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July  2011, 7(3): 655-676. doi: 10.3934/jimo.2011.7.655

Analysis of the finite source retrial queues with server breakdowns and repairs

Jinting Wang 1, , Linfei Zhao 1, and  Feng Zhang 1,

1. 

Department of Mathematics, Beijing Jiaotong University, 100044 Beijing, China, China, China

Received  September 2010 Revised  May 2011 Published  June 2011

This paper is concerned with the queueing analysis as well as reliability evaluation of an $M/G/1//K$ retrial queue with a finite number of sources in which the server is subject to breakdowns and repairs. The server has a exponentially distributed life time and a generally distributed repair time. Our analysis extends previous work on this topic and includes the analysis of the arriving customer's distribution, the busy period, the waiting time process and main reliability characteristics. This queueing system and its variants could be used to model magnetic disk memory systems, star-like local area networks and other communication systems with detected or undetected breakdowns.
Keywords: reliability analysis., finite source, discrete transformation, retrial queues, Busy period.
Mathematics Subject Classification: Primary: 65K25; Secondary: 90B2.
Citation: Jinting Wang, Linfei Zhao, Feng Zhang. Analysis of the finite source retrial queues with server breakdowns and repairs. Journal of Industrial & Management Optimization, 2011, 7 (3) : 655-676. doi: 10.3934/jimo.2011.7.655
References:
[1]

A. Aissani, A retrial queue with redundancy and unreliable server,, Queueing Systems, 17 (1995), 443.   Google Scholar

[2]

B. Almási, J. Roszik and J. Sztrik, Homogeneous finite-source retrial queues with server subject to breakdowns and repairs,, Mathematical and Computer Modelling, 42 (2005), 673.  doi: 10.1016/j.mcm.2004.02.046.  Google Scholar

[3]

J. R. Artalejo, New results in retrial queueing systems with breakdown of the servers,, Statistica Neerlandica, 48 (1994), 23.  doi: 10.1111/j.1467-9574.1994.tb01429.x.  Google Scholar

[4]

J. R. Artalejo, Retrial queue with a finite number of sources,, J. Korean Math, 35 (1998), 503.   Google Scholar

[5]

J. R. Artalejo, A classified bibliography of research on retrial queues: Progress in 1990-1999,, Top, 7 (1999), 187.  doi: 10.1007/BF02564721.  Google Scholar

[6]

J. R. Artalejo and A. Gómez-Corral, Modelling communication systems with phase type service and retrial times,, IEEE Communications Letters, 11 (2007), 955.  doi: 10.1109/LCOMM.2007.070742.  Google Scholar

[7]

J. R. Artalejo and M. J. Lopez-Herrero, A simulation study of a discrete-time multiserver retrial queue with finite population,, Journal of Statistical Planning and Inference, 137 (2007), 2536.  doi: 10.1016/j.jspi.2006.04.018.  Google Scholar

[8]

J. R. Artalejo and A. Gómez-Corral, "Retrial Queueing Systems. A Computational Approach,", Springer-Verlag, (2008).  doi: 10.1007/978-3-540-78725-9.  Google Scholar

[9]

I. Atencia, I. Fortes, P. Moreno and S. Sánchez, An $M$/$G$/$1$ retrial queue with active breakdowns and Bernoulli schedule in the server,, International Journal of Information and Management Sciences, 17 (2006), 1.   Google Scholar

[10]

V. G. Kulkarni and B. D. Choi, Retrial queues with server subject to breakdowns and repairs,, Queueing Systems Theory Appl., 7 (1990), 191.  doi: 10.1007/BF01158474.  Google Scholar

[11]

G. I. Falin and J. R. Artalejo, A finite source retrial queue,, European Journal of Operational Research, 108 (1998), 409.  doi: 10.1016/S0377-2217(97)00170-7.  Google Scholar

[12]

G. I. Falin and J. G. C. Templeton, "Retrial Queues,", Chapman & Hall, (1997).   Google Scholar

[13]

