Skip to main content
SpringerLink
Log in

A heavy-traffic analysis of a closed queueing system with a GI/∞ service center

  • Published:
Queueing Systems Aims and scope Submit manuscript

Abstract

This paper studies the heavy-traffic behavior of a closed system consisting of two service stations. The first station is an infinite server and the second is a single server whose service rate depends on the size of the queue at the station. We consider the regime when both the number of customers, n, and the service rate at the single-server station go to infinity while the service rate at the infinite-server station is held fixed. We show that, as n→∞, the process of the number of customers at the infinite-server station normalized by n converges in probability to a deterministic function satisfying a Volterra integral equation. The deviations of the normalized queue from its deterministic limit multiplied by √n converge in distribution to the solution of a stochastic Volterra equation. The proof uses a new approach to studying infinite-server queues in heavy traffic whose main novelty is to express the number of customers at the infinite server as a time-space integral with respect to a time-changed sequential empirical process. This gives a new insight into the structure of the limit processes and makes the end results easy to interpret. Also the approach allows us to give a version of the classical heavy-traffic limit theorem for the G/GI/∞ queue which, in particular, reconciles the limits obtained earlier by Iglehart and Borovkov.

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

Similar content being viewed by others

References

  1. P.J. Bickel and M.J. Wichura, Convergence criteria for multiparameter stochastic processes and some applications, Ann. Math. Statist. 42 (1971) 1656–1670.

    Google Scholar 

  2. P. Billingsley, Convergence of Probability Measures (Wiley, 1968).

  3. A. Birman and Y. Kogan, Asymptotic evaluation of closed queueing networks with many stations, Comm. Statist. Stochastic Models 8 (1992) 543–563.

    Google Scholar 

  4. A.A. Borovkov, On limit laws for service processes in multi-channel systems (in Russian), Siberian Math. J. 8 (1967) 746–763.

    Article  Google Scholar 

  5. A.A. Borovkov, Asymptotic Methods in Queueing Theory (in Russian: Nauka, 1980, English translation: Wiley, 1984).

  6. P. Brémaud, Point Processes and Queues. Martingale Dynamics (Springer, 1981).

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

    Article  Google Scholar 

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

    Google Scholar 

  9. E.G. Coffman, Jr., A.A. Pukhalskii and M.I. Reiman, Storage-limited queues in heavy traffic, Probab. Engrg. Inform. Sci. 5 (1991) 499–522.

    Article  Google Scholar 

  10. M. Csörgo and P. Révéz, Strong Approximations in Probability and Statistics (Akadémiai Kiadó, 1981).

  11. C. Dellacherie, Capacités et Processus Stochastiques (Springer, 1972).

  12. R.J. Elliott, Stochastic Calculus and Applications (Springer, 1982).

  13. S.N. Ethier and T.G. Kurtz, Markov Processes. Characterisation and Convergence (Wiley, 1986).

  14. P. Gänssler and W. Stute, Empirical processes: a survey of results for independent and identically distributed random variables, Ann. Probab. 7 (1979) 193–243.

    Google Scholar 

  15. P.W. Glynn, On the Markov property of the GI/G/∞Gaussian limit, Adv. Appl. Probab. 14 (1982) 191–194.

    Article  Google Scholar 

  16. P.W. Glynn, Diffusion approximations, in: Handbooks on OR & MS, Vol. 2, Stochastic Models, eds. D. P. Heyman and M.J. Sobel (North-Holland, 1990) pp. 145–198.

  17. P.W. Glynn and W. Whitt, A new view of the heavy-traffic limit theorem for infinite-server queues, Adv. Appl. Probab. 23 (1991) 188–209.

    Article  Google Scholar 

  18. D.L. Iglehart, Limit diffusion approximations for the many server queue and the repairman problem, J. Appl. Probab. 2 (1965) 429–441.

    Article  Google Scholar 

  19. D.L. Iglehart, Weak convergence of compound stochastic processes, Stoch. Proc. Appl. 1 (1973) 11–31.

    Article  Google Scholar 

  20. D.L. Iglehart, Weak convergence in queueing theory, Adv. Appl. Probab. 5 (1973) 570–594.

    Article  Google Scholar 

  21. J. Jacod and A.N. Shiryaev, Limit Theorems for Stochastic Processes (Springer, 1987).

  22. A.N. Kolmogorov and S.V. Fomin, Introduction to Real Analysis (Dover, 1975).

  23. Y. Kogan, Another approach to asymptotic expansions for large closed queueing networks, Oper. Res. Lett. (1992) 317–321.

