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Stationary Analysis and Optimal Control Under Multiple Working Vacation Policy in a GI/M(a,b)/1 Queue

Abstract

This paper considers an infinite buffer renewal input queue with multiple working vacation policy wherein customers are served by a single server according to general bulk service (a, b)-rule (1 ≤ ab). If the number of waiting customers in the system at a service completion epoch (during a normal busy period) is lower than ‘a’, then the server starts a vacation. During a vacation if the number of waiting customers reaches the minimum threshold size ‘a’, then the server starts serving this batch with a lower rate than that of the normal busy period. After completion of a batch service during working vacation, if the server finds less than ‘a’ customers accumulated in the system, then the server takes another vacation, otherwise the server continues to serve the available batch with that lower rate. The maximum allowed size of a batch in service is ‘b’. The authors derive both queue-length and system-length distributions at pre-arrival epoch using both embedded Markov chain approach and the roots method. The arbitrary epoch probabilities are obtained using the classical argument based on renewal theory. Several performance measures like average queue and system-length, mean waiting-time, cost and profit optimization are studied and numerically computed.

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Acknowledgements

The authors would like to thank Dr. N K Jana, School of Mathematical Sciences, National Institute of Science Education and Research Bhubaneswar for his help to improve the language of presentation of the paper. We would also like to thank three anonymous referees for their insightful and constructive comments that helped us to upgrade the standard of the paper. Also, the authors would like to thank Dr. A K Ojha, School of Basic Sciences, Indian Institute of Technology Bhubaneswar for his thorough language editing before submitting the second revised version of this paper.

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Correspondence to Gopinath Panda.

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This paper was recommended for publication by Editor WANG Shouyang.

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Panda, G., Banik, A.D. & Guha, D. Stationary Analysis and Optimal Control Under Multiple Working Vacation Policy in a GI/M(a,b)/1 Queue. J Syst Sci Complex 31, 1003–1023 (2018). https://doi.org/10.1007/s11424-017-6172-y

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  • DOI: https://doi.org/10.1007/s11424-017-6172-y

Keywords

  • Bulk service
  • cost optimization
  • mean idle period
  • multiple working vacations
  • roots method
  • system-length