Skip to main content
SpringerLink
Log in

Heavy-traffic asymptotics for networks of parallel queues with Markov-modulated service speeds

  • Published:
Queueing Systems Aims and scope Submit manuscript

Abstract

We study a network of parallel single-server queues, where the speeds of the servers are varying over time and governed by a single continuous-time Markov chain. We obtain heavy-traffic limits for the distributions of the joint workload, waiting-time and queue length processes. We do so by using a functional central limit theorem approach, which requires the interchange of steady-state and heavy-traffic limits. The marginals of these limiting distributions are shown to be exponential with rates that can be computed by matrix-analytic methods. Moreover, we show how to numerically compute the joint distributions, by viewing the limit processes as multi-dimensional semi-martingale reflected Brownian motions in the non-negative orthant.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1

Similar content being viewed by others

References

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

    Google Scholar 

  2. Ata, B., Shneorson, S.: Dynamic control of an M/M/1 service system with adjustable arrival and service rates. Manag. Sci. 52, 1778–1791 (2006)

    Article  Google Scholar 

  3. Bekker, R.: Queues with State-Dependent Rates. PhD thesis, Eindhoven University of Technology (2005)

  4. Bertoin, J.: Lévy Processes. Cambridge University Press, Cambridge (1996)

    Google Scholar 

  5. Budhiraja, A., Lee, C.: Stationary distribution convergence for generalized Jackson networks in heavy traffic. Math. Oper. Res. 34, 45–56 (2009)

    Article  Google Scholar 

  6. Chen, H., Whitt, W.: Diffusion approximations for open queueing networks with service interruptions. Queueing Syst. 13, 335–359 (1993)

    Article  Google Scholar 

  7. Chen, H., Yao, D.D.: Fundamentals of Queueing Networks. Springer, New York (2001)

    Book  Google Scholar 

  8. Choudhury, G.L., Mandelbaum, A., Reiman, M.I., Whitt, W.: Fluid and diffusion limits for queues in slowly changing environments. Commun. Stat. Stoch. Models 13, 121–146 (1997)

    Article  Google Scholar 

  9. Dai, J.G., Harrison, J.M.: Reflected Brownian motion in an orthant: numerical methods for steady-state analysis. Ann. Appl. Probab. 2, 65–86 (1992)

    Article  Google Scholar 

  10. Debicki, K., Kosiński, K.M., Mandjes, M.: Gaussian queues in light and heavy traffic. Queueing Syst. 71, 137–149 (2012)

    Article  Google Scholar 

  11. Dieker, A.B., Moriarty, J.: Reflected Brownian motion in a wedge: sum-of-exponential stationary densities. Electron. Commun. Probab. 14, 1–16 (2009)

    Article  Google Scholar 

  12. Dorsman, J.L., Bhulai, S., Vlasiou, M.: Dynamic server assignment in an extended machine-repair model. IIE Trans. (2014). doi:10.1080/0740817X.2014.928962

  13. Dorsman, J.L., Boxma, O.J., Vlasiou, M.: Marginal queue length approximations for a two-layered network with correlated queues. Queueing Syst. 75, 29–63 (2013)

    Article  Google Scholar 

  14. Dorsman, J.L., van der Mei, R.D., Vlasiou, M.: Analysis of a two-layered network by means of the power-series algorithm. Perform. Eval. 70, 1072–1089 (2013)

    Article  Google Scholar 

  15. Gamarnik, D., Zeevi, A.: Validity of heavy traffic steady-state approximation in generalized Jackson networks. Ann. Appl. Probab. 16, 56–90 (2006)

    Article  Google Scholar 

  16. George, J.M., Harrison, J.M.: Dynamic control of a queue with adjustable service rate. Oper. Res. 49, 720–731 (2001)

    Article  Google Scholar 

  17. Halfin, S.: Steady-state distribution for the buffer content of an M/G/1 queue with varying service rate. SIAM J. Appl. Math. 23, 356–363 (1972)

    Article  Google Scholar 

  18. Harrison, J.M., Williams, R.J.: Brownian models of open queueing networks with homogeneous customer populations. Stochastics 22, 77–115 (1987)

    Article  Google Scholar 

  19. Harrison, J.M., Williams, R.J.: Multidimensional reflected Brownian motions having exponential stationary distributions. Ann. Probab. 15, 115–137 (1987)

    Article  Google Scholar 

  20. Hopp, W.J., Iravani, S.M.R., Yuen, G.J.: Operations systems with discretionary task completion. Manag. Sci. 53, 61–77 (2006)

    Article  Google Scholar 

  21. Ivanovs, J., Boxma, O.J., Mandjes, M.R.H.: Singularities of the matrix exponent of a Markov additive process with one-sided jumps. Stoch. Process. Their Appl. 120, 1776–1794 (2010)

    Article  Google Scholar 

  22. Jelenković, P.R., Momčilović, P., Zwart, B.: Reduced load equivalence under subexponentiality. Queueing Syst. 46, 97–112 (2004)

    Article  Google Scholar 

  23. Kella, O., Whitt, W.: Diffusion approximations for queues with server vacations. Adv. Appl. Probab. 22, 706–729 (1990)

    Article  Google Scholar 

  24. Kingman, J.F.C.: The single server queue in heavy traffic. Math. Proc. Camb. Philos. Soc. 57, 902–904 (1961)

    Article  Google Scholar 

  25. Kingman, J.F.C.: The heavy traffic approximation in the theory of queues. In: Smith, W.L., Wilkinson, W.E. (eds.) Proceedings of the Symposium on Congestion Theory, pp. 137–159. University of North Carolina Press, Chapel Hill (1965)

