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Analyzing discrete-time bulk-service Geo/Geob/m queue

Published online by Cambridge University Press:  08 November 2006

Veena Goswami
Department of Computer Science, Kalinga Institute of Industrial Technology, Bhubaneswar-751024, India;
Umesh C. Gupta
Department of Mathematics, Indian Institute of Technology, Kharagpur-721302, India;;
Sujit K. Samanta
Department of Mathematics, Indian Institute of Technology, Kharagpur-721302, India;;
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This paper analyzes a discrete-time multi-server queue in which service capacity of each server is a minimum of one and a maximum of b customers. The interarrival- and service-times are assumed to be independent and geometrically distributed. The queue is analyzed under the assumptions of early arrival system and late arrival system with delayed access. Besides, obtaining state probabilities at arbitrary and outside observer's observation epochs, some performance measures and waiting-time distribution in the queue have also been discussed. Finally, it is shown that in limiting case the results obtained in this paper tend to the continuous-time counterpart.

Research Article
© EDP Sciences, 2006

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Artalejo, J.R. and Hern, O.ández-Lerma, Performance analysis and optimal control of the Geo/Geo/c queue. Perform. Evaluation 1013 (2002) 125.
H. Bruneel and B.G. Kim, Discrete-Time Models for Communication Systems Including ATM. Kluwer Academic Publishers, Boston (1983).
W.C. Chan and D.Y. Maa, The GI/Geom/N queue in discrete-time. INFOR 16 (3) (1978) 232–252.
Chaudhry, M.L. and Chang, S.H., Analysis of the discrete-time bulk-service queue Geo/GY/1/N + B. Oper. Res. Lett. 32 (2004) 355363. CrossRef
M.L. Chaudhry and U.C. Gupta, Transient behaviour of the discrete-time Geom/Geom/m/m Erlang loss model, in Proc. of Probability Models and Statistics, edited by A.C. Borthakur and H. Choudhury. A J. Medhi Festschrift, New age international limited, publishers, New Delhi (1996) 133–145.
Chaudhry, M.L. and Gupta, U.C., Algorithmic discussions of distributions of numbers of busy channels for GI/Geom/m/m queues. INFOR. 38 (2000) 5163.
M.L. Chaudhry and U.C. Gupta, Numerical evaluation of state probabilities at different epochs in multiserver GI/Geom/m queue, in Proc. of Advances on Methodological and Applied Aspects of Probability and Statistics, edited by N. Balakrishnan. Gordon and Breach Science Publishers (2001) 31–46.
Chaudhry, M.L. and Kim, N.M., A complete and simple solution for a discrete-time multi-server queue with bulk arrivals and deterministic service times. Oper. Res. Lett. 31 (2003) 101107. CrossRef
M.L. Chaudhry, U.C. Gupta and V. Goswami, Modelling and analysis of discrete-time multiserver queues with batch arrivals: GIX/Geom/m. Inform. J. Comput. 13 (3) (2001) 172–180.
M.L. Chaudhry, U.C. Gupta and V. Goswami, On discrete-time multiserver queue with finite buffer: GI/Geom/m/N. Comput. Oper. Res. 31 (2004) 2137–2150. CrossRef
P. Gao, S. Wittevrongel and H. Bruneel, Discrete-time multiserver queues with geometric service times. Comput. Oper. Res. 31 (2004) 81–99.
Gravey, A. and G. Hébuterne, Simultaneity in discrete time single server queues with Bernoulli inputs. Perform. Evaluation 14 (1992) 123131. CrossRef
U.C. Gupta and V. Goswami, Performance analysis of finite buffer discrete-time queue with bulk service. Comput. Oper. Res. 29 (2002) 1331–1341.
Gupta, U.C., Samanta, S.K. and Sharma, R.K., Computing queueing length and waiting time distributions in finite-buffer discrete-time multi-server queues with late and early arrivals. Comput. Math. Appl. 48 (2004) 15571573. CrossRef
J.J. Hunter, Mathematical Techniques of Applied Probability, Vol-II, Discrete Time Models: Techniques and Applications. New York, Academic Press (1983).
J. Medhi, Stochastic Models in Queueing Theory. Academic Press, Inc. (1991).
Rubin, I. and Zhang, Z., Message delay and queue size analysis for circuit-switched TDMA systems. IEEE Trans. Comm. 39 (1991) 905913. CrossRef
R.M. Spiegel, Schaum's outline of theory and problems of calculus of finite differences and difference equations. Mcgraw Hill Inc. (1971).
S. Wittevrongel, H. Bruneel and B. Vinck, Analysis of the Discrete-Time G(G)/Geom/c Queueing Model, in Proc. of Networking 2002-Lecture Notes in Computer Science 2345, edited by E. Gregori, M. Conti, A.T. Campbell, G. Omidyar and M. Zukerman. Pisa, Italy (2002) 757–768.
M.E. Woodward, Communication and Computer Networks: Modelling with Discrete-Time Queues. Los Alamitos, CA: California IEEE Computer Society Press (1994).