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Preventive maintenance in an unreliable M/G/1 retrial queue with persistent and impatient customers

  • S.I.: Retrial Queues and Related Models
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Abstract

In this paper we study a new version of an unreliable retrial queue with persistent and impatient customers. The considered model takes into account corrective and preventive maintenances. If a preventive action occurs in a busy period, then it is postponed to an ulterior date. We give the necessary and sufficient condition for the system to be stable and obtain the joint distribution of the server state and the number of orbiting customers in the system in terms of generating functions. Some performance measures are derived. From the reliability view point, we analyze the time to the first failure of the server. The effect of the system parameters on the performance measures is shown through an illustrative numerical example.

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References

  • Aissani, A. (1988). On the M/G/1/1 queueing system with repeated orders and unreliable server. Journal of Technology, 6, 98–123.

    Google Scholar 

  • Aissani, A. (1994). A retrial queue with redundancy and unreliable server. Queueing Systems, 17, 431–449.

    Article  Google Scholar 

  • Aissani, A., & Artalejo, J. R. (1998). On the single server retrial queue subject to breakdowns. Queueing Systems, 30, 309–321.

    Article  Google Scholar 

  • Artalejo, J. R. (1994). New results in retrial queueing systems with breakdown of the servers. Statistica Neerlandica, 48(1), 23–36.

    Article  Google Scholar 

  • Artalejo, J. R. (1999a). A classified bibliography of research on retrial queues: Progress in 1990–1999. Top, 7(2), 187–211.

  • Artalejo, J. R. (1999b). Accessible bibliography on retrial queues. Mathematical and Computer Modelling, 30(3), 1–6.

    Article  Google Scholar 

  • Artalejo, J. R. (2010). Accessible bibliography on retrial queues: Progress in 2000–2009. Mathematical and Computer Modelling, 51(9–10), 1071–1081.

    Article  Google Scholar 

  • Artalejo, J. R., & Gomez-Corral, A. (2008). Retrial queueing systems: A computational approach. Heidelberg: Springer.

    Book  Google Scholar 

  • Artalejo, J. R., Krishnamoorthy, A., & Lopez-Herrero, M. J. (2006). Numerical analysis of (s, S) inventory systems with repeated attempts. Annals of Operations Research, 141(1), 67–83.

    Article  Google Scholar 

  • Artalejo, J. R., & Pozo, M. (2002). Numerical calculation of the stationary distribution of the main multiserver retrial queue. Annals of Operations Research, 116, 41–56.

    Article  Google Scholar 

  • Atencia, I., Bouza, G., & Moreno, P. (2008). An \(\text{ M }^{\rm[X]}\)/G/1 retrial queue with server breakdowns and constant rate of repeated attempts. Annals of Operations Research, 157, 225–243.

    Article  Google Scholar 

  • Atencia, I., Fortes, I., Moreno, P., & Sanchez, S. (2006). An M/G/1 retrial queue with active breakdowns and Bernoulli schedule in the server. Information and Management Sciences, 17(1), 1–17.

    Google Scholar 

  • Atencia, I., & Moreno, P. (2006). A discrete-time Geo/G/1 retrial queue with server subject to starting failures. Annals of Operations Research, 141(1), 85–107.

    Article  Google Scholar 

  • Avram, F., & Gomez-Corral, A. (2006). On bulk service MAP/\(\text{ PH }^{\rm LN}\)/1/N G-queues with repeated attempts. Annals of Operations Research, 141(1), 109–137.

    Article  Google Scholar 

  • Awi, F., & So, K. C. (1990). Optimal maintenance policies for single server queueing systems subject to breakdowns. Operations Research, 38, 330–343.

    Article  Google Scholar 

  • Ayyappan, G., & Sathiya, K. (2013). Transient analysis of batch arrival feedback retrial queue with starting failure and Bernoulli vacation. Mathematical Theory and Modeling, 3(8), 60–67.

    Google Scholar 

  • Bagyam, E. A., & Chandrika, K. U. (2013). \(\text{ M }/(\text{ G }_{1}\), \(\text{ G }_{2}\),., \(\text{ G }_{k})\)/1 retrial queueing system with server breakdown, delayed repair and reserved time. International Journal of Emerging Technology and Advanced Engineering, 3(8), 587–597.

