Asymptotic behavior of the stationary distributions in the GI/PH/c queue with heterogeneous servers

  • Marcel F. Neuts
  • Yukio Takahashi
Article

Summary

This paper deals with the stablec-server queue with renewal input. The service time distributions may be different for the various servers. They are however all probability distributions of phase type. It is shown that the stationary distribution of the queue length at arrivals has an exact geometric tail of rate η, 0<η<1. It is further shown that the stationary waiting time distribution at arrivals has an exact exponential tail of decay parameter ξ>0. The quantities η and ξ may be evaluated together by an elementary algorithm. For both distributions, the multiplicative constants which arise in the asymptotic forms may be fully characterized. These constants are however difficult to compute in general.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Bellman, R.: Introduction to Matrix Analysis. New York: McGraw Hill 1960Google Scholar
  2. 2.
    Gantmacher, F.R.: The Theory of Matrices. New York: Chelsea 1959Google Scholar
  3. 3.
    Lavenberg, S.S.: Stability and Maximum Departure Rate of Certain Open Queueing Networks having Finite Capacity Constraints. R.A.I.R.O. Informatique/Computer Science12, 353–370 (1978)Google Scholar
  4. 4.
    Neuts, M.F.: Probability Distributions of Phase Type. Liber Amicorum Prof. Emeritus H. Florin. Dept. Math., Univ. Louvain, Belgium, 173–206 (1975)Google Scholar
  5. 5.
    Neuts, M.F.: Renewal Processes of Phase Type. Naval. Res. Logist. Quart.25, 445–454 (1978)Google Scholar
  6. 6.
    Neuts, M.F.: Markov Chains with Applications in Queueing Theory, which have a Matrix-geometric Invariant Probability Vector. Advances in Appl. Probability10, 185–212 (1978)Google Scholar
  7. 7.
    Neuts, M.F.: The Probabilistic Significance of the Rate Matrix in Matrix-geometric Invariant Vectors. J. Appl. Probability17, 291–296 (1980)Google Scholar
  8. 8.
    Neuts, M.F.: Matrix-geometric Solutions in Stochastic Models. An Algorithmic Approach. Baltimore, MD.: Johns Hopkins University Press 1981Google Scholar
  9. 9.
    Neuts, M.F.: Stationary Waiting Time Distributions in the GI/PH/1 Queue. J. Appl. Probability18 (1981)Google Scholar
  10. 10.
    Takahashi, Y., Takami, Y.: A Numerical Method for the Steady-state Probabilities of a GI/G/c Queueing System in a General Class. J. Operations Res. Soc. Japan19, 147–157 (1976)Google Scholar
  11. 11.
    Takahashi, Y.: Asymptotic Exponentiality of the Tail of the Waiting Time Distribution in a PH/PH/c Queue. Advances in Appl. Probability13 (1981)Google Scholar

Copyright information

© Springer-Verlag 1981

Authors and Affiliations

  • Marcel F. Neuts
    • 1
  • Yukio Takahashi
    • 2
  1. 1.Department of Mathematical SciencesUniversity of DelawareNewarkUSA
  2. 2.Department of EconomicsTohoku UniversityKawauchi-SendaiJapan

Personalised recommendations