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Optimal bulking threshold of batch service queues

Part of: Markov processes Special processes

Published online by Cambridge University Press:  22 June 2017

Yun Zeng  and
Cathy Honghui Xia
Yun Zeng*
Affiliation:
The Ohio State University
Cathy Honghui Xia*
Affiliation:
The Ohio State University
*
* Postal address: Department of Integrated Systems Engineering, The Ohio State University, 1971 Neil Avenue, Columbus, OH 43210, USA.
* Postal address: Department of Integrated Systems Engineering, The Ohio State University, 1971 Neil Avenue, Columbus, OH 43210, USA.
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Abstract

Batch service has a wide application in manufacturing, communication networks, and cloud computing. In batch service queues with limited resources, one critical issue is to properly schedule the service so as to ensure the quality of service. In this paper we consider an M/G[a,b]/1/N batch service queue with bulking threshold a, max service capacity b, and buffer capacity N, where N can be finite or infinite. Through renewal theory, busy period analysis and decomposition techniques, we demonstrate explicitly how the bulking threshold influences the system performance such as the mean waiting time and time-averaged number of loss customers in batch service queues. We then establish a necessary and sufficient condition on the optimal bulking threshold that minimizes the expected waiting time. Enabled by this condition, we propose a simple algorithm which guarantees to find the optimal threshold in polynomial time. The performance of the algorithm is also demonstrated by numerical examples.

Keywords

Batch serviceoptimal bulking thresholdqueueingPoisson arrival

MSC classification

Primary: 60K25: Queueing theory
Secondary: 60J10: Markov chains (discrete-time Markov processes on discrete state spaces) 60K05: Renewal theory
Type
Research Papers
Information
Journal of Applied Probability , Volume 54 , Issue 2 , June 2017 , pp. 409 - 423
Copyright
Copyright © Applied Probability Trust 2017 

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References

[1] Arumuganathan, R. and Jeyakumar, S. (2005). Steady state analysis of a bulk queue with multiple vacations, setup times with N-policy and closedown times. Appl. Math. Modelling 29, 972–986. CrossRefGoogle Scholar
[2] Banerjee, A., Chakravarthy, S. R. and Gupta, U. C. (2015). Analysis of a finite-buffer bulk-service queue under Markovian arrival process with batch-size-dependent service. Comput. Operat. Res. 60, 138–149. CrossRefGoogle Scholar
[3] Banik, A. D. (2009). Queueing analysis and optimal control of BMAP/G (a,b) /1/N and BMAP/MSP (a,b) /1/N systems. Comput. Indust. Eng. 57, 748–761. CrossRefGoogle Scholar
[4] Banik, A. D. (2015). Single server queues with a batch Markovian arrival process and bulk renewal or non-renewal service. J. Systems Sci. Systems Eng. 24, 337–363. CrossRefGoogle Scholar
[5] Bar-Lev, S. K. et al. (2007). Applications of bulk queues to group testing models with incomplete identification. Europ. J. Operat. Res. 183, 226–237. CrossRefGoogle Scholar
[6] Chakravarthy, S. (1993). Analysis of a finite MAP/G/1 queue with group services. Queueing Systems 13, 385–407. CrossRefGoogle Scholar
[7] Chaudhry, M. L. and Gupta, U. C. (1999). Modelling and analysis of M/G a,b /1/N queue—a simple alternative approach. Queueing Systems 31, 95–100. CrossRefGoogle Scholar
[8] Chaudhry, M. L. and Templeton, J. G. C. (1983). A First Course on Bulk Queues. John Wiley, New York. Google Scholar
[9] Deb, R. K. and Serfozo, R. F. (1973). Optimal control of batch service queues. Adv. Appl. Prob. 5, 340–361. CrossRefGoogle Scholar
[10] Gold, H. and Tran-Gia, P. (1993). Performance analysis of a batch service queue arising out of manufacturing system modelling. Queueing Systems 14, 413–426. CrossRefGoogle Scholar
[11] Goswami, V., Patra, S. S. and Mund, G. B. (2012). Performance analysis of cloud computing centers for bulk services. Internat. J. Cloud Appl. Comput. 2, 53–65. Google Scholar
[12] Gupta, U. C. and Banerjee, A. (2011). New results on bulk service queue with finite-buffer: M/G(a,b) /1/N . Opsearch 48, 279–296. CrossRefGoogle Scholar
[13] Meyer, C. D. (1989). Stochastic complementation, uncoupling Markov chains, and the theory of nearly reducible systems. SIAM Rev 31, 240–272. CrossRefGoogle Scholar
[14] Neuts, M. F. (1967). A general class of bulk queues with Poisson input. Ann. Math. Statist. 38, 759–770. CrossRefGoogle Scholar
[15] Sikdar, K. and Gupta, U. C. (2005). Analytic and numerical aspects of batch service queues with single vacation. Comput. Operat. Res. 32, 943–966. CrossRefGoogle Scholar
[16] Tadj, L. and Choudhury, G. (2005). Optimal design and control of queues. Top 13, 359–412. CrossRefGoogle Scholar
[17] Tadj, L. and Ke, J.-C. (2003). Control policy of a hysteretic queueing system. Math. Meth. Operat. Res. 57, 367–376. CrossRefGoogle Scholar
[18] Tadj, L. and Ke, J.-C. (2005). Control policy of a hysteretic bulk queueing system. Math. Comput. Modelling 41, 571–579. CrossRefGoogle Scholar
[19] Tadj, L. and Tadj, C. (2003). On an M/D r 1 queueing system. J. Statist. Theory Appl. 2, 17–32. Google Scholar