Skip to main content
SpringerLink
Log in

Tail probabilities with statistical multiplexing and effective bandwidths in multi-class queues

  • Published:
Telecommunication Systems Aims and scope Submit manuscript

Abstract

Our primary purpose in this paper is to contribute to the design of admission control schemes for multi-class service systems. We are motivated by emerging highspeed networks exploiting asynchronous transfer mode (ATM) technology, but there may be other applications. We develop a simple criterion for feasibility of a set of sources in terms of “effective bandwidths”. These effective bandwidths are based on asymptotic decay rates of steady-state distributions in queueing models. We show how to compute asymptotic decay rates of steady-state queue length and workload tail probabilities in general infinite-capacity multi-channel queues. The model hasm independent heterogeneous servers that are independent of an arrival process which is a superposition ofn independent general arrival processes. The contribution of each component arrival process to the overall asymptotic decay rates can be determined from the asymptotic decay rates produced by this arrival process alone in a G/D/1 queue (as a function of the arrival rate). Similarly, the contribution of each service process to the overall asymptotic decay rates can be determined from the asymptotic decay rates produced by this service process alone in a D/G/1 queue. These contributions are characterized in terms of single-channel asymptotic decay-rate functions, which can be estimated from data or determined analytically from models. The asymptotic decay-rate functions map potential decay rates of the queue length into associated decay rates of the workload. Combining these relationships for the arrival and service channels determines the asymptotic decay rates themselves. The asymptotic decay-rate functions are the time-average limits of logarithmic moment generating functions. We give analytical formulas for the asymptotic decay-rate functions of a large class of stochastic point processes, including batch Markovian arrival processes. The Markov modulated Poisson process is a special case. Finally, we try to put our work in perspective with the related literature.

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. J. Abate, G.L. Choudhury and W. Whitt, Exponential approximation for tail probabilities in queues, I: waiting times, Oper. Res., to appear.

  2. J. Abate, G.L. Choudhury and W. Whitt, Exponential approximations for tail probabilities in queues, II: sojourn time and workload (1992), submitted.

  3. J. Abate, G.L. Choudhury and W. Whitt, Asymptotics for steady-state tail probabilities in structured Markov queueing models, Stochastic Models 10 (1994), to appear.

  4. J. Abate and W. Whitt, A heavy-traffic expansion for asymptotic decay rates in multi-channel queues, Oper. Res. Lett., to appear.

  5. D. Anick, D. Mitra and M.M. Sondhi, Stochastic theory of a data-handling system with multiple sources, Bell Syst. Tech. J. 61(1982)1871–1894.

    Google Scholar 

  6. S. Asmussen,Applied Probability and Queues (Wiley, New York, 1987).

    Google Scholar 

  7. S. Asmussen and D. Perry, On cycle maxima, first passage problems and extreme value theory for queues, Stochastic Models 8(1992)421–458.

    Google Scholar 

  8. A.W. Berger and W. Whitt, The pros and cons of a job buffer in a token-bank rate-control throttle, IEEE Trans. Commun., to appear.

  9. A.A. Borovkov,Stochastic Processes in Queueing Theory (Springer, New York, 1976).

    Google Scholar 

  10. J.A. Bucklew,Large Deviation Techniques in Decision, Simulation and Estimation (Wiley, New York, 1990).

    Google Scholar 

  11. C.S. Chang, Stability, queue length and delay of deterministic and stochastic queueing networks, IEEE Trans. Aut. Cont., to appear. (Preliminary version in CDC'92.)

  12. C.S. Chang, P. Heidelberger, S. Juneja and P. Shahabuddin, Effective bandwidth and fast simulation of ATM in tree networks, IBM T.J. Watson Research Center, Yorktown Heights, NY (1992).

    Google Scholar 

  13. G.L. Choudhury and D.M. Lucantoni, Numerical computation of the moments of a probability distribution from its transform, Oper. Res., to appear.

  14. G.L. Choudhury, D.M. Lucantoni and W. Whitt, Squeezing the most out of ATM (1993), submitted.

  15. G.L. Choudhury, D.M. Lucantoni and W. Whitt, An algorithm for a large class of G/G/1 queues, in preparation.

  16. G.L. Choudhury and W. Whitt, Heavy-traffic asymptotic expansions for the asymptotic decay rate in BMAP/G/1 queues. Stochastic Models, to appear.

