Abstract
A queueing-inventory system where the server is subject to breakdown is examined. The products in inventory are provided by production. Two different types of customers are considered. Each customer type arrives in the system according to a Poisson process with different parameters. There is only one server and the service time follows an exponential distribution. Service may be interrupted randomly due to breakdowns. There may be minor or major breakdowns and after the recovery process, the priority customer class gets service first. The described system is formulated as a level-dependent quasi-birth-death process (LDQBD). The stationarity condition is provided and the steady-state probabilities are calculated. A comprehensive numerical study is presented that examines the sensitivity of the performance measures to the change of system parameters and also investigates the optimal inventory policy according to the considered cost function.
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References
Anilkumar, M. P., & Jose, K. P. (2020). A \(Geo/Geo/1\) inventory priority queue with self induced interruption. International Journal of Applied Computational Mathematics, 6, 99. https://doi.org/10.1007/s40819-020-00857-8
Baek, J. W., & Moon, S. K. (2014). The \(M/M/1\) queue with a production-inventory system and lost sales. Applied Mathematics and Computation, 233, 534–544. https://doi.org/10.1016/j.amc.2014.02.033
Baek, J. W., & Moon, S. K. (2016). A production-inventory system with a Markovian service queue and lost sales. Journal of the Korean Statistical Society, 45(1), 14–24. https://doi.org/10.1016/j.jkss.2015.05.002
Baek, J. W., Bae, Y. H., Lee, H. W., & Ahn, S. (2018). Continuous-type \((s, Q)\)-inventory model with an attached \(M/M/1\) queue and lost sales. Performance Evaluation, 125, 68–79. https://doi.org/10.1016/j.peva.2018.07.003
Baumann, H., & Sandman, W. (2013). Computing stationary expectations in level-dependent QBD processes. Journal of Applied Probability, 50, 151–165. https://doi.org/10.1239/jap/1363784430
Berman, O., Kaplan, E. H., & Shevishak, D. G. (1993). Deterministic approximations for inventory management at service facilities. IIE Transactions, 25(5), 98–104. https://doi.org/10.1080/07408179308964320
Berman, O., & Kim, E. (1999). Stochastic models for inventory management at service facilities. Communications in Statistics. Stochastic Models, 15(4), 695–718. https://doi.org/10.1080/15326349908807558
Berman, O., & Sapna, K. P. (2000). Inventory management at service facilities for systems with arbitrarily distributed service times. Communications in Statistics. Stochastic Models, 16(3–4), 343–360. https://doi.org/10.1080/15326340008807592
Berman, O., & Kim, E. (2001). Dynamic order replenishment policy in internet-based supply chains. Mathematical Methods of Operations Research, 53, 371–390.
Berman, O., & Sapna, K. P. (2001). Optimal control of service for facilities holding inventory. Computers & Operations Research, 28, 429–441. https://doi.org/10.1016/S0305-0548(99)00128-8
Berman, O., & Kim, E. (2004). Dynamic inventory strategies for profit maximization in a service facility with stochastic service, demand and lead time. Mathematical Methods of Operations Research, 60, 497–521. https://doi.org/10.1007/s001860400369
Bright, L., & Taylor, P. G. (1995). Calculating the equilibrium distribution in level dependent quasi-birth-and-death processes. Communications in Statistics. Stochastic Models, 11(3), 497–525. https://doi.org/10.1080/15326349508807357
Chakravarthy, S. R., Maity, A., & Gupta, U. C. (2017). An \((s, S)\) inventory in a queueing system with batch service facility. Annals of Operations Research, 258, 263–283. https://doi.org/10.1007/s10479-015-2041-z
