Abstract
In many service systems, the service rates of the servers or the machines degrade over time due to a variety of reasons like fatigue, deficiency (due to manufacturing defect or energy loss, deformation, or excessive heat), and lack of proper maintenance. To make a service system beneficial for both customers and management points of view, it is imperative that models are developed to understand the impact of degradation. In this paper, we study \(MAP/PH/1-\)type queueing models incorporating degradation, failures/breakdowns and repairs. The degradation and to restore to normalcy in the service rate we look at two scenarios. In the first one, the service is restored to normalcy immediately after the server becomes idle or a fixed number of services is offered. In the second one, the restoration takes a random time that is modeled using a phase type distribution. Also, there are a variety of reasons that lead to failures/breakdowns of the server. We model these using (possibly) different phase type distributions. Both the models are studied analytically using matrix-analytic methods and illustrative numerical examples bringing out the qualitative behavior of the impact of degrading services are discussed. Two cost optimization problems whose solutions are obtained using particle swarm optimization technique along with a few illustrative examples are presented.
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References
Alfa, A. (2014). Some decomposition results for a class of vacation queues. Operations Research Letters, 42, 140–144. https://doi.org/10.1016/j.orl.2014.01.005
Artalejo, J. R., & G’omez-Corral, A. (2010). Markovian arrivals in stochastic modelling: A survey and some new results (invited article with discussion: Rafael P’erez-Oc’on, Miklos Telek and Yiqiang Q. Zhao). SORT-Statistics and Operations Research Transactions, 34, 101–156.
Baba, Y. (2005). Analysis of a GI/M/1 queue with multiple working vacations. Operations Research Letters, 33, 201–209. https://doi.org/10.1016/j.orl.2004.05.006
Bouchentouf, A. A., & Guendouzi, A. (2019). Cost optimization analysis for an \( M^{X}/M/c \) vacation queueing system with waiting servers and impatient customers. SeMA Journal, 76(2), 309–341. https://doi.org/10.1007/s40324-018-0180-2
Bouchentouf, A. A., Cherfaoui, M., & Boualem, M. (2019). Performance and economic analysis of a single server feedback queueing model with vacation and impatient customers. Opsearch, 56(1), 300–323. https://doi.org/10.1007/s12597-019-00357-4
Chakravarthy, S. R. (2001). The batch Markovian arrival process: A review and future work. In Krishnamoorthy, A. et al. (Eds.) Advances in probability theory and stochastic processes (pp. 21–39). Notable Publications Inc.
Chakravarthy, S. R. (2007). A multi-server synchronous vacation model with thresholds and a probabilistic decision rule. European Journal of Operational Research, 182, 305–320. https://doi.org/10.1016/j.ejor.2006.07.037
Chakravarthy, S. R. (2009). Analysis of a multi-server queue with Markovian arrivals and synchronous phase type vacations. Asia-Pacific Journal of Operational Research, 26, 85–113. https://doi.org/10.1142/S0217595909002134
Chakravarthy, S. R. (2010). Markovian Arrival Process. Wiley Encyclopedia of Operations Research and Management Science. https://doi.org/10.1002/9780470400531.eorms0499.
Chakravarthy, S. R. (2013a). Analysis of a multi-server queueing model with vacations and optional secondary services. Mathematica Applicanda, 41, 127–151. https://doi.org/10.14708/ma.v41i2.389.
Chakravarthy, S. R. (2013). Analysis of MAP/PH1, PH2/1 queue with vacations and optional secondary services. Applied Mathematical Modelling, 37, 8886–8902. https://doi.org/10.1016/j.apm.2013.04.012
Chakravarthy, S. R. (2014). A multi-server queueing model with server consultations. European Journal of Operational Research, 233(3), 625–639. https://doi.org/10.1016/j.ejor.2013.10.008
Chakravarthy, S. R. (2015). Matrix-analytic queueing models. InChapter 8 in Bhat U.N. In: An Introduction to Queueing Theory: Modeling and Analysis in Applications (pp. 177-199). Birkhäuser. https://doi.org/10.1007/978-0-8176-4725-4.
