Abstract
This paper analyzes a discrete-time multiple vacations finite-buffer queueing system with batch renewal input in which inter-arrival time of batches are arbitrarily distributed. Service and vacation times are mutually independent and geometrically distributed. The server takes vacations when the system does not have any waiting jobs at a service completion epoch or a vacation completion epoch. The system is analyzed under the assumptions of late arrival system with delayed access and early arrival system. Using the supplementary variable and the imbedded Markov chain techniques, the authors obtain the queue-length distributions at pre-arrival, arbitrary and outside observer’s observation epochs for partial-batch rejection policy. The blocking probability of the first-, an arbitraryand the last-job in a batch have been discussed. The analysis of actual waiting-time distributions measured in slots of the first-, an arbitrary- and the last-job in an accepted batch, and other performance measures along with some numerical results have also been investigated.
Similar content being viewed by others
References
H. Bruneel and B. G. Kim, Discrete-Time Models for Communication Systems Including ATM, Kluwer Academic Publishers, Massachusetts, 1993.
M. E. Woodward, Communication and Computer Networks: Modelling with Discrete-Time Queues, Los Alamitos, CA: California IEEE Computer Society Press, 1994.
H. Takagi, Queueing Analysis: A Foundation of Performance Evaluation. Vol. 3. Discrete-Time Systems, North-Holland Publishing, Amsterdam, 1993.
J. J. Hunter, Mathematical Techniques of Applied Probability, Vol. II, Discrete Time Models: Techniques and Applications, Academic Press, New York, 1983.
U. Chatterjee and S. P. Mukerjee, GI/M/1 queue with server vacations, Journal of the Operational Research Society, 1990, 41: 83–87.
F. Karaesmen and S. M. Gupta, The finite capacity GI/M/1 with server vacations, Journal of the Operational Research Society, 1996, 47: 817–828.
N. Tian, D. Zhang, and C. Cao, The GI/M/1 queue with exponential vacations, Queueing Systems, 1989, 5: 331–344.
N. Tian and Z. G. Zhang, Vacation Queueing Models: Theory and Applications, Springer-Verlag, New York, 2006.
N. Tian and Z. G. Zhang, The discrete-time GI/Geo/1 queue with multiple vacations, Queueing Systems, 2002, 40: 283–294.
S. K. Samanta, U. C. Gupta, and R. K. Sharma, Analysis of finite capacity discrete-time GI/Geo/1 queueing system with multiple vacations, Journal of the Operational Research Society, 2007, 58: 368–377.
W. Sun, N. Tian, and S. Li, Steady state analysis of the batch arrival GI/Geo/1 queue with multiple adaptive vacations, International Journal of Management Science and Engineering Management, 2007, 2: 83–97.
A. S. Alfa, Vacation models in discrete time, Queueing Systems, 2003, 44: 5–30.
P. Moreno, A discrete-time single-server queueing system under multiple vacations and setup-closedown times, Stochastic Analysis and Applications, 2009, 27: 221–239.
M. L. Chaudhry and U. C. Gupta, Performance analysis of the discrete-time GI/Geom/1/N queue, Journal of Applied Probability, 1996, 33: 239–255.
M. L. Chaudhry and U. C. Gupta, Performance analysis of discrete-time finite-buffer batch-arrival GIX/Geom/1/N Queues, Discrete Event Dynamic Systems, 1998, 8(1): 55–70.
K. Sikdar, U. C. Gupta, and R. K. Sharma, The analysis of finite-buffer general input queue with batch arrival and exponential multiple vacations, International Journal of Operational Research, 2008, 3: 219–234.
G. Latouche and V. Ramaswami, Introduction to Matrix Analytic Methods in Stochastic Modelling, ASA-SIAM Series on Statistics and Applied Probability, Society for Industrial and Applied Mathematics, Pennsylvania, 1999.
P. J. Burke, Delays in single-server queues with batch input, Bell Syst. Tech. J., 1975, 54: 830–833.
Author information
Authors and Affiliations
Corresponding author
Additional information
This paper was recommended for publication by Editor Hanqin ZHANG.
Rights and permissions
About this article
Cite this article
Goswami, V., Mund, G.B. Analysis of discrete-time queues with batch renewal input and multiple vacations. J Syst Sci Complex 25, 486–503 (2012). https://doi.org/10.1007/s11424-012-0057-x
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11424-012-0057-x