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Queueing processes in bulk systems under the D-policy

Part of: Special processes

Published online by Cambridge University Press:  14 July 2016

Jewgeni H. Dshalalow
Jewgeni H. Dshalalow*
Affiliation:
Florida Institute of Technology
*
∗Postal address: Department of Applied Mathematics, Florida Institute of Technology, Melbourne, FL 32901, USA. Email address: eugene@winnie.FIT.edu
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Abstract

This paper studies the queueing process in a class of D-policy models with Poisson bulk input, general service time, and four different vacation scenarios, among them a multiple vacation, single vacation and idle server. The D-policy specifies a busy period discipline, which requires an idle or vacationing server to resume his service when the workload process crosses some fixed level D. The analysis of the queueing process is based on the theory of fluctuations for three-dimensional marked counting processes presented in the paper. For all models, we derive the stationary distributions for the embedded and continuous time parameter queueing processes in closed analytic forms and illustrate the results by a number of examples and applications.

Keywords

Queueingqueueing processvacationsD-policyKendall's formulafluctuation theoryfirst excess level theoryfirst passage timesemi-regenerative techniquesembedded Markov chain

MSC classification

Primary: 60K10: Applications (reliability, demand theory, etc.) 60K15: Markov renewal processes, semi-Markov processes 60K25: Queueing theory
Type
Research Papers
Information
Journal of Applied Probability , Volume 35 , Issue 4 , December 1998 , pp. 976 - 989
Copyright
Copyright © Applied Probability Trust 1998 

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References

Abolnikov, L., and Dukhovny, A. (1991). Markov chains with transition delta-matrix: ergodicity conditions, invariant probability measures and applications. J. Appl. Math. Stoch. Anal. 4, 335–355.Google Scholar
Balachandran, K. R. (1971). Queue length dependent priority queues. Management Sci. 17, 463–471.Google Scholar
Balachandran, K. R. (1973). Control policies for a single server system. Management Sci. 19, 1013–1018.Google Scholar
Balachandran, K. R., and Tijms, H. (1975). On the D-policy for the M/G/1 queue. Management Sci. 21, 1073–1076.Google Scholar
Boxma, O. J. (1976). Note on a control problem of Balachandran and Tijms. Management Sci. 22, 916–917.Google Scholar
Çinlar, E. (1975). Introduction to Stochastic Processes. Prentice-Hall, Englewood Cliffs, NJ.Google Scholar
Dshalalow, J. H. (1995). Excess level processes in queueing. In Advances in Queueing. Ed. Dshalalow, J. H., CRC Press, Boca Raton, FL, pp. 243–262.Google Scholar
Dshalalow, J. H. (1997). Queueing systems with state dependent parameters. In Frontiers in Queueing. Ed. Dshalalow, J. D. CRC Press, Boca Raton, FL, pp. 61–116.Google Scholar
Li, J., and Niu, S-C. (1992). The waiting time distribution for the GI/G/1 queue under D-policy. Prob. Eng. Inf. Sci. 6, 287–308.Google Scholar
Rubin, I., and Zhang, Z. (1988). Switch-on policies for communications and queueing systems. In Proc. Third International Conference on Data Communication, Elsevier North-Holland, Amsterdam, pp. 329–339.Google Scholar
Teghem, J. Jr. (1985). Optimal control of queues: removable servers [Tutorial paper XIX]. Belgian J. Oper. Res. Stat. Comp. Sci. 25, 99–128.Google Scholar