Analysis of a queueing system in random environment with an unreliable server and geometric abandonments
RAIRO - Operations Research - Recherche Opérationnelle, Tome 52 (2018) no. 3, pp. 903-922.

This paper studies a single server queueing model in a multi-phase random environment with server breakdowns and geometric abandonments, where server breakdowns only occur while the server is in operation. At a server breakdown instant (i.e., an abandonment opportunity epoch), all present customers adopt the so-called geometric abandonments, that is, the customers decide sequentially whether they will leave the system or not. In the meantime, the server abandons the service and a repair process starts immediately. After the server is repaired, the server resumes its service, and the system enters into the operative phase i with probability q i , i = 1 , 2 , ... , d . Using probability generating functions and matrix geometric approach, we obtain the steady state distribution and various performance measures. In addition, some numerical examples are presented to show the impact of parameters on the performance measures.

DOI : 10.1051/ro/2018021
Classification : 68M20, 60K20, 90B22
Mots-clés : Server breakdown, geometric abandonments, sojourn time, cycle analysis
Jiang, Tao 1 ; Xin, Baogui 1 ; Chang, Baoxian 1 ; Liu, Liwei 1

1 
@article{RO_2018__52_3_903_0,
     author = {Jiang, Tao and Xin, Baogui and Chang, Baoxian and Liu, Liwei},
     title = {Analysis of a queueing system in random environment with an unreliable server and geometric abandonments},
     journal = {RAIRO - Operations Research - Recherche Op\'erationnelle},
     pages = {903--922},
     publisher = {EDP-Sciences},
     volume = {52},
     number = {3},
     year = {2018},
     doi = {10.1051/ro/2018021},
     zbl = {1408.90073},
     mrnumber = {3868451},
     language = {en},
     url = {http://www.numdam.org/articles/10.1051/ro/2018021/}
}
TY  - JOUR
AU  - Jiang, Tao
AU  - Xin, Baogui
AU  - Chang, Baoxian
AU  - Liu, Liwei
TI  - Analysis of a queueing system in random environment with an unreliable server and geometric abandonments
JO  - RAIRO - Operations Research - Recherche Opérationnelle
PY  - 2018
SP  - 903
EP  - 922
VL  - 52
IS  - 3
PB  - EDP-Sciences
UR  - http://www.numdam.org/articles/10.1051/ro/2018021/
DO  - 10.1051/ro/2018021
LA  - en
ID  - RO_2018__52_3_903_0
ER  - 
%0 Journal Article
%A Jiang, Tao
%A Xin, Baogui
%A Chang, Baoxian
%A Liu, Liwei
%T Analysis of a queueing system in random environment with an unreliable server and geometric abandonments
%J RAIRO - Operations Research - Recherche Opérationnelle
%D 2018
%P 903-922
%V 52
%N 3
%I EDP-Sciences
%U http://www.numdam.org/articles/10.1051/ro/2018021/
%R 10.1051/ro/2018021
%G en
%F RO_2018__52_3_903_0
Jiang, Tao; Xin, Baogui; Chang, Baoxian; Liu, Liwei. Analysis of a queueing system in random environment with an unreliable server and geometric abandonments. RAIRO - Operations Research - Recherche Opérationnelle, Tome 52 (2018) no. 3, pp. 903-922. doi : 10.1051/ro/2018021. http://www.numdam.org/articles/10.1051/ro/2018021/

[1] J.R. Artalejo, A. Economou and M.J. Lopez-Herrero, Evaluating growth measures in an immigration process subject to binomial and geometric catastrophes. Math. Biosci. Eng. 4 (2007) 573–594. | DOI | MR | Zbl

[2] J. Cao and K. Cheng, Analysis of M/G/1 queueing system with repairable service station. Acta Math. Appl. Sin. 5 (1982) 113–127. | MR | Zbl

[3] G. Choudhury and K. Deka, An M/G/1 retrial queueing system with two phases of service subject to the server breakdown and repair. Perform. Eval. 65 (2008) 714–724. | DOI

[4] G. Choudhury and M. Deka, A single server queueing system with two phases of service subject to server breakdown and Bernoulli vacation. Appl. Math. Model. 36 (2012) 6050–6060. | DOI | MR | Zbl

