Skip to main content
SpringerLink
Log in

Approximations for multi-server queues: System interpolations

  • Invited Paper
  • Published:
Queueing Systems Aims and scope Submit manuscript

Abstract

This paper provides a unifying method of generating and/or evaluating approximations for the principal congestion measures in aGI/G/s queueing system. The main focus is on the mean waiting time, but approximations are also developed for the queue-length distribution, the waiting-time distribution and the delay probability for the Poisson arrival case. The approximations have closed forms that combine analytical solutions of simpler systems, and hence they are referred to as system-interpolation approximations or, simply, system interpolations. The method in this paper is consistent with and generalizes system interpolations previously presented for the mean waiting time in theGI/G/s queue.

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. U.N. Bhat, M. Shalaby and M.J. Fischer, Approximation techniques in the solution of queueing problems, Naval Res. Log. Quarterly 26 (1979) 311–326.

    Google Scholar 

  2. M. Björklund and A. Elldin, A practical method of calculation for certain types of complex common control systens, Ericsson Technics 20 (1964) 3–75.

    Google Scholar 

  3. O.J. Boxma, J.W. Cohen and N. Huffels, Approximations of the mean waiting time in anM/G/s queueing system, Oper. Res. 27 (1979) 1115–1127.

    Google Scholar 

  4. D.Y. Burman and D.R. Smith, A light traffic limit theorem for multi-server queues, Math. Oper. Res. 8 (1983) 15–25.

    Google Scholar 

  5. D.Y. Burman and D.R. Smith, Asymptotic analysis of a queueing model with bursty traffic, Bell Syst. Techn. J. 62 (1983) 1433–1453.

    Google Scholar 

  6. D.Y. Burman and D.R. Smith, Asymptotic analysis of a queueing model with Markov modulated arrivals, Oper. Res. 34 (1986) 105–119.

    Google Scholar 

  7. G.P. Cosmetatos, Approximate equilibrium results for the multi-server queue (GI/M/r), Oper. Res. Quarterly 25 (1974) 625–634.

    Google Scholar 

  8. G.P. Cosmetatos, Approximate explicit formulae for the average queueing time in the processes (M/D/r) and (D/M/r), INFOR 13 (1975) 328–332.

    Google Scholar 

  9. G.P. Cosmetatos, Some approximate equilibrium results for the multi-server queue (M/G/r), Oper. Res. Quarterly 27 (1976) 615–620.

    Google Scholar 

  10. G.P. Cosmetatos, Some approximate equilibrium results for the multi-server queue (E m /E k /r), Opsearch 14 (1977) 108–117.

    Google Scholar 

  11. G.P. Cosmetatos, Cobham's model on nonpreemptive multi-server queueing systems: a heuristic method for its generalization, Eur. J. Oper. Res. 1 (1977) 262–264.

    Google Scholar 

  12. G.P. Cosmetatos, Some practical considerations on multi-server queues with multiple Poisson arrivals, OMEGA 6 (1978) 443–448.

    Google Scholar 

  13. G.P. Cosmetatos, On the implementation of Page's approximation for waiting times in general multi-server queues. J. Oper. Res. Soc. 33 (1982) 1158–1159.

    Google Scholar 

  14. G.P. Cosmetatos and S.A. Godsave, Approximations in the multi-server queue with hyper exponential inter-arrival times and exponential service times, J. Oper. Res. Soc. 31 (1980) 57–62.

    Google Scholar 

  15. C.D. Crommelin, Delay probability formulae when the holding times are constant, Post Office Electrical Eng. J. 25 (1932) 41–50.

    Google Scholar 

  16. C.D. Crommelin, Delay probability formulae, Post Office Electrical Eng. J. 26 (1934) 266–274.

    Google Scholar 

  17. A.M. Eikeboom and H.C. Tijms, Waiting time percentiles in the multi-serverM x/G/c queue with batch arrivals, Prob. Eng. Int. Sci. 1 (1987) 75–98.

    Google Scholar 

  18. A.A. Fredericks, A class of approximations for the waiting time distribution in aGI/G/1 queueing system, Bell Syst. Techn. J. 61 (1982) 295–325.

