The M X /M/1 queue with working breakdown
RAIRO - Operations Research - Recherche Opérationnelle, Tome 48 (2014) no. 3, pp. 399-413.

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.

DOI : 10.1051/ro/2014014
Classification : 68M20, 60K20, 90B22
Mots-clés : MX/M/1 queue, working breakdown, probability generating function (PGF), Laplace-Stieltjes transform (LST), waiting time distribution, stochastic decomposition 
@article{RO_2014__48_3_399_0,
     author = {Liu, Zaiming and Song, Yang},
     title = {The $M^{X}/M/1$ queue with working breakdown},
     journal = {RAIRO - Operations Research - Recherche Op\'erationnelle},
     pages = {399--413},
     publisher = {EDP-Sciences},
     volume = {48},
     number = {3},
     year = {2014},
     doi = {10.1051/ro/2014014},
     zbl = {1297.90019},
     language = {en},
     url = {http://www.numdam.org/articles/10.1051/ro/2014014/}
}
TY  - JOUR
AU  - Liu, Zaiming
AU  - Song, Yang
TI  - The $M^{X}/M/1$ queue with working breakdown
JO  - RAIRO - Operations Research - Recherche Opérationnelle
PY  - 2014
SP  - 399
EP  - 413
VL  - 48
IS  - 3
PB  - EDP-Sciences
UR  - http://www.numdam.org/articles/10.1051/ro/2014014/
DO  - 10.1051/ro/2014014
LA  - en
ID  - RO_2014__48_3_399_0
ER  - 
%0 Journal Article
%A Liu, Zaiming
%A Song, Yang
%T The $M^{X}/M/1$ queue with working breakdown
%J RAIRO - Operations Research - Recherche Opérationnelle
%D 2014
%P 399-413
%V 48
%N 3
%I EDP-Sciences
%U http://www.numdam.org/articles/10.1051/ro/2014014/
%R 10.1051/ro/2014014
%G en
%F RO_2014__48_3_399_0
Liu, Zaiming; Song, Yang. The $M^{X}/M/1$ queue with working breakdown. RAIRO - Operations Research - Recherche Opérationnelle, Tome 48 (2014) no. 3, pp. 399-413. doi : 10.1051/ro/2014014. http://www.numdam.org/articles/10.1051/ro/2014014/

[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. | MR | Zbl

[2] Y. Baba, The MX/M/ 1 queue with multiple working vacation. Am. J. Oper. Res. 2 (2012) 217-224. | Zbl

[3] W.J. Gray, P.P. Wang and M. Scott, A queueing model with multiple types of server breakdowns. Quality Technology and Quantitative Management 1 (2004) 245-255. | MR

[4] G.I. Falin, The M/M/ 1 retrial queue with retrials due to server failures. Queueing Syst. 58 (2008) 155-160. | MR | Zbl

[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. | Zbl

[6] M. Jain and A. Jain, Working vacations queueing model with multiple types of server breakdowns. Appl. Math. Model. 34 (2010) 1-13. | MR | Zbl

[7] J.C. Ke and K.B. Huang, Analysis of batch arrival queue with randomized vacation policy and an unreliable server. J. Syst. Sci. Complexity 25 (2012) 759-777. | MR | Zbl

[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. | MR | Zbl

[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). | MR | Zbl

[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. | MR | Zbl

[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. | MR | Zbl

Cité par Sources :