    Google Scholar 

  • Bhagat, A., & Jain, M. (2013). Unreliable \(\text{ M }^{\rm X}\)/G/1 retrial queue with multi-optional services and impatient customers. International Journal of Operational Research, 17(2), 248–273.

    Article  Google Scholar 

  • Choudhury, G., & Deka, K. (2009). An \(\text{ M }^{\rm X}\) /G/1 unreliable retrial queue with two phases of service and Bernoulli admission mechanism. Applied Mathematics and Computation, 215(3), 936–949.

    Article  Google Scholar 

  • Choudhury, G., & Ke, J. C. (2012). A batch arrival retrial queue with general retrial times under Bernoulli vacation schedule for unreliable server and delaying repair. Applied Mathematical Modelling, 36(1), 255–269.

    Article  Google Scholar 

  • Cooper, R. B. (1981). Introduction to Queueing Theory. New York: North-Holland INC.

    Google Scholar 

  • Dimitriou, I. (2013). A preemptive resume priority retrial queue with state dependent arrivals, unreliable server and negative customers. Top, 21, 542–571.

    Article  Google Scholar 

  • Efrosinin, D., & Winkler, A. (2011). Queueing system with a constant retrial rate, non-reliable server and threshold-based recovery. European Journal of Operational Research, 210(3), 594–605.

    Article  Google Scholar 

  • Falin, G. I. (2008). The M/M/1 retrial queue with retrials due to server failures. Queueing Systems, 58, 155–160.

    Article  Google Scholar 

  • Falin, G. I. (2010). An M/G/1 retrial queue with unreliable server and general repair times. Performance Evaluation, 67(7), 569–582.

    Article  Google Scholar 

  • Falin, G. I., & Templeton, J. G. C. (1997). Retrial queues. London: Chapman & Hall.

    Book  Google Scholar 

  • Ganapathi Subramanian, M., Ayyapan, G., & Sekar, G. (2013). Single server retrial queueing system with partial breakdown by computational method. Service Science and Management Research, 2(2), 26–32.

    Google Scholar 

  • Gaver, D. P. (1962). A waiting line with interrupted service, including priorities. Journal of the Royal Statistical Society B, 24, 73–90.

    Google Scholar 

  • Glazebrook, K. D. (1984). Scheduling stochastic jobs on a single machine subject to breakdowns. Naval Research Logistics Quarterly, 31, 251–264.

    Article  Google Scholar 

  • Gupta, D., Günalay, Y., & Srinivasan, M. (2001). The relationship between preventive maintenance and manufacturing system performance. European Journal of Operational Research, 132, 146–162.

    Article  Google Scholar 

  • Hsu, L. F. (1992). Optimal preventive maintenance policies in an M/G/1 queue like production system. European Journal of Operational Research, 58, 112–122.

    Article  Google Scholar 

  • Hsu, L. F. (1999). Simultaneous determination of preventive maintenance and replacement policies in a queue like production system with minimal repair. Reliability Engineering and Systems Safety, 63, 161–167.

    Article  Google Scholar 

  • Hsu, L. F., & Tapiero, C. S. (1987). Maintenance of unreliable M/G/1 queue like job shop. Queueing Systems, 2, 333–350.

    Article  Google Scholar 

  • Jain, M., & Bhargava, C. (2008). Bulk arrival retrial queue with unreliable server and priority subscribers. International Journal of Operations Research, 5(4), 242–259.

    Google Scholar 

  • Jain, M., & Bhargava, C. (2009). Unreliable server M/G/1 queueing system with Bernoulli feedback, repeated attempts, modified vacation, phase repair and discouragement. Journal of KAU Engineering Science, 20(2), 43–72.

    Google Scholar 

  • Jain, M., & Mishra, A. (2008). Reliability analysis of unreliable server retrial queue with bulk arrivals. Pakistan Journal of Statistics, 24(4), 285–300.

    Google Scholar 

  • Kalyanaraman, R., & Seenivasan, M. (2011). A multiserver retrial queue with breakdown and geometric loss. International Journal of Computational Cognition, 9(1), 44–48.

    Google Scholar 

  • Kim, C. S., Klimenok, V., & Birukov, A. (2006). Optimal multi-threshold control by the BMAP/SM/1 retrial system. Annals of Operations Research, 141(1), 193–210.