    Google Scholar 

  19. D. Gross and C.M. Harris,Fundamentals of Queueing Theory, 2nd ed. (Wiley, 1985).

  20. B. Halachmi and W.R. Franta, A diffusion approximation to the multi-server queue, Manag. Sci. 24 (1978) 522–529.

    Google Scholar 

  21. S. Halfin, Delays in queues, properties and approximations, in:Teletraffic Issues in an Advanced Information Society, ITC11, ed. M. Akiyama, Vol. 1 (North-Holland, 1986) pp. 1.4-3–1-6.

  22. R. Haji and G.F. Newell, A relation between stationary queue and waiting time distributions, J. Appl. Prob. 8 (1971) 617–620.

    Google Scholar 

  23. P. Hokstad, Approximations for theM/G/m queue, Oper. Res. 26 (1978) 510–523.

    Google Scholar 

  24. P. Hokstad, Bounds for the mean queue length of theM/K 2/m queue, Eur. J. Oper. Res. 23 (1986) 108–117.

    Google Scholar 

  25. V.B. Iversen, Decomposition of anM/D/r·k queue with FIFO intok E k /D/r queues with FIFO, Oper. Res. Lett. 2 (1983) 20–21.

    Google Scholar 

  26. T. Kimura, Diffusion approximation for anM/G/m queue, Oper. Res. 31 (1983) 304–321.

    Google Scholar 

  27. T. Kimura, The queueing network analyzer: a survey (1)-(3) [in Japanese], Commun. Oper. Res. Soc. Japan 29 (1984) 366–371, 431–439, 494–500.

    Google Scholar 

  28. T. Kimura, Refining diffusion approximations forGI/G/1 queues: a tight discretization method, in:Teletraffic Issues in an Advanced Information Society, ITC11, ed. M. Akiyama, Vol. 1 (North-Holland, 1986) pp. 3.1A-2–1-7.

  29. T. Kimura, A two-moment approximation for the mean waiting time in theGI/G/s queue, Manag. Sci. 32 (1986) 751–763.

    Google Scholar 

  30. T. Kimura, Approximations for the mean delay in theM/D/s queue,Proc. Seminar on Queueing Theory and Its Applications, Kyoto (1987) pp. 173–184.

  31. T. Kimura, Heuristic approximations for the mean delay in theGI/G/s queue, Econ. J. Hokkaido Univ. 16 (1987) 87–98.

    Google Scholar 

  32. T. Kimura, Approximations for the waiting time in theGI/G/s queue, J. Oper. Res. Soc. Japan 34 (1991) 173–186.

    Google Scholar 

  33. T. Kimura, Refining Cosmetatos' approximation for the mean waiting time in theM/D/s queue, J. Oper. Res. Soc. 42 (1991) 595–603.

    Google Scholar 

  34. T. Kimura, Approximating the mean waiting time in theGI/G/s queue, J. Oper. Res. Soc. 42 (1991) 959–970.

    Google Scholar 

  35. T. Kimura, Interpolation approximations for the mean waiting time in a multi-server queue, J. Oper. Res. Soc. Japan 35 (1992) 77–92.

    Google Scholar 

  36. T. Kimura, Equivalence relations in the approximations for the finite capacityM/G/s queue, Discussion Paper Series A, No. 15, Faculty of Economics, Hokkaido University, Sapporo (1993).

    Google Scholar 

  37. T. Kimura, A transform-free approximation for the queue-length distribution in the finite capacityM/G/s queue, Discussion Paper Series A, No. 18, Faculty of Economics, Hokkaido University, Sapporo (1993).

    Google Scholar 

  38. T. Kimura, Approximations for the delay probability in theM/G/s queue, in:Stochastic Models in Engineering, Technology and Management, eds. S. Osaki and D.N. Pra Murthy (World Scientific, 1993) pp. 277–285.