    Google Scholar 

  26. Kosiński, K.M., Boxma, O.J., Zwart, B.: Convergence of the all-time supremum of a Lévy process in the heavy-traffic regime. Queueing Syst. 67, 295–304 (2011)

    Article  Google Scholar 

  27. Mahabhashyam, S.R., Gautam, N.: On queues with Markov modulated service rates. Queueing Syst. 51, 89–113 (2005)

    Article  Google Scholar 

  28. Mitra, D.: Stochastic theory of a fluid model of producers and consumers coupled by a buffer. Adv. Appl. Probab. 20, 646–676 (1988)

    Article  Google Scholar 

  29. Núñez-Queija, R.: A queueing model with varying service rate for ABR. In: Puigjaner, R., Savino, N.N., Serra, B. (eds.) Proceedings of the 10th International Conference on Computer Performance Evaluation: Modelling Techniques and Tools, pp. 93–104. Springer, Berlin (1998)

    Chapter  Google Scholar 

  30. Purdue, P.: The M/M/1 queue in a Markovian environment. Oper. Res. 22, 562–569 (1974)

    Article  Google Scholar 

  31. Revuz, D., Yor, M.: Continuous Martingales and Brownian Motion. Springer, New York (1999)

    Book  Google Scholar 

  32. Shneer, S., Wachtel, V.: A unified approach to the heavy-traffic analysis of the maximum of random walks. Theory Probab. Appl. 55, 332–341 (2011)

    Article  Google Scholar 

  33. Siebert, F.: Real-time garbage collection in multi-threaded systems on a single processor. In: Proceedings of the 20th IEEE Real-Time Systems Symposium, pp. 277–278 (1999)

  34. Steichen, J.L.: A functional central limit theorem for Markov additive processes with an application to the closed Lu-Kumar network. Stoch. Models 17, 459–489 (2001)

    Article  Google Scholar 

  35. Stidham Jr, S., Weber, R.R.: Monotonic and insensitive optimal policies for control of queues with undiscounted costs. Oper. Res. 37, 611–625 (1989)

    Article  Google Scholar 

  36. Takács, L.: Introduction to the Theory of Queues. Oxford University Press, New York (1962)

    Google Scholar 

  37. Takine, T.: Single-server queues with Markov-modulated arrivals and service speed. Queueing Syst. 49, 7–22 (2005)

    Article  Google Scholar 

  38. Tse, D., Viswanath, P.: Fundamentals of Wireless Communication. Cambridge University Press, Cambridge (2005)

    Book  Google Scholar 

  39. Tzenova, E.I., Adan, I.J.B.F., Kulkarni, V.G.: Fluid models with jumps. Stoch. Models 21, 37–55 (2005)

    Article  Google Scholar 

  40. van der Mei, R.D., Hariharan, R., Reeser, P.K.: Web server performance modeling. Telecommun. Syst. 16, 361–378 (2001)

    Article  Google Scholar 

  41. Weber, R.R., Stidham Jr, S.: Optimal control of service rates in networks of queues. Adv. Appl. Probab. 19, 202–218 (1987)

    Article  Google Scholar 

  42. Whitt, W.: Asymptotic formulas for Markov processes with applications to simulation. Oper. Res. 40, 279–291 (1992)

    Article  Google Scholar 

  43. Whitt, W.: Stochastic-Process Limits. Springer, New York (2002)

    Google Scholar 

  44. Zwart, B.: Heavy-traffic asymptotics for the single-server queue with random order of service. Oper. Res. Lett. 33, 511–518 (2004)

    Article  Google Scholar 

Download references

Acknowledgments

The authors are indebted to Sem Borst and Onno Boxma for providing valuable comments on earlier drafts of this paper. Furthermore, the authors wish to thank an anonymous referee for providing constructive criticism and for making several suggestions that led to an improved exposition of the contents in this paper. Funded in the framework of the STAR-project ‘Multilayered queueing systems’ by the Netherlands Organization for Scientific Research (NWO). The research of Maria Vlasiou is also partly supported by an NWO individual Grant through Project 632.003.002. The research of Bert Zwart is partly supported by an NWO VIDI grant.

Author information

Authors and Affiliations

  1. EURANDOM and Department of Mathematics and Computer Science, Eindhoven University of Technology, Eindhoven, The Netherlands

    Jan-Pieter L. Dorsman, Maria Vlasiou & Bert Zwart

  2. Stochastics, Centrum Wiskunde & Informatica (CWI), Amsterdam, The Netherlands

    Jan-Pieter L. Dorsman, Maria Vlasiou & Bert Zwart

  3. Department of Mathematics, Faculty of Sciences, VU University Amsterdam, Amsterdam, The Netherlands

    Bert Zwart

  4. H. Milton Stewart School of Industrial and Systems Engineering, Georgia Institute of Technology, Atlanta, GA, 30332, USA

    Bert Zwart

Authors
  1. Jan-Pieter L. Dorsman
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Maria Vlasiou
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Bert Zwart
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Jan-Pieter L. Dorsman.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Dorsman, JP.L., Vlasiou, M. & Zwart, B. Heavy-traffic asymptotics for networks of parallel queues with Markov-modulated service speeds. Queueing Syst 79, 293–319 (2015). https://doi.org/10.1007/s11134-014-9422-x

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11134-014-9422-x

Keywords

Mathematics Subject Classification

Search

Navigation