Chakravarthy, S. R., & Ozkar, S. (2016). MAP/PH/1 queueing model with working vacation and crowdsourcing. Mathematica Applicanda, 44, 263–294.https://doi.org/10.14708/ma.v44i2.1244
Chakravarthy, S. R., Shruti, & Kulshrestha, R. (2020). A queueing model with server breakdowns, repairs, vacations and backup server. Operations Research Perspectives, 7, 100–131. https://doi.org/10.1016/j.orp.2019.100131
Chakravarthy, S. R., Meena, R. K., & Choudhary, A. (2020). Queues with Markovian Arrivals, Phase Type Services, Breakdowns, and Repairs. In Distributed Computer and Communication Networks: 23rd International Conference, DCCN 2020, Moscow, Russia, September 14-18, 2020, Revised Selected Papers 23 (pp. 259–281). Springer. https://doi.org/10.1007/978-3-030-66471-8_21.
Chakravarthy SR (2021). A comparative study of vacation models under various vacation policies—A Simulation approach. In Kulshrestha, R. et al. (Eds). Mathematical modeling and computation of real-time problems: An interdisciplinary approach (pp. 3–20). CRC Press. https://doi.org/10.1201/9781003055037.
Chakravarthy, S. R. (2022). Introduction to matrix-analytic methods in queues 1: Analytical and simulation approach - basics. Wiley and ISTE Ltd.
Chakravarthy, S. R. (2022). Introduction to matrix-analytic methods in queues 2: Analytical and simulation approach - queues and simulation. Wiley and ISTE Ltd.
Chandrasekaran, V., Indhira, K., Saravanarajan, M., & Rajadurai, P. (2016). A survey on working vacation queueing models. International Journal of Pure and Applied Mathematics, 106, 33–41. https://doi.org/10.12732/ijpam.v106i6.5
Chang, J., & Wang, J. (2018). Unreliable M/M/1/1 retrial queues with set-up time. Quality Technology & Quantitative Management, 15(5), 589–601. https://doi.org/10.1080/16843703.2017.1320459
Choudhury, G., Tadj, L., & Deka, M. (2015). An unreliable server retrial queue with two phases of service and general retrial times under Bernoulli vacation schedule. Quality Technology & Quantitative Management, 12(4), 437–464. https://doi.org/10.1080/16843703.2015.11673430
Choudhary, A., Chakravarthy, S. R., & Sharma, D. C. (2021). Analysis of MAP/PH/1 queueing system with degrading service rate and phase type vacation. Mathematics, 9(19), 2387. https://doi.org/10.3390/math9192387
Deepa, B., & Kalidass, K. (2018). An M/M/1/N queue with working breakdowns and vacations. International Journal of Pure and Applied Mathematics, 119, 859–873.
Doshi, B. T. (1986). Queueing systems with vacations—A survey. Queueing System, 1, 29–66. https://doi.org/10.1007/BF01149327
Dudin, A. N., Klimenok, V. I., & Vishnevsky, V. M. (2020). The theory of queuing systems with correlated flows. Springer. https://doi.org/10.1007/978-3-030-32072-0
Fiems, D., Maertens, T., & Bruneel, H. (2008). Queueing systems with different types of server interruptions. European Journal of Operational Research, 188(3), 838–845. https://doi.org/10.1016/j.ejor.2007.05.010
Jain, M., Dhibar, S., & Sanga, S. S. (2021). Markovian working vacation queue with imperfect service, balking and retrial. Journal of Ambient Intelligence and Humanized Computing, 13, 1907–1923. https://doi.org/10.1007/s12652-021-02954-y
Jain, M., Sanga, S. S., & Meena, R. K. (2016). Control F-policy for Markovian retrial queue with server breakdowns. IEEE 1st International conference on power electronics, intelligent control and energy systems (pp. 1–5). https://doi.org/10.1109/ICPEICES.2016.7853083.
