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.
Similar content being viewed by others
References
S. Asmussen and G. Koole, Marked point processes as limits Markovian arrival streams, J. Appl. Probab. 30 (1993) 365-372.
R. Bellman, Introduction to Matrix Analysis, 2nd ed. (SIAM, Philadelphia, PA, 1997).
S.G. Eick, W.A. Massey and W. Whitt, M t /G/? queues with sinusoidal arrival rates, Managm. Sci. 39(2) (1993) 241-252.
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.
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.
L. Liu, B.R.K. Kashyap and J.G.C. Templeton, On the GI X /G/? system, J. Appl. Probab. 27 (1990) 671-683.
L. Liu and J.G.C. Templeton, The GR Xn /G n /? system: System size, Queueing Systems 8 (1991) 323-356.
D.M. Lucantoni, New results on the single server queue with a batch Markovian arrival process, Stochastic Models 7 (1991) 1-46.
V. Ramaswami, The N/G/? queue, Technical Report, Department of Mathematics, Drexel University, Philadelphia, PA (1978).
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.
D.N. Shanbhag, On infinite server queues with batch arrivals, J. Appl. Probab. 3 (1966) 274-279.
H.C. Tijms, Stochastic Model: An Algorithmic Approach (Wiley, Chichester, 1994).
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Masuyama, H., Takine, T. Analysis of an Infinite-Server Queue with Batch Markovian Arrival Streams. Queueing Systems 42, 269–296 (2002). https://doi.org/10.1023/A:1020575915095
Issue Date:
DOI: https://doi.org/10.1023/A:1020575915095