G. K. Janssens, The quasi-random input queueing system with repeated attempts as a model for collision-avoidance star local area network,, IEEE Transactions on Communications, 45 (1997), 360.  doi: 10.1109/26.558699.  Google Scholar

[14]

N. Gharbi and M. Ioualalen, GSPN analysis of retrial systems with servers breakdowns and repairs,, Applied Mathematics and Computation, 174 (2006), 1151.  doi: 10.1016/j.amc.2005.06.005.  Google Scholar

[15]

D. J. Houck and W. S. Lai, Traffic modeling and analysis of hybrid fiber-coax systems,, Computer Networks and ISDN Systems, 30 (1998), 821.  doi: 10.1016/S0169-7552(97)00126-8.  Google Scholar

[16]

H. Li and T. Yang, A single-server retrial queue with server vacations and a finite number of input sources,, European Journal of Operational Research, 85 (1995), 149.  doi: 10.1016/0377-2217(94)E0358-I.  Google Scholar

[17]

H. Ohmura and Y. Takahashi, An analysis of repeated call model with a finite number of sources,, Electronics and Communications in Japan, 68 (1985), 112.  doi: 10.1002/ecja.4410680613.  Google Scholar

[18]

J. Sztrik, B. Almási and J. Roszik, Heterogeneous finite-source retrial queues with server subject to breakdowns and repairs,, Journal of Mathematical Sciences, 132 (2006), 677.  doi: 10.1007/s10958-006-0014-0.  Google Scholar

[19]

P. Tran-Gia and M. Mandjes, Modeling of customer retrial phenomenon in cellular mobile networks,, IEEE Journal on Selected Areas in Communications, 15 (1997), 1406.  doi: 10.1109/49.634781.  Google Scholar

[20]

J. Wang, Reliability analysis of $M$/$G$/$1$ queues with general retrial times and server breakdowns,, Progress in Natural Science (English Ed.), 16 (2006), 464.   Google Scholar

[21]

J. Wang, J. Cao and Q. Li, Reliability analysis of the retrial queue with server breakdowns and repairs,, Queueing Systems, 38 (2001), 363.  doi: 10.1023/A:1010918926884.  Google Scholar

[22]

J. Wang, B. Liu and J. Li, Transient analysis of an $M$/$G$/$1$ retrial queue subject to disasters and server failures,, European Journal of Operational Research, 189 (2008), 1118.  doi: 10.1016/j.ejor.2007.04.054.  Google Scholar

[23]

T. Yang and H. Li, The $M$/$G$/$1$ retrial queue with the server subject to starting failure,, Queueing Systems Theory Appl., 16 (1994), 83.  doi: 10.1007/BF01158950.  Google Scholar

show all references

References:
[1]

A. Aissani, A retrial queue with redundancy and unreliable server,, Queueing Systems, 17 (1995), 443.   Google Scholar

[2]

B. Almási, J. Roszik and J. Sztrik, Homogeneous finite-source retrial queues with server subject to breakdowns and repairs,, Mathematical and Computer Modelling, 42 (2005), 673.  doi: 10.1016/j.mcm.2004.02.046.  Google Scholar

[3]

J. R. Artalejo, New results in retrial queueing systems with breakdown of the servers,, Statistica Neerlandica, 48 (1994), 23.  doi: 10.1111/j.1467-9574.1994.tb01429.x.  Google Scholar

[4]

J. R. Artalejo, Retrial queue with a finite number of sources,, J. Korean Math, 35 (1998), 503.   Google Scholar

[5]

J. R. Artalejo, A classified bibliography of research on retrial queues: Progress in 1990-1999,, Top, 7 (1999), 187.  doi: 10.1007/BF02564721.  Google Scholar

[6]

J. R. Artalejo and A. Gómez-Corral, Modelling communication systems with phase type service and retrial times,, IEEE Communications Letters, 11 (2007), 955.  doi: 10.1109/LCOMM.2007.070742.  Google Scholar

[7]