Jain, M., Shekhar, C., & Meena, R. K. (2017). Admission control policy of maintenance for unreliable server machining system with working vacation. Arabian Journal for Science and Engineering, 42(7), 2993–3005. https://doi.org/10.1007/s13369-017-2488-0
Jain, M., Kaur, S., & Singh, P. (2021). Supplementary variable technique (SVT) for non-Markovian single server queue with service interruption (QSI). Operational Research, 21(4), 2203–2246. https://doi.org/10.1007/s12351-019-00519-8
Jacob, V., Chakravarthy, S. R., & Krishnamoorthy, A. (2012). On a customer induced interruption in a service system. Stochastic Analysis and Applications, 30, 949–962. https://doi.org/10.1080/07362994.2012.704845
Jaya, S., & Lakshmi, B. (2021). Queues with interruption in Markovian environment. Journal of Mathematical and Computational Science, 11(4), 4753–4765.
Kim, J., Choi, D., & Chae, K. (2003). Analysis of queue-length distribution of the M/G/1 queue with working vacations. In International conference on statistics and related fields (pp. 1191-1200). Hawaii.
Ke, J. C. (2005). Modified T vacation policy for an M/G/1 queueing system with an unreliable server and startup. Mathematical and Computer Modelling, 41, 1267–1277. https://doi.org/10.1016/j.mcm.2004.08.009
Ke, J. C., Liu, T. H., & Yang, D. Y. (2018). Modeling of machine interference problem with unreliable repairman and standbys imperfect switchover. Reliability Engineering & System Safety, 174, 12–18. https://doi.org/10.1016/j.mcm.2004.08.009
Kennedy, J., & Eberhart, R. C. (1995). Particle Swarm Optimization. In Proceedings of IEEE international conference on neural networks (pp. 1942–1948). Piscataway. https://doi.org/10.1109/ICNN.1995.488968
Kim, C., Klimenok, V. I., & Dudin, A. N. (2017). Analysis of unreliable BMAP/PH/N type queue with Markovian flow of breakdowns. Applied Mathematics and Computation, 314, 154–172. https://doi.org/10.1016/j.amc.2017.06.035
Krishnamoorthy, A., & Divya, V. (2020). (M, MAP)/(PH, PH)/1 queue with non-preemptive priority and working vacation under N-policy. Journal of the Indian Society for Probability and Statistics, 21(1), 1–54. https://doi.org/10.1007/s41096-020-00081-z
Krishnamoorthy, A., Pramod, P. K., & Chakravarthy, S. R. (2013). A note on characterizing service interruptions with phase-type distribution. Stochastic Analysis and Applications, 31(4), 671–683. https://doi.org/10.1080/07362994.2013.800367
Krishnamoorthy, A., Pramod, P., & Chakravarthy, S. R. (2014). Queues with interruptions: A survey. TOP, 22, 290–320. https://doi.org/10.1007/s11750-012-0256-6
Li, J., & Tian, N. (2007). The M/M/1 queue with working vacations and vacation interruptions. Journal of Systems Science and Systems Engineering, 16(1), 121–127. https://doi.org/10.1007/s11518-006-5030-6
Latouche, G., & Ramaswami, V. (1999). Introduction to matrix analytic methods in stochastic modeling. SIAM. https://epubs.siam.org/doi/pdf/10.1137/1.9780898719734.bm.
Lucantoni, D. M. (1991). New results on the single server queue with a batch Markovian arrival process. Communications in Statistics. Stochastic Models, 7, 1–46. https://doi.org/10.1080/15326349108807174
Lucantoni, D. M., Meier-Hellstern, K. S., & Neuts, M. F. (1990). A single server queue with server vacations and a class of non-renewal arrival processes. Advances in Applied Probability, 22, 676–705. https://doi.org/10.2307/1427464
Marcus, M., & Minc, H. (1964). Survey of matrix theory and matrix inequalities. Allyn and Bacon.
Neuts, M. F. (1975). Probability distributions of phase type. Liber Amicorum Prof. Emeritus H.
