Abstract
This paper considers an infinite-buffer batch-service queue with Markovian arrival process, generally distributed and batch-size-dependent service time. We obtain a bivariate vector generating function of queue length and server content distribution at departure epoch of a batch. The complete joint distribution of queue length, server content and phase of the arrival process at departure epoch is extracted in terms of roots of the associated characteristic equation. By employing these probability vectors we also perceive the joint distribution at arbitrary and pre-arrival epochs. Our analytic procedure and results are demonstrated using some numerical examples for phase-type as well as deterministic service time distributions with high and low values of model parameters. The occurrence of multiple roots are also investigated in case of phase-type service time distribution. Finally, we also investigate the influence of correlation of the arrival process on the behavior of the system performance.
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Acknowledgements
The authors thank the anonymous referees for their valuable comments and suggestions that helped to improve the presentation of the paper. The second author acknowledges the Science and Engineering Research Board, New-Delhi, India, for the financial support under the Project Grant SR/S4/MS:789/12.
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Pradhan, S., Gupta, U.C. Analysis of an infinite-buffer batch-size-dependent service queue with Markovian arrival process. Ann Oper Res 277, 161–196 (2019). https://doi.org/10.1007/s10479-017-2476-5
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DOI: https://doi.org/10.1007/s10479-017-2476-5