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Higher order approximations for tandem queueing networks

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Abstract

In this paper a higher order approximation for single server queues and tandem queueing networks is proposed and studied. Different from the most popular two-moment based approximations in the literature, the higher order approximation uses the higher moments of the interarrival and service distributions in evaluating the performance measures for queueing networks. It is built upon the MacLaurin series analysis, a method that is recently developed to analyze single-node queues, along with the idea of decomposition using higher orders of the moments matched to a distribution. The approximation is computationally flexible in that it can use as many moments of the interarrival and service distributions as desired and produce the corresponding moments for the waiting and interdeparture times. Therefore it can also be used to study several interesting issues that arise in the study of queueing network approximations, such as the effects of higher moments and correlations. Numerical results for single server queues and tandem queueing networks show that this approximation is better than the two-moment based approximations in most cases.

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References

  1. G.A. Baker,Essentials of Padé Approximants (Academic Press, 1975).

  2. F. Baskett, K.M. Chandy, R.R. Muntz and F.G. Palacios, Open, closed, and mixed networks of queues with different classes of customers, JACM 22 (1975) 248–260.

    Google Scholar 

  3. G.R. Bitran and S. Dasu, Approximating nonrenewal processes by Markov chains: Use of Super-Erlang (SE) chains, Oper. Res. 41 (1993) 903–923.

    Google Scholar 

  4. G.R. Bitran and S. Dasu, Analysis of the ΣPh i /Ph/1 queue, Oper. Res. 42 (1994) 158–174.

    Google Scholar 

  5. J.A. Buzacott and D.D. Yao, On queueing network models of flexible manufacturing systems, Queueing Systems 1 (1986) 5–27.

    Google Scholar 

  6. J.G. Dai, V. Nguyen and M.I. Reiman, Sequential bottleneck decomposition: An approximation method for generalized Jackson networks, Oper. Res. 42 (1994) 119–136.

    Google Scholar 

  7. D.J. Daley and T. Rolski, Light traffic approximations in queues, Math. Oper. Res. 16 (1991) 57–71.

    Google Scholar 

  8. M.K. Girish and J.Q. Hu, Higher order approximations for queueing networks, Technical Report, Dept. of Manufacturing Engineering, Boston University (April 1995).

  9. M.K. Girish and J.Q. Hu, The departure process of the G/G/1 queue with Markov-modulated arrivals, submitted for publication in Stochas. Models.

  10. W.B. Gong and J.Q. Hu, The MacLaurin series for the GI/G/1 queue, J. Appl. Prob. 29 (1992) 176–184.

    Google Scholar 

  11. W.B. Gong, S. Nananukul and A. Yan, Padé approximation for stochastic discrete event systems, IEEE Trans. Autom. Control, to appear.

  12. W.J. Gordon and G.F. Newell, Closed queueing systems with exponential servers, Oper. Res. 15 (1967) 254–265.

    Google Scholar 

  13. J.M. Harrison and V. Nguyen, The QNET method for two-moment analysis of open queueing networks, Queueing Systems 6 (1990) 1–32.

    Google Scholar 

  14. J.Q. Hu, Analyticity of single-server queues in light traffic, Queueing Systems 19 (1995) 63–80.

    Google Scholar 

  15. J.Q. Hu, The departure process of the GI/G/1 queue and its MacLaurin series, Oper. Res., to appear.

  16. J.R. Jackson, Jobshop-like queueing systems, Manag. Sci. 10 (1963) 131–142.

    Google Scholar 

  17. M.A. Johnson and M.R. Taaffe, Matching moments to phase distributions: Mixtures of Erlang distributions of common order, Stochas. Models 5 (1989) 711–743.

    Google Scholar 

  18. M.A. Johnson and M.R. Taaffe, Matching moments to phase distributions: Density function shapes, Stochas. Models 6 (1990) 283–306.

    Google Scholar 

  19. M.A. Johnson and M.R. Taaffe, An investigation of phase-distribution moment matching algorithms for use in queueing models, Queueing Systems 8 (1991) 129–148.

    Google Scholar 

  20. M.A. Johnson and M.R. Taaffe, A graphical investigation of error bounds for moment-based queueing approximations, Queueing Systems 8 (1991) 295–312.

    Google Scholar 

  21. M.A. Johnson, Selecting parameters of phase distributions: Combining nonlinear programming, heuristics, and Erlang distributions, ORSA J. Comput. 5 (1993) 69–83.

    Google Scholar 

  22. L. Kleinrock,Queueing Systems, Vol. 1 (Wiley Interscience, New York, 1975).

    Google Scholar 

  23. W. Kraemer and M. Langenbach-Belz, Approximate formulae for the delay in the queueing system GI/G/1, Proc. 8th Internat. Teletraffic Congress, Melbourne (1976) 235–248.

  24. P.J. Kuehn, Approximate analysis of general queueing networks by decomposition, IEEE Trans. Commun. COM-27 (1979) 113–126.

    Google Scholar 

  25. M. Livny, B. Melamed and A.K. Tsiolis, The impact of autocorrelation on queueing systems, Manag. Sci. 39 (1993) 322–339.

    Google Scholar 

  26. K.T. Marshall, Some inequalities in queueing, Oper. Res. 16 (1968) 651–665.

    Google Scholar 

  27. P.P. Petrushev and V.A. Popov,Rational Approximation of Real Functions (Cambridge University Press, 1987).

  28. L. Schmickler, MEDA: Mixed Erlang distributions as phase-type representations of empirical distribution functions, Stochas. Models 8 (1992) 131–155.

    Google Scholar 

  29. J.G. Shanthikumar and J.A. Buzacott, Open queueing network models of dynamic job shops, Int. J. Prod. Res. 19 (1981) 225–266.

    Google Scholar 

  30. R. Suri, J.L. Sanders and M. Kamath, Performance evaluation of production networks,Hand-books in OR & MS, Vol. 4 (Elsevier, 1993) pp. 199–286.

  31. H.C. Tijms, Stochastic modelling and analysis: A computational approach (Wiley, New York, 1986).

    Google Scholar 

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

    Google Scholar 

  33. W. Whitt, Performance of the queueing network analyzer, Bell Syst. Tech. J. 62 (1983) 2817–2843.

    Google Scholar 

  34. W. Whitt, Approximating a point process by a renewal process, I: Two basic methods, Oper. Res. 50 (1982) 125–147.

    Google Scholar 

  35. W. Whitt, Approximations for departure processes and queues in series, Naval Res. Logistics Quarterly 31 (1984) 499–521.

    Google Scholar 

  36. W. Whitt, On approximations for queues, III: Mixtures of exponential distributions, AT&T Bell Lab. Tech. J. 63 (1984) 163–175.

    Google Scholar 

  37. D.D. Yao and J.A. Buzacott, The exponentialization approach to flexible manufacturing systems, Ann. Oper. Res. 24 (1986).

  38. Y. Zhu and H. Li, The MacLaurin expansion for a G/G/1 queue with Markov-modulated arrivals and services, Queueing Systems 14 (1993) 125–134.

    Google Scholar 

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Authors and Affiliations

  1. Department of Manufacturing Engineering, Boston University, 15 St. Mary's Street, 02215, Boston, MA, USA

    Muckai K. Girish & Jian -Qiang Hu

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  1. Muckai K. Girish
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  2. Jian -Qiang Hu
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Girish, M.K., Hu, J.Q. Higher order approximations for tandem queueing networks. Queueing Syst 22, 249–276 (1996). https://doi.org/10.1007/BF01149174

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  • DOI: https://doi.org/10.1007/BF01149174

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