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The MX/M/1 queue with working breakdown

Published online by Cambridge University Press:  30 April 2014

Zaiming Liu  and
Yang Song
Yang Song
Affiliation:
School of Mathematics and Statistics, Central South University, Changsha 410075, P.R. China. yy0605xx@126.com
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Abstract

In this paper, we consider a batch arrival MX/M/1 queue model with working breakdown. The server may be subject to a service breakdown when it is busy, rather than completely stoping service, it will decrease its service rate. For this model, we analyze a two-dimensional Markov chain and give its quasi upper triangle transition probability matrix. Under the system stability condition, we derive the probability generating function (PGF) of the stationary queue length, and then obtain its stochastic decomposition, which shows the relationship with that of the classical MX/M/ 1 queue without vacation or breakdown. Besides, we also give the Laplace-Stieltjes transform (LST) of the stationary waiting time distribution of an arbitrary customer in a batch. Finally, some numerical examples are given to illustrate the effect of the parameters on the system performance measures.

Keywords

MX/M/1 queueworking breakdownprobability generating function (PGF)Laplace–Stieltjes transform (LST)waiting time distributionstochastic decomposition
Type
Research Article
Information
RAIRO - Operations Research , Volume 48 , Issue 3 , July 2014 , pp. 399 - 413
Copyright
© EDP Sciences, ROADEF, SMAI, 2014

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References

Atencia, I. and Moreno, P., A discrete-time Geo/G/ 1 retrial queue with server breakdowns. Asia-Pac. J. Oper. Res. 23 (2006) 247–271. Google Scholar
Baba, Y., The M X/M/ 1 queue with multiple working vacation. Am. J. Oper. Res. 2 (2012) 217–224. Google Scholar
Gray, W.J., Wang, P.P. and Scott, M., A queueing model with multiple types of server breakdowns. Quality Technology and Quantitative Management 1 (2004) 245–255. Google Scholar
Falin, G.I., The M/M/ 1 retrial queue with retrials due to server failures. Queueing Syst. 58 (2008) 155–160. Google Scholar
Jayaraman, D., Nadarajan, R. and Sitrarasu, R.M., A general bulk service queue with arrival rate dependent on server breakdowns. Appl. Math. Model. 18 (1994) 156–160. Google Scholar
Jain, M. and Jain, A., Working vacations queueing model with multiple types of server breakdowns. Appl. Math. Model. 34 (2010) 1–13. Google Scholar
Ke, J.C. and Huang, K.B., Analysis of batch arrival queue with randomized vacation policy and an unreliable server. J. Syst. Sci. Complexity 25 (2012) 759–777. Google Scholar
Kalidass, K. and Kasturi, R., A queue with working breakdowns. Comput. Indus. Eng. 63 (2012) 779–783. Google Scholar
Liu, W.Y., Xu, X.L. and Tian, N.S., Stochastic decompositions in the M/M/ 1 queue with working vacations. Oper. Res. Lett. 35 (2007) 596–600. Google Scholar
Servi, L.D. and Finn, S.G., M/M/ 1 queues with working vacations (M/M/ 1 / WV). Perform. Eval. 50 (2002) 41–52. Google Scholar
Sridharan, V. and Jayashree, P.J., Some characteristics on a finite queue with normal, partial and total failures. Microelectron. Reliab. 36 (1996) 265–267. Google Scholar
R.W. Wolff, Stochastic modeling and the theory of queues. Prentice Hall, Inc., New Jersey (1989).
Wang, J.T. and Zhang, P., A discrete-time retrial queue with negative customers and unreliable server. Comput. Indus. Eng. 56 (2009) 1216–1222. Google Scholar
Wang, J.T. and Cao, J.H., Reliability analysis of the retrial queue with server breakdowns and repairs. Queueing Syst. 38 (2001) 363–380. Google Scholar
Xu, X.L. and Zhang, Z.J., Analysis of multi-server queue with a single vacation (e,d)-Policy. Perform. Eval. 63 (2006) 625–638. Google Scholar
Xu, X.L., Zhang, Z.J. and Tian, N.S., Analysis for the M X/M/ 1 working vacation queue. Int. J. Inf. Manage. Sci. 20 (2009) 379–394. Google Scholar