Chakravarthy, S. R., & Rumyantsev, A. (2020). Analytical and simulation studies of queueing-inventory models with MAP demands in batches and positive phase type services. Simulation Modelling Practice and Theory, 103, 1–15. https://doi.org/10.1016/j.simpat.2020.102092
Hanukov, G., Avinadav, T., Chernonog, T., Spiegel, U., & Yechiali, U. (2017). A queueing system with decomposed service and inventoried preliminary services. Applied Mathematical Modelling, 47, 276–293. https://doi.org/10.1016/j.apm.2017.03.008
Hanukov, G., Avinadav, T., Chernonog, T., & Yechiali, U. (2019). A multi-server queueing-inventory system with stock-dependent demand. IFAC-PapersOnLine, 52(13), 671–676. https://doi.org/10.1016/j.ifacol.2019.11.124
Hanukov, G., Avinadav, T., Chernonog, T., & Yechiali, U. (2019). Performance improvement of a service system via stocking perishable preliminary services. European Journal of Operational Research, 274(3), 1000–1011. https://doi.org/10.1016/j.ejor.2018.10.027
Hanukov, G., Avinadav, T., Chernonog, T., & Yechiali, U. (2020). A service system with perishable products where customers are either fastidious or strategic. International Journal of Production Economics. https://doi.org/10.1016/j.ijpe.2020.107696
Jeganathan, K., Reiyas, M. A., Lakshmi, K. P., & Saravanan, S. (2019). Two server Markovian inventory systems with server interruptions: Heterogeneous vs. homogeneous servers. Mathematics and Computers in Simulation, 155, 177–200. https://doi.org/10.1016/j.matcom.2018.03.001
Jose, K. P., & Nair, S. S. (2017). Analysis of two production inventory systems with buffer, retrials and different production rates. Journal of Industrial Engineering International, 13, 369–380. https://doi.org/10.1007/s40092-017-0191-0
Karthikeyan, K., & Sudhesh, R. (2016). Recent review article on queueing inventory systems. Research Journal of Pharmacy and Technology, 9(11), 1451–1461.
Krishnamoorthy, A., Narayanan, V. C., Deepak, T. G., & Vineetha, P. (2006). Control policies for inventory systems. Stochastic Analysis and Applications, 24, 889–899. https://doi.org/10.1080/07362990600753635
Krishnamoorthy, A., Nair, S. S., & Narayanan, V. C. (2010). An inventory model with server interruptions. Proceedings of the 5th international conference on queueing theory and network applications (pp. 132-139). https://doi.org/10.1145/1837856.1837876
Krishnamoorthy, A., Lakshmy, B., & Manikandan, R. (2011). A survey on inventory models with positive service time. OPSEARCH, 48(2), 153–169. https://doi.org/10.1007/s12597-010-0032-z
Krishnamoorthy, A., & Narayanan, V. C. (2011). Production inventory with service time and vacation to the server. IMA Journal of Management Mathematics, 22(1), 33–45. https://doi.org/10.1093/imaman/dpp025
Krishnamoorthy, A., Nair, S. S., & Narayanan, V. C. (2012). An inventory model with server interruptions and retrials. Operational Research- An International Journal, 12, 151–171. https://doi.org/10.1007/s12351-011-0120-8
Krishnamoorthy, A., & Narayanan, V. C. (2013). Stochastic decomposition in production inventory with service time. European Journal of Operational Research, 228, 358–366. https://doi.org/10.1016/j.ejor.2013.01.041
Krishnamoorthy, A., Manikandan, R., & Lakshmy, B. (2015). A revisit to queueing-inventory system with positive service time. Annals of Operations Research, 233, 221–236. https://doi.org/10.1007/s10479-013-1437-x
Krishnamoorthy, A., Nair, S. S., & Narayanan, V. C. (2015). Production inventory with service time and interruptions. International Journal of Systems Science, 46(10), 1800–1816. https://doi.org/10.1080/00207721.2013.837538