  39. J.F.C. Kingman, The single server queue in heavy traffic, Proc. Cambridge Phil. Soc. 57 (1961) 902–904.

    Google Scholar 

  40. J.F.C. Kingman, On queues in heavy traffic, J. Roy. Statist. Soc., Series B 24 (1962) 383–392.

    Google Scholar 

  41. J.F.C. Kingman, The heavy traffic approximation in the theory of queues, in:Proc. Symp. on Congestion Theory, eds. W.L. Smith and R.I. Wilkinson (Chapel Hill, 1964) pp. 137–169.

  42. A.G. de Kok and H.C. Tijms, A two-moment approximation for a buffer design problem requiring a small rejection probability, Perf. Eval. 5 (1985) 77–84.

    Google Scholar 

  43. J. Köllerström, Heavy traffic theory for queues with several servers. I, J. Appl. Prob. 11 (1974) 544–552.

    Google Scholar 

  44. D.D. Kouvatsos and J. Almond, Maximum entropy two-station cyclic queues with multiple general servers, Acta Inf. 26 (1988) 241–267.

    Google Scholar 

  45. W. Krämer and M. Langenbach-Belz, Approximate formulae for the delay in the queueing systemGI/G/1, Proc. 8th Int. Teletraffic Congress, Melbourne (1976) pp. 235.1–8.

  46. A.M. Lee and P.A. Longton, Queueing process associated with airline passenger check-in, Oper. Res. Quarterly 10 (1957) 56–71.

    Google Scholar 

  47. E. Maaløe, Approximation formulae for estimation of waiting-time in multiple-channel queueing systems, Manag. Sci. 19 (1973) 703–710.

    Google Scholar 

  48. M. Miyazawa, Approximation for the queue-length distribution of anM/GI/s queue by the basic equations, J. Appl. Prob. 23 (1986) 443–458.

    Google Scholar 

  49. M. Miyazawa, Rate conservation laws: a survey, Queueing Syst. 15 (1994) 1–58.

    Google Scholar 

  50. E.C. Molina, Application of the theory of probability to telephone trunking problems, Bell Syst. Techn. J. 6 (1927) 461–494.

    Google Scholar 

  51. S.A. Nozaki and S.M. Ross, Approximations in finite-capacity multi-server queues with Poisson arrivals, J. Appl. Prob. 15 (1978) 826–834.

    Google Scholar 

  52. NTT,Queueing Tables (The Electrical Communication Laboratories, Nippon Telegraph and Telephone Public Corporation, 1981).

  53. E. Page,Queueing Theory in OR (Butterworth, 1972).

  54. E. Page, Tables of waiting times forM/M/n, M/D/n andD/M/n and their use to give approximate waiting times in more general queues, J. Oper. Res. Soc. 33 (1982) 453–473.

    Google Scholar 

  55. M.I. Reiman and B. Simon, An interpolation approximation for queueing systems with Poisson input, Oper. Res. 36 (1988) 454–469.

    Google Scholar 

  56. H. Sakasegawa, An approximation formulaL q ≃αρβ/(1-ρ), Ann. Inst. Statist. Math. 29 (1977) Part A, 67–75.

    Google Scholar 

  57. L.P. Seelen, An algorithm forPh/Ph/c queues, Eur. J. Oper. Res. 23 (1986) 118–127.

    Google Scholar 

  58. L.P. Seelen and H.C. Tijms, Approximations for the conditional waiting times in theGI/G/c queue, Oper. Res. Lett. 3 (1984) 183–190.

    Google Scholar 

  59. L.P. Seelen and H.C. Tijms, Approximations to the waiting time percentiles in theM/G/c queue, in:Teletraffic Issues in an Advanced Information Society, ITC11, ed. M. Akiyama, Vol. 1 (North- Holland, 1986) pp. 1.4-4–1-5.

  60. L.P. Seelen, H.C. Tijms and M.H. van Hoorn,Tables for Multi-Server Queues (North-Holland, 1985).

  61. J.G. Shanthikumar and J.A. Buzacott, On the approximations to the single server queue, Int. J. Prod. Res. 18 (1980) 761–773.

