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Perturbation Theory for the Asymptotic Decay Rates in the Queues with Markovian Arrival Process and/or Markovian Service Process

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Abstract

Three kinds of queues with Markovian arrival process and/or Markovian service process, are considered in this paper. In great generality, their basic steady-state distributions have asymptotically exponential tails. We investigate the sensitivity of these asymptotic decay rates to the small entrywise perturbations in the parameter matrices of the Markovian arrival process.

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References

  1. J. Abate, L. Choudhury and W. Whitt, Asymptotics for steady-state tail probabilities in structured Markov queueing models, Commun. Statist. Stochastic Models 1 (1994) 99–143.

    Google Scholar 

  2. J. Abate, L. Choudhury and W. Whitt, Exponential approximation for tail probabilities in queues, I: Waiting times, Oper. Res. 43 (1995) 885–901.

    Google Scholar 

  3. A.S. Alfa and S. Chakravarthy, A discrete queue with the Markovian arrival process and phase type primary and secondary services, Commun. Statist. Stochastic Models 10 (1994) 437–451.

    Google Scholar 

  4. [4]A.S. Alfa, J. Xue and Q. Ye, Entrywise perturbation theory for diagonally dominant M-matrices with application to GI/M/1 Markov chains, Commun. Statist. Stochastic Models, accepted.

  5. G.L. Choudhury, D.M. Lucantoni and W. Whitt, Squeezing the most out of ATM, IEEE Trans. Commun. 44 (2) (1996) 203–217.

    Google Scholar 

  6. L. Elsner, I. Koltracht, M. Neumann and D. Xiao, On accurate computation of the Perron root, SIAM J. Matrix Anal. Appl. 14 (1993) 456–467.

    Google Scholar 

  7. E. Falkenberg, On the asymptotic behaviour of the stationary distribution ofMarkov chains ofM/G/1 type, Commun. Statist. Stochastic Models 10 (1994) 75–97.

    Google Scholar 

  8. W. Feller, Introduction to Probability Theory and its Applications, Vol. II (Wiley, New York, 1966).

    Google Scholar 

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

    Google Scholar 

  10. D.M. Lucantoni, K.S. Meier-Hellstern and M.F. Neuts, A single server queue with server vacations and a class of non-renewal arrivals, Adv. in Appl. Probab. 22 (1990) 675–705.

    Google Scholar 

  11. M.F. Neuts, Matrix Geometric Solutions in Stochastic Models (Johns Hopkins Univ. Press, Baltimore, MD, 1981).

    Google Scholar 

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

    Google Scholar 

  13. M. Neuts, The caudal characteristic curve of queues, Adv. in Appl. Probab. 18 (1986) 221–254.

    Google Scholar 

  14. M.F. Neuts and Y. Takahashi, Asymptotic behavior of the stationary distribution in the GI/PH/c queue with heterogeneous servers, Z. Warhrscheinlichkeits. 57 (1981) 441–452.

    Google Scholar 

  15. [15]W. Whitt, Tail probabilities with statistical multiplexing and effective bandwidths for multi-class queues, Telecommunication Systems, forthcoming.

Download references

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Authors and Affiliations

  1. Department of Industrial and Manufacturing Systems Engineering, University of Windsor, Windsor, ON, Canada, N9B 3P4

    Attahiru Sule Alfa

  2. Department of Mathematics, Fudan University, Shanghai, China

    Jungong Xue

  3. Department of Mathematics, University of Manitoba, Winnipeg, MB, Canada, R3T 2N2

    Qiang Ye

Authors
  1. Attahiru Sule Alfa
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  2. Jungong Xue
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  3. Qiang Ye
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Alfa, A.S., Xue, J. & Ye, Q. Perturbation Theory for the Asymptotic Decay Rates in the Queues with Markovian Arrival Process and/or Markovian Service Process. Queueing Systems 36, 287–301 (2000). https://doi.org/10.1023/A:1011032718715

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