Liu, M., Feng, M., & Wong, C. Y. (2014). Flexible service policies for a Markov inventory System with two demand classes. International Journal of Production Economics, 151, 180–185. https://doi.org/10.1016/j.ijpe.2013.10.010
Melikov, A. Z., & Molchanov, A. A. (1992). Stock optimization in transport/storage systems. Cybernetics and Systems Analysis, 28(3), 484–487. https://doi.org/10.1007/BF01125431
Melikov, A., Mirzayev, R., & Nair, S. S. (2022). Double sources queuing-inventory system with hybrid replenishment policy. Mathematics, 10(14), 2423. https://doi.org/10.3390/math10142423
Nair, A. N., & Jacob, M. J. (2015). An \((s, S)\) Production inventory controlled self-service queuing system. Journal of Probability and Statistics. https://doi.org/10.1155/2015/505082
Nair, A. N., Jacob, M. J., & Krishnamoorthy, A. (2015). The multi server \(M/M/(s, S)\) queueing inventory system. Annals of Operations Research, 233, 321–333. https://doi.org/10.1007/s10479-013-1405-5
Narayanan, V. C., Deepak, T. G., Krishnamoorthy, A., & Krishnakumar, B. (2008). On an \((s, S)\) inventory policy with service time, Vacation to server and correlated lead time. Quality Technology & Quantitative Management, 5(2), 129–143. https://doi.org/10.1080/16843703.2008.11673392
Neuts, M. F. (1981). Matrix-geometric solutions in stochastic models: An algorithmic approach. Baltimore, MD: The Johns Hopkins University Press [1994 version is Dover Edition].
Ozkar, S., & Uzunoglu Kocer, U. (2021). Two-commodity queueing-inventory system with two classes of customers. OPSEARCH, 58, 234–256. https://doi.org/10.1007/s12597-020-00479-0
Saffari, M., Asmussen, S., & Haji, R. (2013). The \(M/M/1\) queue with inventory, lost sale, and general lead times. Queueing Systems, 75, 65–77. https://doi.org/10.1007/s11134-012-9337-3
Schwarz, M., & Daduna, H. (2006). Queueing systems with inventory management with random lead times and with backordering. Mathematical Methods of Operations Research, 64, 383–414. https://doi.org/10.1007/s00186-006-0085-1
Schwarz, M., Sauer, C., Daduna, H., Kulik, R., & Szekli, R. (2006). \(M/M/1\) queueing systems with inventory. Queueing Systems, 54, 55–78. https://doi.org/10.1007/s11134-006-8710-5
Shajin, D., Jacob, J., & Krishnamoorthy, A. (2022). On a queueing inventory problem with necessary and optional inventories. Annals of Operations Research, 315, 2089–2114. https://doi.org/10.1007/s10479-021-03975-8
Sigman, K., & Simchi-Levi, D. (1992). Light traffic heuristic for an \(M/G/1\) queue with limited inventory. Annals of Operations Research, 40, 371–380. https://doi.org/10.1007/BF02060488
Tweedie, R. (1975). Sufficient conditions for egodicity and reccurrence of Markov chains on a general state space. Stochastic Processes and their Applications, 3(4), 385–403. https://doi.org/10.1016/0304-4149(75)90033-2
Wang, K., Jiang, Z., & Xie, W. (2013). Optimal scheduling policies for a production service system with two items. Paper presented at 7th IFAC conference on manufacturing modelling, management, and control (pp. 19-21).
Yadavalli, V. S., Sivakumar, B., & Arivarignan, G. (2008). Inventory system with renewal demands at service facilities. International Journal of Production Economics, 114, 252–264. https://doi.org/10.1016/j.ijpe.2007.12.013
Zhao, N., & Lian, Z. (2011). A queueing-inventory system with two classes of customers. International Journal of Production Economics, 129, 225–231. https://doi.org/10.1016/j.ijpe.2010.10.011
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Kocer, U.U., Ozkar, S. A production queueing-inventory system with two-customer and a server subject to breakdown. Ann Oper Res 331, 1089–1117 (2023). https://doi.org/10.1007/s10479-023-05275-9
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DOI: https://doi.org/10.1007/s10479-023-05275-9