Skip to main content
SpringerLink
Log in

The Analysis of Queues with Time-Varying Rates for Telecommunication Models

  • Published:
Telecommunication Systems Aims and scope Submit manuscript

Abstract

Time dependent behavior has an impact on the performance of telecommunication models. Examples include: staffing a call center, pricing the inventory of private line services for profit maximization, and measuring the time lag between the peak arrivals and peak load for a system. These problems and more motivate the development of a queueing theory with time varying rates. Queueing theory as discussed in this paper is organized and presented from a communications perspective. Canonical queueing models with time-varying rates are given and the necessary mathematical tools are developed to analyze them. Finally, we illustrate the use of these models through various communication applications.

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. Burke, The output of a queueing system, Operations Research 4(6) (1956) 699–704.

    Google Scholar 

  2. F.H. Clarke, Optimization and Nonsmooth Analysis (SIAM Philadelphia, PA, 1990)

    Google Scholar 

  3. R.B. Cooper, Introduction to Queueing Theory (Elsevier, North-Holland, Amsterdam, 1981).

    Google Scholar 

  4. D.J. Daley and D. Vere-Jones, An Introduction to the Theory of Point Processes (Springer, New York, 1988).

    Google Scholar 

  5. J.L. Davis, W.A. Massey and W. Whitt, Sensitivity to the service-time distribution in the nonstationary Erlang loss model, Management Science 41(6) (June 1995) 1107–1116.

    Google Scholar 

  6. N.G. Duffield, W.A. Massey and W. Whitt, A nonstationary offered load model for packet networks, Telecommunication Systems 13(3/4) (March/April 2001) 271–296.

    Google Scholar 

  7. S. Eick, W.A. Massey and W. Whitt, Infinite-server queues with sinusoidal arrival rates, Management Science 39 (January 1993) 241–252.

    Google Scholar 

  8. S. Eick, W.A. Massey and W. Whitt, The physics of the M(t)/G/∞ queue, Operations Research 41 (July/August 1993) 400–408.

    Google Scholar 

  9. A.K. Erlang, Solutions of some problems in the theory of probabilities of significance in automatic telephone exchanges, The Post Office Electrical Engineers Journal 10 (from the 1917 article in Danish in Elektroteknikeren, Vol. 13) (1918) 189–197.

    Google Scholar 

  10. S.N. Ethier and T.G. Kurtz, Markov Process: Characterization and Convergence (Wiley, New York, 1986).

    Google Scholar 

  11. L. Green and P.J. Kolesar, The pointwise stationary approximation for queues with nonstationary arrivals, Management Science (January 1991).

  12. N. Grier, W.A. Massey, T. McKoy and W. Whitt, The time-dependent Erlang loss model with retrials, Telecommunication Systems 7 (1997) 253–265.

    Google Scholar 

  13. R. Hall, Queueing Methods: For Services and Manufacturing (Prentice-Hall, Englewood Cliffs, NJ, 1991).

    Google Scholar 

  14. R.C. Hampshire, W.A. Massey, D. Mitra and Q. Wang, Provisioning for bandwidth trading, in: Telecommunication Network Design and Management (Kluwer Academic, Dordrecht, 2002) pp. 207–226.

    Google Scholar 

  15. R.C. Hampshire, W.A. Massey and Q. Wang, Dynamic pricing for on-demand bandwidth services, Bell Labs Technical Report (2002).

  16. J.R. Jackson, Jobshop-like queueing systems, Managment Science 10(1) (October 1963) 131–142.

    Google Scholar 

  17. D.L. Jagerman, Nonstationary blocking in telephone traffic, Bell System Technical Journal 54 (1975) 625–661.

    Google Scholar 

  18. O.B. Jennings, A. Mandelbaum, W.A. Massey and W. Whitt, Server staffing to meet time-varying demand, Management Science 42(10) (October 1996) 1383–1394.

    Google Scholar 

  19. O.B. Jennings and W.A. Massey, A modified offered load approximation for nonstationary circuit switched networks, in: Selected Proceedings of the 3rd INFORMS Telecommunications Conf., Vol. 7, 1997, pp. 253–265.

    Google Scholar 

  20. O.B. Jennings, W.A. Massey and C. McCalla, Optimal profit for leased lines services, in: Proc. of the 15th Internat. Teletraffic Congress, June 1997, pp. 803–814.

  21. J. Keilson and L.D. Servi, Networks of non-homogeneous M/G/∞ systems, GTE Laboratories, Waltham, MA (1989).

    Google Scholar 

  22. F.P. Kelly, Reversibility and Stochastic Networks (Wiley, New York, 1979).

    Google Scholar 

  23. F.P. Kelly, Notes on effective bandwidths, in: Stochastic Networks: Theory and Applications,eds. F.P. Kelly, S. Zachary and I.E. Ziedins, Royal Statistical Society Lecture Notes Series, Vol. 4 (Oxford Univ. Press, Oxford, 1996) pp. 141–168.

    Google Scholar 

  24. T.G. Kurtz, Strong approximation theorems for density dependent Markov chains, Stochastic Processes and Their Applications 6 (1978) 223–240.

    Google Scholar 

  25. S. Lanning, W.A. Massey, B. Rider and Q. Wang, Optimal pricing in queueing systems with quality of service constraints, in: Teletraffic Engineering in a Competitive World, Proc. of the 16th Internat. Teletraffic Congress, 1999, pp. 747–756.

