# The MX/M/1 queue with working breakdown

RAIRO - Operations Research - Recherche Opérationnelle (2014)

- Volume: 48, Issue: 3, page 399-413
- ISSN: 0399-0559

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topLiu, Zaiming, and Song, Yang. "The MX/M/1 queue with working breakdown." RAIRO - Operations Research - Recherche Opérationnelle 48.3 (2014): 399-413. <http://eudml.org/doc/275051>.

@article{Liu2014,

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.},

author = {Liu, Zaiming, Song, Yang},

journal = {RAIRO - Operations Research - Recherche Opérationnelle},

keywords = {MX/M/1 queue; working breakdown; probability generating function (PGF); Laplace–Stieltjes transform (LST); waiting time distribution; stochastic decomposition; queue; Laplace-Stieltjes transform (LST)},

language = {eng},

number = {3},

pages = {399-413},

publisher = {EDP-Sciences},

title = {The MX/M/1 queue with working breakdown},

url = {http://eudml.org/doc/275051},

volume = {48},

year = {2014},

}

TY - JOUR

AU - Liu, Zaiming

AU - Song, Yang

TI - The MX/M/1 queue with working breakdown

JO - RAIRO - Operations Research - Recherche Opérationnelle

PY - 2014

PB - EDP-Sciences

VL - 48

IS - 3

SP - 399

EP - 413

AB - 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.

LA - eng

KW - MX/M/1 queue; working breakdown; probability generating function (PGF); Laplace–Stieltjes transform (LST); waiting time distribution; stochastic decomposition; queue; Laplace-Stieltjes transform (LST)

UR - http://eudml.org/doc/275051

ER -

## References

top- [1] I. Atencia and P. Moreno, A discrete-time Geo/G/ 1 retrial queue with server breakdowns. Asia-Pac. J. Oper. Res. 23 (2006) 247–271. Zbl1113.90038MR2239052
- [2] Y. Baba, The MX/M/ 1 queue with multiple working vacation. Am. J. Oper. Res.2 (2012) 217–224. Zbl1265.90092
- [3] W.J. Gray, P.P. Wang and M. Scott, A queueing model with multiple types of server breakdowns. Quality Technology and Quantitative Management1 (2004) 245–255. MR2163433
- [4] G.I. Falin, The M/M/ 1 retrial queue with retrials due to server failures. Queueing Syst.58 (2008) 155–160. Zbl1140.60344MR2403098
- [5] D. Jayaraman, R. Nadarajan and R.M. Sitrarasu, A general bulk service queue with arrival rate dependent on server breakdowns. Appl. Math. Model.18 (1994) 156–160. Zbl0797.60085
- [6] M. Jain and A. Jain, Working vacations queueing model with multiple types of server breakdowns. Appl. Math. Model.34 (2010) 1–13. Zbl1185.90048MR2566676
- [7] J.C. Ke and K.B. Huang, Analysis of batch arrival queue with randomized vacation policy and an unreliable server. J. Syst. Sci. Complexity25 (2012) 759–777. Zbl1292.93118MR2957667
- [8] K. Kalidass and R. Kasturi, A queue with working breakdowns. Comput. Indus. Eng.63 (2012) 779–783.
- [9] W.Y. Liu, X.L. Xu and N.S. Tian, Stochastic decompositions in the M/M/ 1 queue with working vacations. Oper. Res. Lett.35 (2007) 596–600. Zbl1129.60085MR2348802
- [10] L.D. Servi and S.G. Finn, M/M/ 1 queues with working vacations (M/M/ 1 /WV). Perform. Eval.50 (2002) 41–52.
- [11] V. Sridharan and P.J. Jayashree, Some characteristics on a finite queue with normal, partial and total failures. Microelectron. Reliab.36 (1996) 265–267.
- [12] R.W. Wolff, Stochastic modeling and the theory of queues. Prentice Hall, Inc., New Jersey (1989). Zbl0701.60083MR1022666
- [13] J.T. Wang and P. Zhang, A discrete-time retrial queue with negative customers and unreliable server. Comput. Indus. Eng.56 (2009) 1216–1222.
- [14] J.T. Wang and J.H. Cao, Reliability analysis of the retrial queue with server breakdowns and repairs. Queueing Syst.38 (2001) 363–380. Zbl1028.90014MR1856543
- [15] X.L. Xu and Z.J. Zhang, Analysis of multi-server queue with a single vacation (e,d)-Policy. Perform. Eval.63 (2006) 625–638.
- [16] X.L. Xu, Z.J. Zhang and N.S. Tian, Analysis for the MX/M/ 1 working vacation queue. Int. J. Inf. Manage. Sci.20 (2009) 379–394. Zbl1184.90052MR2597342

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