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

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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

Abstract
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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 and 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), 443-449.
[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), 673-682. doi: 10.1016/j.mcm.2004.02.046.
[3]	J. R. Artalejo, New results in retrial queueing systems with breakdown of the servers, Statistica Neerlandica, 48 (1994), 23-36. doi: 10.1111/j.1467-9574.1994.tb01429.x.
[4]	J. R. Artalejo, Retrial queue with a finite number of sources, J. Korean Math, Soc., 35 (1998), 503-525.
[5]	J. R. Artalejo, A classified bibliography of research on retrial queues: Progress in 1990-1999, Top, 7 (1999), 187-211. doi: 10.1007/BF02564721.
[6]	J. R. Artalejo and A. Gómez-Corral, Modelling communication systems with phase type service and retrial times, IEEE Communications Letters, 11 (2007), 955-957. doi: 10.1109/LCOMM.2007.070742.
[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), 2536-2542. doi: 10.1016/j.jspi.2006.04.018.
[8]	J. R. Artalejo and A. Gómez-Corral, "Retrial Queueing Systems. A Computational Approach," Springer-Verlag, Berlin, 2008. doi: 10.1007/978-3-540-78725-9.
[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), 1-17.
[10]	V. G. Kulkarni and B. D. Choi, Retrial queues with server subject to breakdowns and repairs, Queueing Systems Theory Appl., 7 (1990), 191-208. doi: 10.1007/BF01158474.
[11]	G. I. Falin and J. R. Artalejo, A finite source retrial queue, European Journal of Operational Research, 108 (1998), 409-424. doi: 10.1016/S0377-2217(97)00170-7.
[12]	G. I. Falin and J. G. C. Templeton, "Retrial Queues," Chapman & Hall, London, 1997.
[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), 360-364. doi: 10.1109/26.558699.
[14]	N. Gharbi and M. Ioualalen, GSPN analysis of retrial systems with servers breakdowns and repairs, Applied Mathematics and Computation, 174 (2006), 1151-1168. doi: 10.1016/j.amc.2005.06.005.
[15]	D. J. Houck and W. S. Lai, Traffic modeling and analysis of hybrid fiber-coax systems, Computer Networks and ISDN Systems, 30 (1998), 821-834. doi: 10.1016/S0169-7552(97)00126-8.
[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), 149-160. doi: 10.1016/0377-2217(94)E0358-I.
[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), 112-121. doi: 10.1002/ecja.4410680613.
[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), 677-685. doi: 10.1007/s10958-006-0014-0.
[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), 1406-1414. doi: 10.1109/49.634781.
[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), 464-473.
[21]	J. Wang, J. Cao and Q. Li, Reliability analysis of the retrial queue with server breakdowns and repairs, Queueing Systems, 38 (2001), 363-380. doi: 10.1023/A:1010918926884.
[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), 1118-1132. doi: 10.1016/j.ejor.2007.04.054.
[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), 83-96. doi: 10.1007/BF01158950.

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References:

[1]	A. Aissani, A retrial queue with redundancy and unreliable server, Queueing Systems, 17 (1995), 443-449.
[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), 673-682. doi: 10.1016/j.mcm.2004.02.046.
[3]	J. R. Artalejo, New results in retrial queueing systems with breakdown of the servers, Statistica Neerlandica, 48 (1994), 23-36. doi: 10.1111/j.1467-9574.1994.tb01429.x.
[4]	J. R. Artalejo, Retrial queue with a finite number of sources, J. Korean Math, Soc., 35 (1998), 503-525.
[5]	J. R. Artalejo, A classified bibliography of research on retrial queues: Progress in 1990-1999, Top, 7 (1999), 187-211. doi: 10.1007/BF02564721.
[6]	J. R. Artalejo and A. Gómez-Corral, Modelling communication systems with phase type service and retrial times, IEEE Communications Letters, 11 (2007), 955-957. doi: 10.1109/LCOMM.2007.070742.
[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), 2536-2542. doi: 10.1016/j.jspi.2006.04.018.
[8]	J. R. Artalejo and A. Gómez-Corral, "Retrial Queueing Systems. A Computational Approach," Springer-Verlag, Berlin, 2008. doi: 10.1007/978-3-540-78725-9.
[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), 1-17.
[10]	V. G. Kulkarni and B. D. Choi, Retrial queues with server subject to breakdowns and repairs, Queueing Systems Theory Appl., 7 (1990), 191-208. doi: 10.1007/BF01158474.
[11]	G. I. Falin and J. R. Artalejo, A finite source retrial queue, European Journal of Operational Research, 108 (1998), 409-424. doi: 10.1016/S0377-2217(97)00170-7.
[12]	G. I. Falin and J. G. C. Templeton, "Retrial Queues," Chapman & Hall, London, 1997.
[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), 360-364. doi: 10.1109/26.558699.
[14]	N. Gharbi and M. Ioualalen, GSPN analysis of retrial systems with servers breakdowns and repairs, Applied Mathematics and Computation, 174 (2006), 1151-1168. doi: 10.1016/j.amc.2005.06.005.
[15]	D. J. Houck and W. S. Lai, Traffic modeling and analysis of hybrid fiber-coax systems, Computer Networks and ISDN Systems, 30 (1998), 821-834. doi: 10.1016/S0169-7552(97)00126-8.
[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), 149-160. doi: 10.1016/0377-2217(94)E0358-I.
[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), 112-121. doi: 10.1002/ecja.4410680613.
[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), 677-685. doi: 10.1007/s10958-006-0014-0.
[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), 1406-1414. doi: 10.1109/49.634781.
[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), 464-473.
[21]	J. Wang, J. Cao and Q. Li, Reliability analysis of the retrial queue with server breakdowns and repairs, Queueing Systems, 38 (2001), 363-380. doi: 10.1023/A:1010918926884.
[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), 1118-1132. doi: 10.1016/j.ejor.2007.04.054.
[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), 83-96. doi: 10.1007/BF01158950.

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