[5] J.D. Cordeiro and J.P. Kharoufeh, The unreliable M/M/1 retrial queue in a random environment. Stoch. Model. 28 (2012) 29–48. | DOI | MR | Zbl

[6] S. Dimou and A. Economou, The single server queue with catastrophes and geometric reneging. Methodol. Comput. Appl. 15 (2013) 595–621. | DOI | MR | Zbl

[7] S. Dimou, A. Economou and D. Fakinos, The single server vacation queueing model with geometric abandonments. J. Stat. Plan. Infer. 141 (2011) 2863–2877. | DOI | MR | Zbl

[8] A. Economou and A. Gomez-Corral, The batch Markovian arrival process subject to renewal generated geometric catastrophes. Stoch. Model. 23 (2007) 211–233. | DOI | MR | Zbl

[9] A. Economou and S. Kanta, Equilibrium balking strategies in the observable single-server queue with breakdowns and repairs. Oper. Res. Lett. 36 (2008) 696–699. | DOI | MR | Zbl

[10] W.J. Gray, P.P. Wang and M. Scott, A queueing model with multiple types of server breakdowns. Qual. Technol. Quant. Manage. 1 (2004) 245–255. | DOI | MR

[11] T. Jiang and L. Liu, Analysis of a GI/M/1 queue in a multi-phase service environment with disasters. RAIRO: OR 51 (2017) 79–100. | DOI | Numdam | MR | Zbl

[12] T. Jiang,L. Liu and J. Li, Analysis of the M/G/1 queue in multi-phase random environment with disasters. J. Math. Anal. Appl. 430 (2015) 857–873. | DOI | MR | Zbl

[13] K. Kalidass and R. Kasturi, A queue with working breakdowns. Comput. Ind. Eng. 63 (2012) 779–783. | DOI

[14] J.C. Ke, On M/G/1 system under NT policies with breakdowns, startup and unreliable server. Appl. Math. Model. 30 (2006) 49–66. | DOI | Zbl

[15] J.C. Ke, Batch arrival queue under vacation policies with server breakdowns and startup/closedown. Appl. Math. Model. 31 (2007) 1282–1292. | DOI | Zbl

[16] B.K. Kim and D.H. Lee, The M/G/1 queue with disasters and working breakdowns. Appl. Math. Model. 38 (2014) 1788–1798. | DOI | MR | Zbl

[17] W. Li, D. Shi and X. Chao, Reliability analysis of M/G/1 queueing systems with server breakdowns and vacations. J. Appl. Probab. 34 (1997) 546–555. | DOI | MR | Zbl

[18] X. Li, J. Wang and F. Zhang, New results on equilibrium balking strategies in the single-server queue with breakdowns and repairs. Appl. Math. Comput. 241 (2014) 380–388. | MR | Zbl

[19] Z. Liu and S. Yu, The M/M/C queueing system in a random environment. J. Math. Anal. Appl. 436 (2016) 556–567. | DOI | MR | Zbl

[20] Z. Liu, J. Wu and G. Yang, An M/G/1 retrial G-queue with preemptive resume and feedback under N-policy subject to the server breakdowns and repairs. Comput. Math. Appl. 58 (2009) 1792–1807. | MR | Zbl

[21] M.F. Neuts, Matrix-Geometric Solutions in Stochastic Models. Johns Hopkins University Press, Baltimore (1981). | MR | Zbl

[22] N. Paz and U. Yechiali, An M/M/1 queue in random environment with disasters. Asia Pac. J. Oper. Res. 31 (2014) 1450016. | DOI | MR | Zbl

[23] N. Perel and U. Yechiali, Queues with slow servers an impatient customers. Eur. J. Oper. Res. 201 (2010) 247–258. | DOI | MR | Zbl

[24] N. Perel and U. Yechiali, The Israeli queue with retrials. Queueing Syst. 78 (2014) 31–56. | DOI | MR | Zbl

[25] J. Wang, An M/G/1 queue with second optional service and server breakdowns. Comput. Math. Appl. 47 (2004) 1713–1723. | MR | Zbl

[26] U. Yechiali, Queues with system disasters and impatient customers when system is down. Queueing Syst. 56 (2007) 195–202. | DOI | MR | Zbl

Cité par Sources :