Skip to main content
SpringerLink
Log in

Single server queues with a batch Markovian arrival process and bulk renewal or non-renewal service

  • Published:
Journal of Systems Science and Systems Engineering Aims and scope Submit manuscript

Abstract

We first consider an infinite-buffer single server queue where arrivals occur according to a batch Markovian arrival process(BMAP). The server serves customers in batches of maximum size ‘b’ with a minimum threshold size ‘a’. The service time of each batch follows general distribution independent of each other as well as the arrival process. The proposed analysis is based on the use of matrix-analytic procedure to obtain queue-length distribution at a post-departure epoch. Next we obtain queue-length distributions at various other epochs such as, pre-arrival, arbitrary and pre-service using relations with post-departure epoch. Later we also obtain the system-length distributions at post-departure and arbitrary epochs using queue-length distribution at post-departure epoch. Some important performance measures, like mean queue-lengths and mean waiting times have been obtained. Total expected cost function per unit time is also derived to determine the locally optimal values of a and b. Secondly, we perform similar analysis for the corresponding infinite-buffer single server queue where arrivals occur according to a BMAP and service process in this case follows a non-renewal one, namely, Markovian service process (MSP).

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Albores-Velasco, F. J. & Tajonar-Sanabria F. S. (2004). Anlysis of the GI/MSP/c/r queueing system. Information Processes, 4: 46–57.

    Google Scholar 

  2. Banik, A. D., Chaudhry M. L. & Gupta U. C. (2008). On the finite buffer queue with renewal input and batch markovian service process: GI/BMSP/1/N. Methodology and Computing in Applied Probability, 10: 559–575.

    Article  MATH  MathSciNet  Google Scholar 

  3. Banik, A. D., Gupta U. C. & Chaudhry M. L. (2009). Finite-buffer bulk service queue under markovian service process: GI/MSP (a,b)/1/N. Stochastic Analysis and Applications, 27: 500–522.

    Article  MATH  MathSciNet  Google Scholar 

  4. Banik, A. D. (2009). The infinite-buffer single server queue with a variant of multiple vacation policy and batch Markovian arrival process. Applied Mathematical Modelling, 33: 3025–3039.

    Article  MATH  MathSciNet  Google Scholar 

  5. Banik, A. D. (2009). Queueing analysis and optimal control of BMAP/G (a,b)/1/N and BMAP/MSP (a,b)/1/N systems. Computers & Industrial Engineering, 57: 748–761.

    Article  Google Scholar 

  6. Banik, A. D. (2011). Analyzing state-dependent arrival in GI/BMSP/1/∞ queues. Mathematical and Computer Modelling, 53: 1229–1246.

    Article  MATH  MathSciNet  Google Scholar 

  7. Bocharov, P. P. (1996). Stationary distribution of a finite queue with recurrent flow and Markovian service. Automation and Remote Control, 57: 66–78.

    MathSciNet  Google Scholar 

  8. Cha, M. S., Koo, P.-H. & Joe, J. (2012). Real-time control strategies of batch processing machines in semiconductor manufacturing. Proceedings of the Asia Pacific Industrial Engineering & Management Systems Conference, Patong Beach, Phuket, Thailand.

    Google Scholar 

  9. Chaudhry, M. L. & Templeton J. G. C. (1983). A First Course in Bulk Queues. New York: Wiley.

    MATH  Google Scholar 

  10. Chaudhry, M. L. & Gupta U. C. (1999). Modelling and analysis of M/G (a,b)/1/N queue — a simple alternative approach. Queueing Systems, 31: 95–100.

    Article  MATH  MathSciNet  Google Scholar 

  11. Chakravarthy, S. (1993). Analysis of a finite MAP/G/1 queue with group services. Queueing Systems, 13: 385–407.

    Article  MATH  MathSciNet  Google Scholar 

  12. Dshalalow, J. H. (1997). Frontiers in Queueing: Models and Applications in Science and Engineering. Boca Raton, FL.: CRC Press.

    MATH  Google Scholar 

  13. Goswami, V., Patra, S. S. & Mund, G. B. (2012). Performance analysis of cloud computing centers for bulk services. International Journal of Cloud Applications and Computing, 2(4): 53–65.

    Article  Google Scholar 

  14. Grassmann, W. K., Taksar M. I. & D. P. Heyman (1985). Regenerative analysis and steady state distributions for Markov chains. Operations Research, 33: 1107–1116.