  17. K.L. Chung,A Course in Probability, 2nd ed. (Academic Press, New York, 1974).

    Google Scholar 

  18. A. Dembo and O. Zeitouni,Large Deviation Techniques and Applications (Jones and Bartlett, Boston, 1993).

    Google Scholar 

  19. A.I. Elwalid and D. Mitra, Effective bandwidth of general Markovian traffic sources and admission control of high speed networks, IEEE Trans. Networks, to appear. (Abbreviated version inINFOCOM'93).

  20. A. I. Elwalid and D. Mitra, Markovian arrival and service communication systems: spectral expansions, separability and Kronecker-product forms, AT&T Bell Laboratories, Murray Hill, NJ (1993).

    Google Scholar 

  21. A.I. Elwalid, D. Mitra and T.E. Stern, Statistical multiplexing of Markov modulated sources: theory and computational algorithms, in:Teletraffic and Data Traffic in a Period of Change, ITC-13, eds. A. Jensen and B. Iversen (Elsevier, Amsterdam, 1991) pp. 495–500.

    Google Scholar 

  22. K.W. Fendick, V.R. Saksena and W. Whitt, Dependence in packet queues, IEEE Trans. Commun. 37(1989)1173–1183.

    Article  Google Scholar 

  23. K.W. Fendick, V.R. Saksena and W. Whitt, Investigating dependence in packet queues with the index of dispersion for work, IEEE Trans. Commun. 39(1991)1231–1244.

    Article  Google Scholar 

  24. K.W. Fendick and W. Whitt, Measurements and approximations to describe the offered traffic and predict the average workload in a single-server queue, Proc. IEEE 77(1989)171–194.

    Article  Google Scholar 

  25. R.J. Gibbens and P.J. Hunt, Effective bandwidths for the multi-type UAS channel, Queueing Systems 9(1991)17–28.

    Article  Google Scholar 

  26. P.W. Glynn and W. Whitt, Logarithmic asymptotics for steady-state tail probabilities in a single-server queue, J. Appl. Prob. 31(1994), to appear.

  27. P.W. Glynn and W. Whitt, Large deviations behavior of counting processes and their inverses (1993), submitted.

  28. R. Guerin, H. Ahmadi and M. Naghshineh, Equivalent capacity and its application to bandwidth allocation in high-speed networks, IEEE J. Select Areas Commun. SAC-9(1991)968–981.

    Article  Google Scholar 

  29. H. Heffes and D.M. Lucantoni, Markov modulated characterization of packetized voice and data traffic and related statistical multiplexer performance, IEEE J. Select. Areas Commun. SAC-4(1986)856–868.

    Article  Google Scholar 

  30. J.Y. Hui, Resource allocation for broadband networks, IEEE J. Select. Areas Commun. SAC-6(1988)1598–1608.

    Article  Google Scholar 

  31. D.L. Iglehart and W. Whitt, Multiple channel queues in heavy traffic, I, Adv. Appl. Prob. 2(1970)150–177.

    Google Scholar 

  32. D.L. Iglehart and W. Whitt, Multiple channel queues in heavy traffic, II: sequences, networks and batches, Adv. Appl. Prob. 2(1970)355–369.

    Google Scholar 

  33. N.L. Johnson and S. Kotz,Discrete Distributions (Wiley, New York, 1969).

    Google Scholar 

  34. F.P. Kelly, Effective bandwidths at multi-class queues, Queueing Systems 9(1991)5–16.

    Article  Google Scholar 

  35. G. Kesidis, J. Walrand and C.S. Chang, Effective bandwidths for multiclass Markov fluids and other ATM sources, Research Report 18588, IBM T.J. Watson Research Center, Yorktown Heights, NY (1992).

    Google Scholar 

  36. J.F.C. Kingman, A convexity property of positive matrices, Quart J. Math. 12(1961)283–284.

    Google Scholar 

  37. D.M. Lucantoni, New results on the single server queue with a batch Markovian arrival process, Stochastic Models 7(1991)1–46.

    Google Scholar 

  38. D.M. Lucantoni, The BMAP/G/1 queue: a tutorial, in:Models and Techniques for Performance Evaluation of Computer and Communication Systems, eds. L. Danatiello and R. Nelson (Springer, New York, 1993).