    Article  Google Scholar 

  • Kim, C. S., Klimenok, V. I., & Orlovsky, D. S. (2008). The BMAP/PH/N retrial queue with Markovian flow of breakdowns. European Journal of Operational Research, 189, 1057–1072.

    Article  Google Scholar 

  • Kim, C. S., Mushko, V. V., & Dudin, A. N. (2012). Computation of the steady state distribution for multiserver retrial queues with phase type service process. Annals of Operations research, 201, 307–323.

    Article  Google Scholar 

  • Koyanagi, J., & Kawai, H. (1997). An optimal maintenance policy for a queuing system server under periodic observation. International Journal of Reliability, Quality and Safety Engineering, 4, 357–367.

    Article  Google Scholar 

  • Koyanagi, J., & Kawai, H. (2003). An optimal age maintenance for an M/G/1 queueing system. Mathematical and computer Modelling, 38, 1333–1338. Please confirm the inserted volume number for the reference Koyanagi and Kawai (2003).

    Article  Google Scholar 

  • Krishna Kumar, B., Pavai Madheswari, S., & Vijayakumar, A. (2002a). The M/G/1 retrial queue with feedback and starting failures. Applied Mathematical Modelling, 26, 1057–1075.

  • Krishna Kumar, B., Vijayakumar, A., & Arivudainambi, D. (2002b). An M/G/1 retrial queueing system with two phases of service and preemptive resume. Annals of Operations Research, 113, 61–79.

  • Krishnamoorthy, A., Pramod, P. K., & Chakravarthy, S. R. (2014). Queues with interruptions: A survey. Top, 22(1), 290–320.

    Article  Google Scholar 

  • Kulkarni, V. G., & Choi, B. D. (1990). Retrial queues with server subject to breakdowns and repairs. Queueing Systems, 7, 191–208.

    Article  Google Scholar 

  • Li, J., & Wang, J. (2006). An M/G/1 retrial queue with second multi-optional service, feedback and unreliable server. Applied Mathematics. A Journal of Chinese Universities, Series B, 21(3), 252–262.

    Article  Google Scholar 

  • Li, Q. L., Ying, Y., & Zhao, Y. Q. (2006). A BMAP/G/1 retrial queue with a server subject to breakdowns and repairs. Annals of Operations Research, 141, 233–270.

    Article  Google Scholar 

  • Mokaddis, G. S., Metwally, S. A., & Zaki, B. M. (2007). A feedback retrial queuing system with failures and single vacation. Tamkang Journal of Science and Engineering, 10(3), 183–192.

    Google Scholar 

  • Osaki, S. (1972). An intermittently used system with preventive maintenance. Journal of the Operations Research Society of Japan, 15, 102–111.

    Google Scholar 

  • Perry, D., & Posner, M. J. M. (2000). A correlated M/G/1type queue with randomized server repair and maintenance modes. Operations Research Letters, 26, 137–147.

    Article  Google Scholar 

  • Phung-Duc, T., Masuyama, H., Kasahara, S., & Takahashi, Y. (2013). A matrix continued fraction approach to multiserver retrial queues. Annals of Operations Research, 202(1), 161–183.

    Article  Google Scholar 

  • Phung-Duc, T., Rogiest, W., Takahashi, Y., & Bruneel, H. (2014). Retrial queues with balanced call blending: Analysis of single server and multiserver model. Annals of Operations Research,. doi:10.1007/s10479-014-1598-2.

    Google Scholar 

  • Purohit, G. N., Jain, M., & Rani, S. (2012). M/M/1 retrial queue with constant retrial policy, unreliable server, threshold recovery and state dependent arrival rates. Applied Mathematical Sciences, 6(37), 1837–1846.

    Google Scholar 

  • Sennot, L. I., Humblet, P. A., & Tweedie, R. L. (1983). Mean drift and the non-ergodicity of Markov chains. Operations Research, 31, 783–788.

    Article  Google Scholar 

  • Senthil, M., & Arumuganathan, R. (2013). An \(\text{ M }^{\rm X}\)/G/1 unreliable retrial queue with two phase service and persistence behavior of customers in service. RAIRO Operations Research, 47(1), 9–32.

