In this paper, we study a single server GI/M/1 queue in a multi-phase service environment with disasters, where the disasters occur only when the server is busy serving customers. Whenever a disaster occurs in an operative service phase, all present customers are forced to leave the system simultaneously, the server abandons the service and an exponential repair time is set on. After the system is repaired, the server resumes his service and moves to service phase immediately with probability . Using the matrix analytic approach and semi-Markov process, we obtain the stationary queue length distribution at both arrival and arbitrary epochs. After introducing tagged customers and the concept of a cycle, we also derive the sojourn time distribution, the duration of a cycle, and the length of the server’s working time in a service cycle. In addition, numerical examples are presented to illustrate the impact of some critical model parameters on performance measures.
Accepté le :
DOI : 10.1051/ro/2016005
Mots clés : GI/M/1 queue, matrix analytic approach, multi-phase service environment, disasters, cycle analysis
@article{RO_2017__51_1_79_0,
author = {Jiang, Tao and Liu, Liwei},
title = {Analysis of a {GI/M/1} queue in a multi-phase service environment with disasters},
journal = {RAIRO - Operations Research - Recherche Op\'erationnelle},
pages = {79--100},
publisher = {EDP-Sciences},
volume = {51},
number = {1},
year = {2017},
doi = {10.1051/ro/2016005},
zbl = {1364.90122},
mrnumber = {3590463},
language = {en},
url = {http://www.numdam.org/articles/10.1051/ro/2016005/}
}
TY - JOUR AU - Jiang, Tao AU - Liu, Liwei TI - Analysis of a GI/M/1 queue in a multi-phase service environment with disasters JO - RAIRO - Operations Research - Recherche Opérationnelle PY - 2017 SP - 79 EP - 100 VL - 51 IS - 1 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/ro/2016005/ DO - 10.1051/ro/2016005 LA - en ID - RO_2017__51_1_79_0 ER -
%0 Journal Article %A Jiang, Tao %A Liu, Liwei %T Analysis of a GI/M/1 queue in a multi-phase service environment with disasters %J RAIRO - Operations Research - Recherche Opérationnelle %D 2017 %P 79-100 %V 51 %N 1 %I EDP-Sciences %U http://www.numdam.org/articles/10.1051/ro/2016005/ %R 10.1051/ro/2016005 %G en %F RO_2017__51_1_79_0
Jiang, Tao; Liu, Liwei. Analysis of a GI/M/1 queue in a multi-phase service environment with disasters. RAIRO - Operations Research - Recherche Opérationnelle, Tome 51 (2017) no. 1, pp. 79-100. doi : 10.1051/ro/2016005. http://www.numdam.org/articles/10.1051/ro/2016005/
and , The discrete-time Geo/Geo/1 queue with negative customers and disasters. Comput. Oper. Res. 31 (2004) 1537–1548. | DOI | Zbl
and , Steady state analysis of lever dependent quasi-birth-and-death processes with catastrophes. Comput. Oper. Res. 39 (2012) 413–423. | DOI | MR | Zbl
and , Optimal and equilibrium balking strategies in the single server Markovian queue with catastrophes. Eur. J. Oper. Res. 218 (2012) 708–715. | DOI | MR | Zbl
and , The effect of catastrophes on the strategic customer behavior in queueing systems. Nav. Res. Logist. 60 (2013) 571–587. | DOI | MR | Zbl
, A disaster queue with Markovian arrivals and impatient customers. Appl. Math. Comput. 214 (2009) 48–59. | MR | Zbl
J.W. Cohen, The single server queue. North-Holland, Amsterdam (1982). | MR | Zbl
and , The single server queue with catastrophes and geometric reneging. Methodol. Comput. Appl. Probab. 15 (2013) 595–621. | DOI | MR | Zbl
and , Equilibrium balking strategies for a clearing system in alternating environment. Ann. Oper. Res. 208 (2013) 489–514. | DOI | MR | Zbl
and , A Pollaczek-Khintchine formula for M/G/1 queues with disasters. J. Appl. Probab. 33 (1996) 1191–1200. | DOI | MR | Zbl
, and , 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
and , The M/G/1 queue with disasters and working breakdowns. Appl. Math. Model. 38 (2014) 1788–1798. | DOI | MR | Zbl
and , The N-policy of a discrete time Geo/G/1 queue with disasters and its application to wireless sensor networks. Appl. Math. Model. 37 (2013) 9722–9731. | DOI | MR | Zbl
, and , Geo/G/1 queues with disasters and general repair times. Appl. Math. Model. 35 (2011) 1561–1570. | DOI | MR | Zbl
, and , The GI/M/1 queue with Bernoulli-schedule-controlled vacation and vacation interruption. Appl. Math. Model. 37 (2013) 3724–3735. | DOI | MR | Zbl
and , Performance analysis of a GI/M/1 queue with single working vacation. Appl. Math. Model. 217 (2011) 4960–4971. | MR | Zbl
and , An M/G/1 queueing system with disasters and repairs under a multiple adapted vacation policy. Nav. Res. 62 (2015) 171–189. | DOI | MR | Zbl
M. Neuts, Matrix-Geometric Solutions in Stochastic Models. Johns Hopkins University Press, Baltimore (1981). | MR | Zbl
, and , The Geo/G/1 with negative customers and disasters. Stoch. Models 25 (2009) 673–688. | DOI | MR | Zbl
, and , Analysis of the GI/Geo/1 queue with disasters. Stoch. Anal. Appl. 28 (2010) 44–53. | DOI | MR | Zbl
and , An M/M/1 queue in random environment with disasters. Asia-Pac. J. Oper. Res. 31 (2014) 1450017. | MR | Zbl
, Transient analysis of a queue with system disasters and customer impatience. Queueing Syst. 66 (2010) 95–105. | DOI | MR | Zbl
and , A single server priority queue with server failures and queue flushing. Oper. Res. Lett. 10 (1991) 353–362. | DOI | MR | Zbl
and , A note on the GI/M/1 queue with Poisson negative arrivals. J. Appl. Probab. 38 (2001) 1081–1085. | DOI | MR | Zbl
, Queues with system disasters and impatient customers when system is down. Queueing Syst. 56 (2007) 195–202. | DOI | MR | Zbl
Cité par Sources :