    Google Scholar 

  62. H. Shore, Simple approximations for theGI/G/c queue — I: The steady-state probabilities, J. Oper. Res. Soc. 39 (1988) 279–284.

    Google Scholar 

  63. H. Shore, Simple approximations for theGI/G/c queue — II: The moments, the inverse distribution function and the loss function of the number in the system and of the queue delay, J. Oper. Res. Soc. 39 (1988) 381–391.

    Google Scholar 

  64. V.L. Smith, Approximating the distribution of customers inM/E n /s queues, J. Oper. Res. Soc. 36 (1985) 327–332.

    Google Scholar 

  65. D. Stoyan, Approximations forM/G/s queues, Math. Operationsforsch. Statistik 7 (1976) 587–594.

    Google Scholar 

  66. U. Sumita and M. Rieders, A new algorithm for computing the ergodic probability vector for large Markov chains: replacement process approach, Prob. Eng. Inf. Sci. 4 (1990) 89–116.

    Google Scholar 

  67. Y. Takahashi and Y. Takami, A numerical method for the steady-state probabilities of aGI/G/c queueing system in a general class, J. Oper. Res. Soc. Japan 19 (1976) 147–157.

    Google Scholar 

  68. Y. Takahashi, An approximation formula for the mean waiting time of anM/G/c queue, J. Oper. Res. Soc. Japan 20 (1977) 150–163.

    Google Scholar 

  69. H.C. Tijms,Stochastic Modelling and Analysis: A Computational Approach (Wiley, 1986).

  70. H.C. Tijms, A quick and practical approximation to the waiting time distribution in the multiserver queue with priorities, in:Computer Performance and Reliability, eds. G. Iazeolla, P.J. Courtois and O.J. Boxma (North-Holland, 1987) pp. 161–169.

  71. H.C. Tijms, M.H. van Hoorn and A. Federgruen, Approximations for the steady-state probabilities in theM/G/c queue, Adv. Appl. Prob. 13 (1981) 186–206.

    Google Scholar 

  72. M.H. van Hoorn,Algorithms and Approximations for Queueing Systems, CWI Tract No. 8 (CWI, 1983).

  73. M.H. van Hoorn and H.C. Tijms, Approximations for the waiting time distribution of theM/G/c queue, Perf. Eval. 2 (1982) 22–28.

    Google Scholar 

  74. W. Whitt, The queueing network analyzer, Bell Syst. Techn. J. 62 (1983) 2779–2815.

    Google Scholar 

  75. W. Whitt, Comparison conjectures about theM/G/s queue, Oper. Res. Lett. 2 (1983) 203–209.

    Google Scholar 

  76. W. Whitt, Approximations for theGI/G/m queue, preprint (1985).

  77. W. Whitt, An interpolation approximation for the mean workload in theGI/G/1 queue, Oper. Res. 37 (1989) 936–952.

    Google Scholar 

  78. R.W. Wolff, Poisson arrivals see time averages, Oper. Res. 30 (1982) 223–231.

    Google Scholar 

  79. J.-S. Wu, Refining the diffusion approximation for theG/G/c queue, Comp. Math. Appl. 20 (1990) 31–36.

    Google Scholar 

  80. J.-S. Wu and W.C. Chan, Maximum entropy analysis of multi-server queueing systems, J. Oper. Res. Soc. 40 (1989) 815–825.

    Google Scholar 

  81. D.D. Yao, Refining the diffusion approximation for theM/G/m queue, Oper. Res. 33 (1985) 1266–1277.

    Google Scholar 

  82. U. Yechiali, On the relative waiting time in theGI/M/s andGI/M/1 queueing systems, Oper. Res. Quarterly 28 (1977) 325–337.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Business Administration, Hokkaido University, Nishi 7, Kita 9, Kita-ku, 060, Sapporo, Japan

    Toshikazu Kimura

Authors
  1. Toshikazu Kimura
    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

Kimura, T. Approximations for multi-server queues: System interpolations. Queueing Syst 17, 347–382 (1994). https://doi.org/10.1007/BF01158699

Download citation

  • Received:

  • Revised:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF01158699

Keywords

Search

Navigation