  26. K. Leung, W.A. Massey and W. Whitt, Traffic models for wireless communications networks, IEEE Journal on Selected Areas in Communications 12 (October 1994) 1353–1364.

    Google Scholar 

  27. A. Mandelbaum and W.A. Massey, Strong approximations for time dependent queues, MOR 20(1) (February 1995) 33–64.

    Google Scholar 

  28. A. Mandelbaum, W.A. Massey and M.I. Reiman, Strong approximations for Markovian service networks, Queueing Systems 30 (1998) 149–201.

    Google Scholar 

  29. A. Mandelbaum, W.A. Massey, M.I. Reiman and B. Rider, Time varying multiserver queues with abandonment and retrials, in: Teletraffic Engineering in a Competitive World, Proc. of the 16th Internat. Teletraffic Congress, 1999, pp. 355–364.

  30. A. Mandelbaum, W.A. Massey, M.I. Reiman, B. Rider and A. Stolyar, Queue lengths and waiting times for multiserver queues with abandonment and retrials, in: Selected Proceedings of the 5th IN-FORMS Telecommunications Conf., to appear.

  31. W.A. Massey, Nonstationary queues, Stanford University (1981) Ph.D. thesis.

  32. W.A. Massey, Asymptotic analysis of the time dependent M/M/1 queue, MOR 10 (May 1985) 305–327.

    Google Scholar 

  33. W.A. Massey, G.A. Parker and W. Whitt, Estimating the parameters of a nonhomogeneous Poisson processes with linear rate, Telecommunication Systems 5 (1996) 361–388.

    Google Scholar 

  34. W.A. Massey and W. Whitt, Networks of infinite-server queues with nonstationary Poisson input, Queueing Systems 13(1) (May 1993) 183–250.

    Google Scholar 

  35. W.A. Massey and W. Whitt, An analysis of the modified offered load approximation for the nonstationary Erlang loss model, Annals of Applied Probability 4(4) (November 1994) 1145–1160.

    Google Scholar 

  36. W.A. Massey and W. Whitt, A stochastic model to capture space and time dynamics in wireless communication systems, Probability in the Engineering and Informational Sciences 8 (1994) 541–569.

    Google Scholar 

  37. G.F. Newell, Queues with time-dependent arrival rates (parts I-III), Journal of Applied Probability 5 (1968) 436–451 (I), 579-590 (II), 591-606 (III).

    Google Scholar 

  38. G.F. Newell, Approximate stochastic behavior of n-server service systems with large n, in: Lecture Notes in Economics and Mathematical Systems, Vol. 87 (Springer, Berlin, 1973).

    Google Scholar 

  39. G.F. Newell, Applications of Queueing Theory (Chapman and Hall, London, 1982).

    Google Scholar 

  40. K.L. Ong and M.R. Taaffe, Approximating nonstationary Pht/Mt/S/C queueing systems, Annals of Operations Research 8 (1987) 103–116.

    Google Scholar 

  41. K.L. Ong and M.R. Taaffe, Nonstationary queues with interrupted Poisson arrivals and unreliable/ repairable servers, Queueing Systems 4 (1989) 27–46.

    Google Scholar 

  42. C. Palm, Intensity variations in telephone traffic, Ericsson Technics 44 (1943) 1–189 (in German); English translation (North-Holland, Amsterdam, 1988).

    Google Scholar 

  43. A. Prékopa, On Poisson and composed Poisson stochastic set functions, Studies in Mathematics 16 (1957) 142–155.

    Google Scholar 

  44. A. Prékopa, On secondary processes generated by a random point distribution of Poisson type, Annales Univ. Sci. Budapest de Eötvös Nom. Sectio Math. 1 (1958) 153–170.

    Google Scholar 

  45. R.T. Rockafellar, Convex Analysis (Princeton Univ. Press, Princeton, 1970) (paperback edition 1997).

    Google Scholar 

  46. T. Rolski, Queues with nonstationary inputs, Queueing Systems 5(1-3) (1989) 113–129.

    Google Scholar 

  47. M.H. Rothkopf and S.S. Oren, A closure approximation for the nonstationary M/M/s queue, Management Science 25 (June 1979) 522–534.

    Google Scholar 

  48. R. Serfozo, Introduction to Stochastic Networks (Springer, Berlin, 1999).

    Google Scholar 

  49. A. Shwartz and A. Weiss, Large Deviations for Performance Analysis, Queues, Communication, and Computing (Chapman and Hall, New York, 1995).

    Google Scholar 

  50. W. Willinger and V. Paxson, Where mathematics meets the Internet, Notices of the American Mathematical Society 5(8) (September 1998) 961–970; http://www.ams.org/notices/199808/paxson.pdf.

    Google Scholar 

  51. G. Yin and Q. Zhang, Continuous-Time Markov Chains and Applications: A Singular Perturbation Approach (Springer, New York, 1998).

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Operations Research and Financial Engineering, Princeton University, Princeton, NJ, 08544, USA

    William A. Massey

Authors
  1. William A. Massey
    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

Massey, W.A. The Analysis of Queues with Time-Varying Rates for Telecommunication Models. Telecommunication Systems 21, 173–204 (2002). https://doi.org/10.1023/A:1020990313587

Download citation

  • Issue Date:

  • DOI: https://doi.org/10.1023/A:1020990313587

Search

Navigation