queueing theory. Oper. Res. Lett. 34(3), 355–360 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  2. Bertsimas, D., Mourtzinou, G.: Multiclass queueing systems in heavy traffic: an asymptotic approach based on distributional and conservation laws. Oper. Res. 45(3), 470–487 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  3. Blanc, J.: The power-series algorithm applied to cyclic polling systems. Stoch. Models 7(4), 527–545 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  4. Boon, M.A., Winands, E.M., Adan, I.J., Van Wijk, A.: Closed-form waiting time approximations for polling systems. Perform. Eval. 68(3), 290–306 (2011)

    Article  Google Scholar 

  5. Boon, M.A.A., Winands, E.M.M.: Heavy-traffic analysis of k-limited polling systems. Probab. Eng. Inf. Sci. 28(4), 451–471 (2014)

    Article  MathSciNet  Google Scholar 

  6. Boxma, O.J., Down, D.G.: Dynamic server assignment in a two-queue model. Eur. J. Oper. Res. 103(3), 595–609 (1997)

    Article  MATH  Google Scholar 

  7. Boxma, O.J., Koole, G.M., Mitrani, I.: A two-queue polling model with a threshold service policy. In: Modeling, Analysis, and Simulation of Computer and Telecommunication Systems, 1995. MASCOTS’95., Proceedings of the Third International Workshop on IEEE, pp. 84–88 (1995)

  8. Deng, Y., Song, S., Tan, J.: Non-preemptive priority queueing model with changeover times and switching threshold. Commun. Appl. Math. Comput. 15, 28–40 (2001)

    MathSciNet  Google Scholar 

  9. Deng, Y., Tan, J.: Priority queueing model with changeover times and switching threshold. J. Appl. Probab. 38, 263–273 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  10. Dongbo: Homotopy methods for mixed trigonometric polynomial systems. Ph.D. thesis, Dalian University of Technology (2008)

  11. Feng, W., Kowada, M., Adachi, K.: Performance analysis of a two-queue model with an (M, N)-threshold service schedule. J. Oper. Res. Soc. Japan-Keiei Kagaku 44(2), 101–124 (2001)

    MathSciNet  MATH  Google Scholar 

  12. Feng, W., Umemura, M.: Analysis of a finite buffer model with two servers and two nonpreemptive priority classes. Eur. J. Oper. Res. 192(1), 151–172 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  13. Fuhrmann, S., Cooper, R.B.: Stochastic decompositions in the m/g/1 queue with generalized vacations. Oper. Res. 33(5), 1117–1129 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  14. Kella, O., Yechiali, U.: Priorities in m/g/1 queue with server vacations. Nav. Res. Logist. 35(1), 23–34 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  15. Landry, R., Stavrakakis, I.: Queueing study of a 3-priority policy with distinct service strategies. IEEE/ACM Trans. Netw. (TON) 1(5), 576–589 (1993)

    Article  Google Scholar 

  16. Lee, D.S., Sengupta, B.: Queueing analysis of a threshold based priority scheme for ATM networks. IEEE/ACM Trans. Netw. (TON) 1(6), 709–717 (1993)

    Article  Google Scholar 

  17. Li, H., Zhao, Y.Q.: Exact tail asymptotics in a priority queuecharacterizations of the preemptive model. Queueing Syst. 63(1–4), 355–381 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  18. Morrison, J.A., Borst, S.C.: Interacting queues in heavy traffic. Queueing Syst. 65(2), 135–156 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  19. Reiman, M.I., Simon, B.: An interpolation approximation for queueing systems with poisson input. Oper. Res. 36(3), 454–469 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  20. Zazanis, M.A.: Analyticity of poisson-driven stochastic systems. Adv. Appl. Probab. 24, 532–541 (1992)

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgments

This research is partially supported by the National Natural Science Foundation of China (11271373, 11201489, 11371374) and Independent Innovation Project of Graduate School of Central South University (2014zzts009). The authors would like to thank the anonymous reviewers for their valuable comments and suggestions.

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Authors and Affiliations

  1. School of Mathematics and Statistics, Central South University, Changsha, 410083, Hunan, People’s Republic of China

    Zaiming Liu, Yuqing Chu & **biao Wu

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  1. Zaiming Liu
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  2. Yuqing Chu
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  3. **biao Wu
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Correspondence to Yuqing Chu.

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Liu, Z., Chu, Y. & Wu, J. On the three-queue priority polling system with threshold service policy. J. Appl. Math. Comput. 53, 445–470 (2017). https://doi.org/10.1007/s12190-015-0976-5

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  • DOI: https://doi.org/10.1007/s12190-015-0976-5

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