    Article  Google Scholar 

  • Sherman, N. P., Kharoufeh, J. P., & Abramson, M. A. (2009). An M/G/1 retrial queue with unreliable server for streaming multimedia applications. Probability in the Engineering and Informational Sciences, 23(2), 281–304.

    Article  Google Scholar 

  • Sumitha, D., & Chandrika, U. K. (2012). Retrial queueing system with starting failure, single vacation and orbital search. International Journal of Computer Applications, 40(13), 29–33.

    Article  Google Scholar 

  • Taleb, S., & Aissani, A. (2010). Unreliable M/G/1 retrial queue: Monotonicity and comparability. Queueing Systems, 64, 227–252.

    Article  Google Scholar 

  • Taleb, S., Saggou, H., & Aissani, A. (2010). Unreliable M/G/1 retrial queue with geometric loss and random reserved time. International Journal of Operational Research, 7(2), 171–191.

    Article  Google Scholar 

  • Wang, J., Cao, J., & Li, Q. (2001). Reliability analysis of the retrial queue with server breakdowns and repairs. Queueing Systems, 38, 363–380.

    Article  Google Scholar 

  • Wang, J., & Li, J. (2008). A repairable M/G/1 retrial queue with Bernoulli vacation and two-phase. Quality Technology & Quantitative Management, 5(2), 179–192.

    Article  Google Scholar 

  • Wang, J., & Zhang, P. (2009). A discrete time retrial queue with negative customers and unreliable server. Computers and Industrial Engineering, 56(4), 1216–1222.

    Article  Google Scholar 

  • Wang, J., & Zhao, Q. (2007). A discrete-time Geo/G/1 retrial queue with general retrial times and starting failures. Mathematical and Computer Modelling, 45, 853–863.

    Article  Google Scholar 

  • Wang, J., & Zhou, P. F. (2010). A batch arrival retrial queue with starting failures, feedback and admission control. Journal of Systems Science and Systems Engineering, 19(3), 306–320.

    Article  Google Scholar 

  • Wu, X., Brill, P., Hlynka, M., & Wang, J. (2005). An M/G/1 retrial queue with balking and retrials. International Journal of Operational Research, 1(1/2), 30–51.

    Article  Google Scholar 

  • Wu, J., & Lian, Z. (2013). A single server retrial G-queue with priority and unreliable server under Bernoulli vacation schedule. Computers & Industrial Engineering, 64(1), 84–93.

    Article  Google Scholar 

  • Wu, J., & Yin, X. (2011). An M/G/1 retrial G-queue with non-exhaustive random vacations and an unreliable server. Computers & Mathematics with Applications, 62(5), 2314–2329.

    Article  Google Scholar 

  • Yang, T., & Li, H. (1994). The M/G/1 retrial queue with the server subject to starting failures. Queueing Systems, 16, 83–96.

    Article  Google Scholar 

  • Zhang, F., & Wang, J. (2012). On the optimal and equilibrium retrial rates in an unreliable retrial queue with vacations. Journal of Industrial and Management Optimization, 8(4), 861–875.

    Article  Google Scholar 

  • Zhang, Z., Wang, J., & Zhang, F. (2014). Equilibrium customer strategies in the single server constant retrial queue with breakdowns and repairs. Mathematical Problems in Engineering, Article ID 379572, pp. 1–14.

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Acknowledgments

The authors thank the referees for their valuable comments and suggestions to improve the paper.

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Authors and Affiliations

  1. Department of Probability and Statistics, Faculty of Mathematics, University of Science and Technology Houari Boumediene USTHB, El-Alia, Bab Ezzouar, B.P. 32, 16111, Algiers, Algeria

    Samira Taleb

  2. Department of Computer Sciences, Faculty of Electronics and Computer Sciences, University of Science and Technology Houari Boumediene USTHB, El-Alia, Bab Ezzouar, B.P. 32, 16111, Algiers, Algeria

    Amar Aissani

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  1. Samira Taleb
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  2. Amar Aissani
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Correspondence to Samira Taleb.

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Taleb, S., Aissani, A. Preventive maintenance in an unreliable M/G/1 retrial queue with persistent and impatient customers. Ann Oper Res 247, 291–317 (2016). https://doi.org/10.1007/s10479-016-2217-1

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