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Stability of multi-class queueing networks with infinite virtual queues

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Abstract

We generalize the standard multi-class queueing network model by allowing both standard queues and infinite virtual queues which have an infinite supply of work. We pose the general problem of finding policies which allow some of the nodes of the network to work with full utilization, and yet keep all the standard queues in the system stable. Toward this end we show that re-entrant lines, systems of two re-entrant lines through two service stations, and rings of service stations can be stabilized with priority policies under certain parameter restrictions. The analysis throughout the paper depends on model and policy and illustrates the difficulty in solving the general problem.

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References

  1. Adan, I.J.B.F., Weiss, G.: Analysis of a simple Markovian re-entrant line with infinite supply of work under the LBFS policy. Queueing Syst. 54(3), 169–183 (2006)

    Article  Google Scholar 

  2. Asmussen, S.: Applied Probability and Queues. Springer, New York (2003)

    Google Scholar 

  3. Baccelli, F., Foss, S.: Ergodicity of Jackson-type queueing networks. Queueing Syst. 17(1), 5–72 (1994)

    Article  Google Scholar 

  4. Bramson, M.: State space collapse with application to heavy traffic limits for multiclass queueing networks. Queueing Syst. 30(1–2), 89–148 (1998)

    Article  Google Scholar 

  5. Bramson, M.: Stability of Queueing Networks. Springer, Berlin (2008)

    Google Scholar 

  6. Chen, H., Harrison, J.M., Mandelbaum, A., Van Ackere, A., Wein, L.M.: Empirical evaluation of a queueing network model for semiconductor wafer fabrication. Oper. Res. 37(2), 202–215 (1988)

    Article  Google Scholar 

  7. Chen, H., Mandelbaum, A.: Discrete flow networks: bottleneck analysis and fluid approximations. Math. Oper. Res. 16(2), 408–446 (1991)

    Article  Google Scholar 

  8. Chen, H., Mandelbaum, A.: Stochastic discrete flow networks: diffusion approximations and bottlenecks. Ann. Probab. 19(4), 1463–1519 (1991)

    Article  Google Scholar 

  9. Chen, H., Yao, D.D.: Fundamentals of Queueing Networks: Performance, Asymptotics, and Optimization. Springer, New York (2001)

    Book  Google Scholar 

  10. Dai, J.G.: On positive Harris recurrence of multiclass queueing networks: a unified approach via fluid limit models. Ann. Appl. Probab. 5(1), 49–77 (1995)

    Article  Google Scholar 

  11. Dai, J.G., Lin, W.: Maximum pressure policies in stochastic processing networks. Oper. Res. 53(2), 197–218 (2005)

    Article  Google Scholar 

  12. Dai, J.G., Weiss, G.: Stability and instability of fluid models for re-entrant lines. Math. Oper. Res. 21(1), 115–134 (1996)

    Article  Google Scholar 

  13. Davis, M.H.A.: Piecewise-deterministic Markov processes: a general class of non-diffusion stochastic models. J. R. Stat. Soc. Ser B 46(3), 353–388 (1984)

    Google Scholar 

  14. Foss, S.G.: Ergodicity of queuing networks. Sib. Math. J. 32(4), 184–203 (1991)

    Google Scholar 

  15. Harrison, J.M.: Brownian models of queueing networks with heterogeneous customer populations. In: Fleming, W., Lions, P.-L. (eds.) Stochastic Differential Systems, Stochastic Control Theory and Applications, pp 147–186. Springer, New York (1988)

  16. Harrison, J.M.: Brownian models of open processing networks: canonical representation of workload. Ann. Appl. Probab. 10(1), 75–103 (2000)

    Article  Google Scholar 

  17. Henderson, S.G., Meyn, S.P., Tadic, V.B.: Performance evaluation and policy selection in multiclass networks. Discr. Event Dyn. Syst. 13(1–2), 149–189 (2003)

    Article  Google Scholar 

  18. Kelly, F.: Reversibility and Stochastic Networks. Wiley, New York (1979)

    Google Scholar 

  19. Kopzon, A., Weiss, G.: A push pull queueing system. Oper. Res. Lett. 30(6), 351–359 (2002)

    Article  Google Scholar 

  20. Kopzon, A., Nazarathy, Y., Weiss, G.: A push-pull network with infinite supply of work. Queueing Syst. 62(1), 75–111 (2009)

    Article  Google Scholar 

  21. Kumar, P.R.: Re-entrant lines. Queueing Syst. 13(1), 87–110 (1993)

    Article  Google Scholar 

  22. Kumar, P.R., Seidman, T.I.: Dynamic instabilities and stabilization methods in distributed real-time scheduling of manufacturing systems. IEEE Trans. Autom. Control AC–35(3), 289–298 (1990)

    Article  Google Scholar 

  23. Lu, S.H., Kumar, P.R.: Distributed scheduling based on due dates and buffer priorities. IEEE Trans. Autom. Control 36(12), 1406–1416 (1991)

    Article  Google Scholar 

  24. Meyn, S.P.: Control Techniques for Complex Networks. Cambridge University Press, Cambridge (2008)

    Google Scholar 

  25. Meyn, S.P., Down, D.: Stability of generalized Jackson networks. Ann. Appl. Probab. 4(1), 124–148 (1994)

    Article  Google Scholar 

  26. Nazarathy, Y., Weiss, G.: Near optimal control of queueing networks over a finite time horizon. Ann. Oper. Res. 170(1), 233–249 (2009)

    Article  Google Scholar 

  27. Nazarathy, Y., Weiss, G.: Positive Harris recurrence and diffusion scale analysis of a push pull queueing network. Perform. Eval. 67(4), 201–217 (2010)

    Article  Google Scholar 

  28. Nazarathy, Y., Rojas-Nandayapa, L., Salisbury, T.S.: Non-existence of stabilizing policies for the critical push-pull network and generalizations. Oper. Res. Lett. 41(3), 265–270 (2013)

    Article  Google Scholar 

  29. Rybko, A.N., Stolyar, A.L.: Ergodicity of stochastic processes describing the operation of open queueing networks. Probl. Inf. Transm. 28(3), 3–26 (1992)

    Google Scholar 

  30. Tassiulas, L., Ephremides, A.: Stability properties of constrained queueing systems and scheduling policies for maximum throughput in multihop radio networks. IEEE Trans. Autom. Control 37(12), 1936–1948 (1992)

    Article  Google Scholar 

  31. Tekin, S., Andradóttir, S., Down, D.G.: Dynamic server allocation for unstable queueing networks with flexible servers. Queueing Syst. 70(1), 45–49 (2012)

    Article  Google Scholar 

  32. Weiss, G.: Stability of a simple re-entrant line with infinite supply of work—the case of exponential processing times. J. Oper. Res. Soc. Jpn. 47(4), 304–313 (2004)

    Google Scholar 

  33. Williams, R.J.: Diffusion approximations for open multiclass queueing networks: sufficient conditions involving state space collapse. Queueing Syst. 30(1–2), 27–88 (1998)

    Article  Google Scholar 

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Acknowledgments

Yongjiang Guo is supported by the NSFC grants 10901023 and 11101050. Erjen Lefeber is supported by NWO-VIDI grant 639.072.072. The bulk of the work of Yoni Nazarathy on this paper was while he was affiliated with EURANDOM/Eindhoven University of Technology and partly supported by the same grant. Gideon Weiss is supported by Israel Science Foundation Grants 454/05 and 711/09.

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Authors and Affiliations

  1. School of Science, Beijing University of Posts and Telecommunications, Beijing, China

    Yongjiang Guo

  2. Eindhoven University of Technology, Eindhoven, The Netherlands

    Erjen Lefeber

  3. The School of Mathematics and Physics, The University of Queensland, Brisbane, Australia

    Yoni Nazarathy

  4. The University of Haifa, Haifa, Israel

    Gideon Weiss

  5. Business School, National University of Singapore, Singapore, Singapore

    Hanqin Zhang

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  1. Yongjiang Guo
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  2. Erjen Lefeber
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  3. Yoni Nazarathy
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  4. Gideon Weiss
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  5. Hanqin Zhang
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Corresponding author

Correspondence to Gideon Weiss.

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Guo, Y., Lefeber, E., Nazarathy, Y. et al. Stability of multi-class queueing networks with infinite virtual queues. Queueing Syst 76, 309–342 (2014). https://doi.org/10.1007/s11134-013-9362-x

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  • DOI: https://doi.org/10.1007/s11134-013-9362-x

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