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Queueing Systems

, Volume 42, Issue 3, pp 269–296 | Cite as

Analysis of an Infinite-Server Queue with Batch Markovian Arrival Streams

  • H. Masuyama
  • T. Takine
Article

Abstract

This paper considers an infinite-server queue with multiple batch Markovian arrival streams. The service time distribution of customers may be different for different arrival streams, and simultaneous batch arrivals from more than one stream are allowed. For this queue, we first derive a system of ordinary differential equations for the time-dependent matrix joint generating function of the number of customers in the system. Next assuming phase-type service times, we derive explicit and numerically feasible formulas for the time-dependent and limiting joint binomial moments. Further, some numerical examples are provided to discuss the impact of system parameters on the performance.

infinite-server queue Markovian arrival stream batch arrivals time-dependent analysis limiting distribution 

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References

  1. [1]
    S. Asmussen and G. Koole, Marked point processes as limits Markovian arrival streams, J. Appl. Probab. 30 (1993) 365-372.Google Scholar
  2. [2]
    R. Bellman, Introduction to Matrix Analysis, 2nd ed. (SIAM, Philadelphia, PA, 1997).Google Scholar
  3. [3]
    S.G. Eick, W.A. Massey and W. Whitt, M t /G/? queues with sinusoidal arrival rates, Managm. Sci. 39(2) (1993) 241-252.Google Scholar
  4. [4]
    D.F. Holman, M.L. Chaudhry and B.R.K. Kashyap, On the number in the system GI X /M/?, Sankhy? Ser. A 44 Pt. 1 (1982) 294-297.Google Scholar
  5. [5]
    D.F. Holman, M.L. Chaudhry and B.R.K. Kashyap, On the service system M X /G/?, European J. Oper. Res. 13 (1983) 142-145.Google Scholar
  6. [6]
    L. Liu, B.R.K. Kashyap and J.G.C. Templeton, On the GI X /G/? system, J. Appl. Probab. 27 (1990) 671-683.Google Scholar
  7. [7]
    L. Liu and J.G.C. Templeton, The GR Xn /G n /? system: System size, Queueing Systems 8 (1991) 323-356.Google Scholar
  8. [8]
    D.M. Lucantoni, New results on the single server queue with a batch Markovian arrival process, Stochastic Models 7 (1991) 1-46.Google Scholar
  9. [9]
    V. Ramaswami, The N/G/? queue, Technical Report, Department of Mathematics, Drexel University, Philadelphia, PA (1978).Google Scholar
  10. [10]
    V. Ramaswami and M.F. Neuts, Some explicit formulas and computational methods for infinite-server queues with phase-type arrivals, J. Appl. Probab. 17 (1980) 498-514.Google Scholar
  11. [11]
    D.N. Shanbhag, On infinite server queues with batch arrivals, J. Appl. Probab. 3 (1966) 274-279.Google Scholar
  12. [12]
    H.C. Tijms, Stochastic Model: An Algorithmic Approach (Wiley, Chichester, 1994).Google Scholar

Copyright information

© Kluwer Academic Publishers 2002

Authors and Affiliations

  • H. Masuyama
    • 1
  • T. Takine
    • 1
  1. 1.Department of Applied Mathematics and Physics, Graduate School of InformaticsKyoto UniversityKyotoJapan

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