J. R. Artalejo and M. J. Lopez-Herrero, A simulation study of a discrete-time multiserver retrial queue with finite population,, Journal of Statistical Planning and Inference, 137 (2007), 2536.  doi: 10.1016/j.jspi.2006.04.018.  Google Scholar

[8]

J. R. Artalejo and A. Gómez-Corral, "Retrial Queueing Systems. A Computational Approach,", Springer-Verlag, (2008).  doi: 10.1007/978-3-540-78725-9.  Google Scholar

[9]

I. Atencia, I. Fortes, P. Moreno and S. Sánchez, An $M$/$G$/$1$ retrial queue with active breakdowns and Bernoulli schedule in the server,, International Journal of Information and Management Sciences, 17 (2006), 1.   Google Scholar

[10]

V. G. Kulkarni and B. D. Choi, Retrial queues with server subject to breakdowns and repairs,, Queueing Systems Theory Appl., 7 (1990), 191.  doi: 10.1007/BF01158474.  Google Scholar

[11]

G. I. Falin and J. R. Artalejo, A finite source retrial queue,, European Journal of Operational Research, 108 (1998), 409.  doi: 10.1016/S0377-2217(97)00170-7.  Google Scholar

[12]

G. I. Falin and J. G. C. Templeton, "Retrial Queues,", Chapman & Hall, (1997).   Google Scholar

[13]

G. K. Janssens, The quasi-random input queueing system with repeated attempts as a model for collision-avoidance star local area network,, IEEE Transactions on Communications, 45 (1997), 360.  doi: 10.1109/26.558699.  Google Scholar

[14]

N. Gharbi and M. Ioualalen, GSPN analysis of retrial systems with servers breakdowns and repairs,, Applied Mathematics and Computation, 174 (2006), 1151.  doi: 10.1016/j.amc.2005.06.005.  Google Scholar

[15]

D. J. Houck and W. S. Lai, Traffic modeling and analysis of hybrid fiber-coax systems,, Computer Networks and ISDN Systems, 30 (1998), 821.  doi: 10.1016/S0169-7552(97)00126-8.  Google Scholar

[16]

H. Li and T. Yang, A single-server retrial queue with server vacations and a finite number of input sources,, European Journal of Operational Research, 85 (1995), 149.  doi: 10.1016/0377-2217(94)E0358-I.  Google Scholar

[17]

H. Ohmura and Y. Takahashi, An analysis of repeated call model with a finite number of sources,, Electronics and Communications in Japan, 68 (1985), 112.  doi: 10.1002/ecja.4410680613.  Google Scholar

[18]

J. Sztrik, B. Almási and J. Roszik, Heterogeneous finite-source retrial queues with server subject to breakdowns and repairs,, Journal of Mathematical Sciences, 132 (2006), 677.  doi: 10.1007/s10958-006-0014-0.  Google Scholar

[19]

P. Tran-Gia and M. Mandjes, Modeling of customer retrial phenomenon in cellular mobile networks,, IEEE Journal on Selected Areas in Communications, 15 (1997), 1406.  doi: 10.1109/49.634781.  Google Scholar

[20]

J. Wang, Reliability analysis of $M$/$G$/$1$ queues with general retrial times and server breakdowns,, Progress in Natural Science (English Ed.), 16 (2006), 464.   Google Scholar

[21]

J. Wang, J. Cao and Q. Li, Reliability analysis of the retrial queue with server breakdowns and repairs,, Queueing Systems, 38 (2001), 363.  doi: 10.1023/A:1010918926884.  Google Scholar

[22]

J. Wang, B. Liu and J. Li, Transient analysis of an $M$/$G$/$1$ retrial queue subject to disasters and server failures,, European Journal of Operational Research, 189 (2008), 1118.  doi: 10.1016/j.ejor.2007.04.054.  Google Scholar

[23]

T. Yang and H. Li, The $M$/$G$/$1$ retrial queue with the server subject to starting failure,, Queueing Systems Theory Appl., 16 (1994), 83.  doi: 10.1007/BF01158950.  Google Scholar

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