  24. Y. Kogan and A. Birman, Asymptotic analysis of closed queueing networks with bottlenecks, in: Proc. Int. Conf. on Performance of Distributed Systems and Integrated Communication Networks, eds. T. Hasegawa, H. Takagi and Y. Takahashi (Kyoto, 1991) pp. 237–252.

  25. Y. Kogan and R. Sh. Liptser, Limit non-stationary behavior of large closed queueing networks with bottlenecks, Queueing Systems 14 (1993) 33–55.

    Article  Google Scholar 

  26. Y. Kogan, R. Sh. Liptser and A.V. Smorodinskii, Gaussian diffusion approximation of closed Markov models of computer networks, Problems Inform. Transmission 22 (1986) 38–51.

    Google Scholar 

  27. E.V. Krichagina, A diffusion approximation for the many-server queue with phase service (in Russian), Avtomatika i Telemekhanika 3 (1989) 73–83.

    Google Scholar 

  28. E.V. Krichagina, Asymptotic analysis of queueing networks (martingale approach), Stochastics 40 (1992) 43–76.

    Google Scholar 

  29. T. Lindvall, Weak convergence of probability measures and random functions in the function space D[0,1), J. Appl. Probab. 10 (1973) 109–121.

    Article  Google Scholar 

  30. R. Sh. Liptser and A.N. Shiryaev, Statistics of Random Processes I (Springer, 1978).

  31. R. Sh. Liptser and A.N. Shiryaev, Theory of Martingales (Kluwer, 1989).

  32. M. Loève, Probability Theory (Van Nostrand, 1955).

  33. G. Louchard, Large finite population queueing systems. Part I: The infinite server model, Comm. Statist. Stochastic Models 4(3) (1988) 473–505.

    Google Scholar 

  34. J. McKenna and D. Mitra, Integral representations and asymptotic expansions for closed Markovian queueing networks: normal usage, Bell Syst. Tech. J. 61 (1982) 661–683.

    Google Scholar 

  35. J. McKenna and D. Mitra, Asymptotic expansions and integral representations of moments of queue lengths in closed Markovian networks, J. ACM 31 (1984) 346–360.

    Article  Google Scholar 

  36. J. McKenna, D. Mitra and K.G. Ramakrishnan, A class of closed Markovian queueing networks: Integral representation, asymptotic expansions and generalizations, Bell Syst. Tech. J. 60 (1981) 599–641.

    Google Scholar 

  37. G. Neuhaus, On weak convergence of stochastic processes with multidimensional time parameter, Ann. Math. Statist. 42 (1971) 1285–1295.

    Google Scholar 

  38. A.V. Smorodinskii, Asymptotic distribution of the queue length of a service system (in Russian), Avtomatika i Telemekhanika 2 (1986).

  39. M.L. Straf, Weak convergence of stochastic processes with several parameters, in: Proc. Sixth Berkeley Symp. Math. Statist. Prob., Vol. 2 (Univ. of California Press, 1971) pp. 187–221.

    Google Scholar 

  40. W. Whitt, On the heavy-traffic limit theorem for GI/G/∞ queues, Adv. Appl. Probab. 14 (1982) 171–190.

    Article  Google Scholar 

  41. W. Whitt, Open and closed models for networks of queues, AT&T Bell Lab. Tech. J. 63 (1984) 1911–1979.

    Google Scholar 

  42. W. Whitt, Departures from a queue with many busy servers, Math. Oper. Res. 9 (1984) 534–544.

    Google Scholar 

Download references

Authors
  1. E.V. Krichagina
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. A.A. Puhalskii
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

About this article

Cite this article

Krichagina, E., Puhalskii, A. A heavy-traffic analysis of a closed queueing system with a GI/∞ service center. Queueing Systems 25, 235–280 (1997). https://doi.org/10.1023/A:1019108502933

Download citation

  • Issue Date:

  • DOI: https://doi.org/10.1023/A:1019108502933

Search

Navigation