    Google Scholar 

  39. D.M. Lucantoni, K.S. Meier-Hellstern and M.F. Neuts, A single-server queue with server vacations and a class on non-renewal arrival processes, Adv. Appl. Prob. 22(1990)676–705.

    Google Scholar 

  40. D.M. Lucantoni and M.F. Neuts, The customer delay in a single server queue with a batch Markovian arrival process, AT&T Bell Laboratories, Holmdel, NJ (1993).

    Google Scholar 

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

    Google Scholar 

  42. D. Mitra, R.J. Gibbens and B.D. Huang, State dependent routing on symmetric loss networks with trunk reservations, I, IEEE Trans. Commun. 40(1992)477–482.

    Article  Google Scholar 

  43. M.F. Neuts, A versatile Markovian point process, J. Appl. Prob. 16(1979)764–779.

    Google Scholar 

  44. M.F. Neuts,Matrix-Geometric Solutions in Stochastic Models (The Johns Hopkins University Press, Baltimore. 1981).

    Google Scholar 

  45. M.F. Neuts, Stationary waiting-time distributions in the GI/PH/1 queue, J. Appl. Prob. 18(1981)901–912.

    Google Scholar 

  46. M.F. Neuts, The caudal characteristic curve of queues, Adv. Appl. Prob. 18(1986)221–254.

    Google Scholar 

  47. M.F. Neuts,Structured Stochastic Matrices of M/G/1 Type and Their Applications (Marcel Dekker, New York, 1989).

    Google Scholar 

  48. M.F. Neuts and Y. Takahashi, Asymptotic behavior of the stationary distributions in the GI/PH/C queue with heterogeneous servers, Z. Wahrscheinlichkeitsth. 57(1981)441–452.

    Article  Google Scholar 

  49. V. Ramaswami, The N/G/1 queue and its detailed analysis, Adv. Appl. Prob. 12(1980)222–261.

    Google Scholar 

  50. J.W. Roberts,Performance Evaluation and Design of Multiservice Networks, COST 224 Final Report (Commission of the European Communities, Luxembourg, 1992).

    Google Scholar 

  51. E. Seneta,Nonnegative Matrices and Markov Chains, 2nd ed. (Springer, New York, 1981).

    Google Scholar 

  52. W.L. Smith, On the distribution of queueing times, Proc. Cambridge Phil. Soc. 49(1953)449–461.

    Google Scholar 

  53. K. Sohraby, On the asymptotic behavior of heterogeneous statistical multiplexer with applications,INFOCOM'92, Florence, Italy.

  54. K. Sohraby, On the theory of general on-off sources with applications in high-speed networks,INFOCOM'93, San Francisco, CA.

  55. K. Sriram and W. Whitt, Characterizing superposition arrival processes in packet multiplexers for voice and data, IEEE J. Select Areas Commun. SAC-4(1986)833–846.

    Article  Google Scholar 

  56. T.E. Stern and A.I. Elwalid, Analysis of separable Markov-modulated rate models for information-handling systems, Adv. Appl. Prob. 23(1991)105–139.

    Google Scholar 

  57. Y. Takahashi, Asymptotic exponentiality of the tail of the waiting-time distribution in a PH/PH/c queue, Adv. Appl. Prob. 13(1981)619–630.

    Google Scholar 

  58. H.C. Tijms,Stochastic Modeling and Analysis: A Computational Approach (Wiley, New York, 1986).

    Google Scholar 

  59. J.C.W. Van Ommeren, Exponential expansion for the tail of the waiting-time probability in the single-server queue with batch arrivals, Adv. Appl. Prob. 20(1988)880–895.

    Google Scholar 

  60. W. Whitt, Some useful functions for functional limit theorems, Math. Oper. Res. 5(1980)67–85.

    Google Scholar 

  61. W. Whitt, Queues with superposition arrival processes in heavy traffic, Stoch. Proc. Appl. 21(1985)81–91.

    Article  Google Scholar 

  62. W. Whitt, A review ofL=λW, Queueing Systems 9(1991)235–268.

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. AT&T Bell Laboratories, 07974-0636, Murray Hill, NJ, USA

    Ward Whitt

Authors
  1. Ward Whitt
    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

Whitt, W. Tail probabilities with statistical multiplexing and effective bandwidths in multi-class queues. Telecommunication Systems 2, 71–107 (1993). https://doi.org/10.1007/BF02109851

Download citation

  • Received:

  • Revised:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF02109851

Keywords

Search

Navigation