    Article  MATH  MathSciNet  Google Scholar 

  15. Gupta, U. C. & Banik A. D. (2007). Complete analysis of finite and infinite buffer GI/MSP/1 queue — a computational approach. Operations Research Letters, 35: 273–280.

    Article  MATH  MathSciNet  Google Scholar 

  16. Harris, C. M. & Marchal, W. G. (1998). Distribution estimation using Laplace transforms. INFORMS journal on Computing, 10: 448–458.

    Article  MathSciNet  Google Scholar 

  17. Lee, H. S. & Srinivasan, M. M. (1989). Control policies for the M [X]/G/1 queueing system. Management Science, 35: 708–721.

    Article  MATH  MathSciNet  Google Scholar 

  18. Lee, H. W. & Park, J. O. (1997). Optimal strategy in N-policy production system with early set-up. Journal Operational Research Society, 48: 306–313.

    Article  MATH  Google Scholar 

  19. Lucantoni, D. M. (1991). New results on the single server queue with a batch Markovian arrival process. Stochactic Models, 7: 1–46.

    MATH  MathSciNet  Google Scholar 

  20. Lucantoni, D. M. & Neuts, M. F. (1994). Some steady-state distributions for the MAP/SM/1 queue. Stochastic Models, 10: 575–598.

    Article  MATH  MathSciNet  Google Scholar 

  21. Lucantoni, D. M., Meier-Hellstern, K. S. & Neuts, M. F. (1990). A single-server queue with server vacations and a class of non-renewal arrival process. Advances in Applied Probability, 22: 676–705.

    Article  MATH  MathSciNet  Google Scholar 

  22. Neuts, M. F. (1979). A versatile Markovian point process. Journal of Applied Probability, 16: 764–779.

    Article  MATH  MathSciNet  Google Scholar 

  23. Neuts, M. F. (1981). Matrix-Geometric Solutions in Stochastic Models: An Algorithmic Approach. Baltimore, MD: Johns Hopkins Univ. Press.

    MATH  Google Scholar 

  24. Neuts, M. F. (1989). Structured stochastic matrices of M/G/1 type and their applications. Marcel Dekker, New York.

    MATH  Google Scholar 

  25. Neuts, M. F. (1992). Models based on the Markovian arrival process. IEICE Transactions on Communication, E75-B: 1255–1265.

    Google Scholar 

  26. Ramaswami, V. (1988). A stable recursion for the steady state vector in Markov chain M/G/1 type. Stochastic Models, 4: 183–188.

    Article  MATH  MathSciNet  Google Scholar 

  27. Shanthikumar, J. G., Ding, S. & Zhang, M. T. (2007). Queueing theory for semiconductor manufacturing systems: a survey and open problems. IEEE Transactions on Automation Science and Engineering, 4: 513–522.

    Article  Google Scholar 

  28. Sikdar, K. & Gupta, U. C. (2005). The queue length distributions in the finite buffer bulk-service MAP/G/1 queue with multiple vacations. TOP, 13(1): 75–103.

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. School of Basic Sciences, Indian Institute of Technology, Samantapuri, Nandan Kanan Road, Bhubaneswar, 751 013, India

    A. D. Banik

Authors
  1. A. D. Banik
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to A. D. Banik.

Additional information

A. D. Banik is presently an assistant professor in the Indian Institute of Technology, Bhubaneswar, India. He received the B.Sc. and M.Sc. degrees in mathematics from the Jadavpur University, West Bengal, India in the year 1998 and 2000, respectively. He received his Ph.D. from the Department of Mathematics, Indian Institute of Technology, Kharagpur, India in the year 2007. He visited a few places as research associate and earned research fellowships, viz., INRIA, France one year post-doc fellowship at IRISA, Rennes, France, two year visiting fellowships at TIFR, Mumbai, India, one year FCT, Portugal post-doc fellowship at IST, Lisbon, Portugal etc. His main research interests include continuous-time queueing theory and its applications. He has published research articles in Operations Research Letters, Applied Mathematics and Computation, Journal of Applied Mathematics and Stochastic Analysis, Applied Mathematical Modelling, Methodology and Computing in Applied Probability, Stochastic Analysis and Applications etc. He is a life member of the operational research society of India (ORSI).

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Banik, A.D. Single server queues with a batch Markovian arrival process and bulk renewal or non-renewal service. J. Syst. Sci. Syst. Eng. 24, 337–363 (2015). https://doi.org/10.1007/s11518-015-5268-y

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11518-015-5268-y

Keywords

Search

Navigation