American Institute of Mathematical Sciences

Advanced
  • Previous Article
    Analysis of globally gated Markovian limited cyclic polling model and its application to uplink traffic in the IEEE 802.16 network
  • JIMO Home
  • This Issue
  • Next Article
    Performance analysis of a Geom/Geom/1 queueing system with variable input probability
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

[1]

Tuan Phung-Duc, Wouter Rogiest, Sabine Wittevrongel. Single server retrial queues with speed scaling: Analysis and performance evaluation. Journal of Industrial & Management Optimization, 2017, 13 (4) : 1927-1943. doi: 10.3934/jimo.2017025
[2]

Tuan Phung-Duc, Ken’ichi Kawanishi. Multiserver retrial queues with after-call work. Numerical Algebra, Control & Optimization, 2011, 1 (4) : 639-656. doi: 10.3934/naco.2011.1.639
[3]

Tuan Phung-Duc. Single server retrial queues with setup time. Journal of Industrial & Management Optimization, 2017, (3) : 1329-1345. doi: 10.3934/jimo.2016075
[4]

Jesus R. Artalejo, Tuan Phung-Duc. Markovian retrial queues with two way communication. Journal of Industrial & Management Optimization, 2012, 8 (4) : 781-806. doi: 10.3934/jimo.2012.8.781
[5]

Tuan Phung-Duc, Hiroyuki Masuyama, Shoji Kasahara, Yutaka Takahashi. M/M/3/3 and M/M/4/4 retrial queues. Journal of Industrial & Management Optimization, 2009, 5 (3) : 431-451. doi: 10.3934/jimo.2009.5.431
[6]

Arnaud Devos, Joris Walraevens, Tuan Phung-Duc, Herwig Bruneel. Analysis of the queue lengths in a priority retrial queue with constant retrial policy. Journal of Industrial & Management Optimization, 2017, 13 (5) : 1-30. doi: 10.3934/jimo.2019082
[7]

Matúš Tibenský, Angela Handlovičová. Convergence analysis of the discrete duality finite volume scheme for the regularised Heston model. Discrete & Continuous Dynamical Systems - S, 2018, 0 (0) : 0-0. doi: 10.3934/dcdss.2020226
[8]

Tuan Phung-Duc, Hiroyuki Masuyama, Shoji Kasahara, Yutaka Takahashi. State-dependent M/M/c/c + r retrial queues with Bernoulli abandonment. Journal of Industrial & Management Optimization, 2010, 6 (3) : 517-540. doi: 10.3934/jimo.2010.6.517
[9]

Shaojun Lan, Yinghui Tang. Performance analysis of a discrete-time $ Geo/G/1$ retrial queue with non-preemptive priority, working vacations and vacation interruption. Journal of Industrial & Management Optimization, 2019, 15 (3) : 1421-1446. doi: 10.3934/jimo.2018102
[10]

Bailey Kacsmar, Douglas R. Stinson. A network reliability approach to the analysis of combinatorial repairable threshold schemes. Advances in Mathematics of Communications, 2019, 13 (4) : 601-612. doi: 10.3934/amc.2019037
[11]

Sofian De Clercq, Wouter Rogiest, Bart Steyaert, Herwig Bruneel. Stochastic decomposition in discrete-time queues with generalized vacations and applications. Journal of Industrial & Management Optimization, 2012, 8 (4) : 925-938. doi: 10.3934/jimo.2012.8.925
[12]

Veena Goswami, Gopinath Panda. Optimal information policy in discrete-time queues with strategic customers. Journal of Industrial & Management Optimization, 2019, 15 (2) : 689-703. doi: 10.3934/jimo.2018065
[13]

Veena Goswami, Gopinath Panda. Optimal customer behavior in observable and unobservable discrete-time queues. Journal of Industrial & Management Optimization, 2017, 13 (5) : 0-0. doi: 10.3934/jimo.2019112
[14]

Gang Chen, Zaiming Liu, Jinbiao Wu. Optimal threshold control of a retrial queueing system with finite buffer. Journal of Industrial & Management Optimization, 2017, 13 (3) : 1537-1552. doi: 10.3934/jimo.2017006
[15]

Tetsuji Hirayama. Analysis of multiclass feedback queues and its application to a packet scheduling problem. Journal of Industrial & Management Optimization, 2010, 6 (3) : 541-568. doi: 10.3934/jimo.2010.6.541
[16]

M. R. Hassan. Maximizing reliability of the capacity vector for multi-source multi-sink stochastic-flow networks subject to an assignment budget. Journal of Industrial & Management Optimization, 2017, 13 (5) : 0-0. doi: 10.3934/jimo.2020020
[17]

Samitha Samaranayake, Axel Parmentier, Ethan Xuan, Alexandre Bayen. A mathematical framework for delay analysis in single source networks. Networks & Heterogeneous Media, 2017, 12 (1) : 113-145. doi: 10.3934/nhm.2017005
[18]

Madhu Jain, Sudeep Singh Sanga. Admission control for finite capacity queueing model with general retrial times and state-dependent rates. Journal of Industrial & Management Optimization, 2017, 13 (5) : 1-25. doi: 10.3934/jimo.2019073
[19]

Yuncherl Choi, Jongmin Han, Chun-Hsiung Hsia. Bifurcation analysis of the damped Kuramoto-Sivashinsky equation with respect to the period. Discrete & Continuous Dynamical Systems - B, 2015, 20 (7) : 1933-1957. doi: 10.3934/dcdsb.2015.20.1933
[20]

Thierry Cazenave, Yvan Martel, Lifeng Zhao. Finite-time blowup for a Schrödinger equation with nonlinear source term. Discrete & Continuous Dynamical Systems - A, 2019, 39 (2) : 1171-1183. doi: 10.3934/dcds.2019050

2018 Impact Factor: 1.025

Tools

Metrics

  • PDF downloads (14)
  • HTML views (0)
  • Cited by (16)

Other articles
by authors

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences