Queueing Systems

, Volume 79, Issue 3–4, pp 293–319 | Cite as

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

  • Jan-Pieter L. Dorsman
  • Maria Vlasiou
  • Bert Zwart


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.


Functional central limit theorem Layered queueing networks Machine-repair model Semi-martingale reflected Brownian motion 

Mathematics Subject Classification

60K25 68M20 90B22 



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.


  1. 1.
    Asmussen, S.: Applied Probability and Queues. Springer, New York (2003)Google Scholar
  2. 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)CrossRefGoogle Scholar
  3. 3.
    Bekker, R.: Queues with State-Dependent Rates. PhD thesis, Eindhoven University of Technology (2005)Google Scholar
  4. 4.
    Bertoin, J.: Lévy Processes. Cambridge University Press, Cambridge (1996)Google Scholar
  5. 5.
    Budhiraja, A., Lee, C.: Stationary distribution convergence for generalized Jackson networks in heavy traffic. Math. Oper. Res. 34, 45–56 (2009)CrossRefGoogle Scholar
  6. 6.
    Chen, H., Whitt, W.: Diffusion approximations for open queueing networks with service interruptions. Queueing Syst. 13, 335–359 (1993)CrossRefGoogle Scholar
  7. 7.
    Chen, H., Yao, D.D.: Fundamentals of Queueing Networks. Springer, New York (2001)CrossRefGoogle Scholar
  8. 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)CrossRefGoogle Scholar
  9. 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)CrossRefGoogle Scholar
  10. 10.
    Debicki, K., Kosiński, K.M., Mandjes, M.: Gaussian queues in light and heavy traffic. Queueing Syst. 71, 137–149 (2012)CrossRefGoogle Scholar
  11. 11.
    Dieker, A.B., Moriarty, J.: Reflected Brownian motion in a wedge: sum-of-exponential stationary densities. Electron. Commun. Probab. 14, 1–16 (2009)CrossRefGoogle Scholar
  12. 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. 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)CrossRefGoogle Scholar
  14. 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)CrossRefGoogle Scholar
  15. 15.
    Gamarnik, D., Zeevi, A.: Validity of heavy traffic steady-state approximation in generalized Jackson networks. Ann. Appl. Probab. 16, 56–90 (2006)CrossRefGoogle Scholar
  16. 16.
    George, J.M., Harrison, J.M.: Dynamic control of a queue with adjustable service rate. Oper. Res. 49, 720–731 (2001)CrossRefGoogle Scholar
  17. 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)CrossRefGoogle Scholar
  18. 18.
    Harrison, J.M., Williams, R.J.: Brownian models of open queueing networks with homogeneous customer populations. Stochastics 22, 77–115 (1987)CrossRefGoogle Scholar
  19. 19.
    Harrison, J.M., Williams, R.J.: Multidimensional reflected Brownian motions having exponential stationary distributions. Ann. Probab. 15, 115–137 (1987)CrossRefGoogle Scholar
  20. 20.
    Hopp, W.J., Iravani, S.M.R., Yuen, G.J.: Operations systems with discretionary task completion. Manag. Sci. 53, 61–77 (2006)CrossRefGoogle Scholar
  21. 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)CrossRefGoogle Scholar
  22. 22.
    Jelenković, P.R., Momčilović, P., Zwart, B.: Reduced load equivalence under subexponentiality. Queueing Syst. 46, 97–112 (2004)CrossRefGoogle Scholar
  23. 23.
    Kella, O., Whitt, W.: Diffusion approximations for queues with server vacations. Adv. Appl. Probab. 22, 706–729 (1990)CrossRefGoogle Scholar
  24. 24.
    Kingman, J.F.C.: The single server queue in heavy traffic. Math. Proc. Camb. Philos. Soc. 57, 902–904 (1961)CrossRefGoogle Scholar
  25. 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. 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)CrossRefGoogle Scholar
  27. 27.
    Mahabhashyam, S.R., Gautam, N.: On queues with Markov modulated service rates. Queueing Syst. 51, 89–113 (2005)CrossRefGoogle Scholar
  28. 28.
    Mitra, D.: Stochastic theory of a fluid model of producers and consumers coupled by a buffer. Adv. Appl. Probab. 20, 646–676 (1988)CrossRefGoogle Scholar
  29. 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)CrossRefGoogle Scholar
  30. 30.
    Purdue, P.: The M/M/1 queue in a Markovian environment. Oper. Res. 22, 562–569 (1974)CrossRefGoogle Scholar
  31. 31.
    Revuz, D., Yor, M.: Continuous Martingales and Brownian Motion. Springer, New York (1999)CrossRefGoogle Scholar
  32. 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)CrossRefGoogle Scholar
  33. 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)Google Scholar
  34. 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)CrossRefGoogle Scholar
  35. 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)CrossRefGoogle Scholar
  36. 36.
    Takács, L.: Introduction to the Theory of Queues. Oxford University Press, New York (1962)Google Scholar
  37. 37.
    Takine, T.: Single-server queues with Markov-modulated arrivals and service speed. Queueing Syst. 49, 7–22 (2005)CrossRefGoogle Scholar
  38. 38.
    Tse, D., Viswanath, P.: Fundamentals of Wireless Communication. Cambridge University Press, Cambridge (2005)CrossRefGoogle Scholar
  39. 39.
    Tzenova, E.I., Adan, I.J.B.F., Kulkarni, V.G.: Fluid models with jumps. Stoch. Models 21, 37–55 (2005)CrossRefGoogle Scholar
  40. 40.
    van der Mei, R.D., Hariharan, R., Reeser, P.K.: Web server performance modeling. Telecommun. Syst. 16, 361–378 (2001)CrossRefGoogle Scholar
  41. 41.
    Weber, R.R., Stidham Jr, S.: Optimal control of service rates in networks of queues. Adv. Appl. Probab. 19, 202–218 (1987)CrossRefGoogle Scholar
  42. 42.
    Whitt, W.: Asymptotic formulas for Markov processes with applications to simulation. Oper. Res. 40, 279–291 (1992)CrossRefGoogle Scholar
  43. 43.
    Whitt, W.: Stochastic-Process Limits. Springer, New York (2002)Google Scholar
  44. 44.
    Zwart, B.: Heavy-traffic asymptotics for the single-server queue with random order of service. Oper. Res. Lett. 33, 511–518 (2004)CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  • Jan-Pieter L. Dorsman
    • 1
    • 2
  • Maria Vlasiou
    • 1
    • 2
  • Bert Zwart
    • 1
    • 2
    • 3
    • 4
  1. 1.EURANDOM and Department of Mathematics and Computer ScienceEindhoven University of TechnologyEindhovenThe Netherlands
  2. 2.StochasticsCentrum Wiskunde & Informatica (CWI)AmsterdamThe Netherlands
  3. 3.Department of Mathematics, Faculty of SciencesVU University AmsterdamAmsterdamThe Netherlands
  4. 4.H. Milton Stewart School of Industrial and Systems EngineeringGeorgia Institute of TechnologyAtlantaUSA

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