Neuts, M. F. (1979). A versatile Markovian point process. Journal of Applied Probability, 16, 764–779. https://doi.org/10.2307/3213143
Neuts, M. F. (1981). Matrix-geometric solutions in stochastic models: An algorithmic approach. Johns Hopkins University. https://doi.org/10.1090/S0273-0979-1983-15095-4
Ozkar, S., & Kocer, U. U. (2017). \(M/C_k/1\) queue model with multiple working vacations. International Journal of Applied and Computational Mathematics, 3, 2729–2744. https://doi.org/10.1007/s40819-016-0220-5
Servi, L., & Finn, S. (2002). M/M/1 queue with working vacations (M/M/1/WV). Performance Evaluation, 50, 41–52. https://doi.org/10.1016/S0166-5316(02)00057-3
Singh, C. J., & Kaur, S. (2017). Unreliable server retrial queue with optional service and multi-phase repair. International Journal of Operations Research, 14(2), 35–51.
Sharma, R., & Jain, M. (2017). Finite priority queueing system with service interruption. Indian Journal of Industrial and Applied Mathematics, 8(1), 90–106. https://doi.org/10.5958/1945-919X.2017.00008.1
Shekhar, C., Jain, M., Iqbal, J., & Raina, A. A. (2017). Threshold control policy for maintainability of manufacturing system with unreliable workstations. Arabian Journal for Science and Engineering, 42(11), 4833–4851. https://doi.org/10.1007/s13369-017-2636-6
Shi, Y., & Eberhart, R. C. (1998). Parameter selection in particle swarm optimization. In Proceedings of the 7th International Conference on Evolutionary Programming (pp. 591–600). Springer. https://link.springer.com/chapter/10.1007/bfb0040810
Steeb, W. H., & Hardy, Y. (2011). Matrix calculus and Kronecker product: A practical approach to linear and multilinear algebra. World Scientific Publishing. https://searchworks.stanford.edu/view/9317860.
Sreenivasan, C., Chakravarthy, S. R., & Krishnamoorthy, A. (2013). MAP/PH/1 queue with working vacations, vacation interruptions and N policy. Applied Mathematical Modelling, 37, 3879–3893. https://doi.org/10.1016/j.apm.2012.07.054
Tian, N., & Zhang, Z. G. (2006). Vacation queueing models: Theory and applications. Springer. https://doi.org/10.1007/978-0-387-33723-4
Tian, N., Li, J., & Zhang, Z. G. (2009). Matrix-analytic method and working vacation queues-a survey. International Journal of Information and Management Sciences, 20, 603–633.
White, H., & Christie, L. S. (1958). Queuing with preemptive priorities or with breakdown. Operations Research, 6(1), 79–95. https://doi.org/10.1287/opre.6.1.79
Wu, D., & Takagi, H. (2006). M/G/1 queue with multiple working vacations. Performance Evaluation, 63, 654–681. https://doi.org/10.1016/j.peva.2005.05.005
Wu, C. H., & Yang, D. Y. (2020). Dynamic control of a machine repair problem with switching failure and unreliable repairmen. Arabian Journal for Science and Engineering, 45(3), 2219–2234. https://doi.org/10.1007/s13369-019-04196-9
Yang, D. Y., & Wu, C. H. (2015). Cost-minimization analysis of a working vacation queue with N-policy and server breakdowns. Computers & Industrial Engineering, 82, 151–158. https://doi.org/10.1016/j.cie.2015.01.017
Yoshida, H., Kawata, K., Fukuyama, Y., & Nakanishi, Y. (2000). A particle swarm optimization for reactive power and voltage control considering voltage security assessment. IEEE Transactions on Power Systems, 15, 1232–1239. https://doi.org/10.1109/59.898095
Zhang, M., & Hou, Z. (2011). Performance analysis of MAP/G/1 queue with working vacations and vacation interruption. Applied Mathematical Modelling, 35, 1551–1560. https://doi.org/10.1016/j.apm.2010.09.031
Acknowledgements
Alka Choudhary is also thankful to CSIR (Council of Scientific and Industrial Research), India for the financial support, file number is 09/1131(0007)/2017-EMR-I.
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Choudhary, A., Chakravarthy, S.R. & Sharma, D.C. Impact of the degradation in service rate in MAP/PH/1 queueing system with phase type vacations, breakdowns, and repairs. Ann Oper Res 331, 1207–1248 (2023). https://doi.org/10.1007/s10479-023-05628-4
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DOI: https://doi.org/10.1007/s10479-023-05628-4