Abstract
We consider a single server queueing-inventory system under (s,Q) replenishment strategy with working vacations, vacation interruptions and lost sales. The server provides service at a lower rate during working vacations than while in normal mode of service. If a working vacation realizes while providing service in that mode, then the server continues in the present status until the current service is completed. Upon that it switches to the normal mode of service, provided there is at least one customer waiting. If no customer is waiting at this point of time, it goes for vacation. We also assume that if there are customers in the system at a service completion epoch during a working vacation, the server will come back to the normal working mode, otherwise the server will stay in the working vacation mode. With the system having infinite capacity, the stability condition for the system is obtained, followed by computation of the steady-state probability vector and discussion. Various performance measures are evaluated. In addition, the busy period analysis is provided and the stationary waiting time distribution in the queue is derived. Numerical illustrations are provided to illustrate the system performance, and an optimization problem is also discussed.
Similar content being viewed by others
References
Baek, J.W. and Moon, S.K., The M/M/1 Queue with a Production-Inventory System and Lost Sales, Appl. Math. Comput., 2014, vol. 233, pp. 534–544.
Baek, J.W. and Moon, S.K., A Production-Inventory System with a Markovian Service Queue and Lost Sales, J. Korean Stat. Soc., 2016, vol. 45, pp. 14–24.
Baek, J.W., Bae, Y.H., Lee, H.W., and Ahn, S., Continuous-Type (s,Q)-Inventory Model with an Attached M/M/1 Queue and Lost Sales, Perform. Eval., 2018, vol. 125, pp. 68–79.
Jayaraman, B., Sivakumar, B., and Arivarignan, G., A Perishable Inventory System with Postponed Demands and Multiple Server Vacations, Model. Simul. Eng., 2012, vol. 2012, article ID 620960.
Koroliuk, V.S., Melikov, A.Z., Ponomarenko, L.A., and Rustamov, A.M., Models of Perishable Queueing-Inventory Systems with Server Vacations, Cybern. Syst. Anal., 2018, vol. 54, no. 1, pp. 31–44.
Koroliuk, V.S., Melikov, A.Z., Ponomarenko, L.A., and Rustamov, A.M., Asymptotic Analysis of the System with Server Vacation and Perishable Inventory, Cybern. Syst. Anal., 2017, vol. 53, no. 4, pp. 543–553.
Krenzler, R. and Daduna, H., Loss Systems in a Random Environment: Steady-State Analysis, Queueing Syst., 2014, vol. 80, pp. 127–153.
Krishnamoorthy, A. and Viswanath C. Narayanan, Stochastic Decomposition in Production Inventory with Service Time, Eur. J. Operat. Res., 2013, vol. 228, pp. 358–366.
Krishnamoorthy, A. and Viswanath C. Narayanan, Production Inventory with Service Time and Vacation to the Server, IMA J. Manage. Math., 2010. https://doi.org/10.1093/imaman/dpp025
Krishnamoorthy, A., Lakshmy, B., and Manikandan, R., A Survey on Inventory Models with Positive Service Time, OPSEARCH, 2011, vol. 48(2), pp. 153–169.
Krishnamoorthy, A., Shajin, D., and Viswanath C. Narayanan, Inventory with Positive Service Time: A Survey, in Advanced Trends in Queueing Theory, Anisimov, V. and Limnios, N., Eds., London: ISTE & Wiley, 2019.
Laxmi, P.V. and Soujanya, M.L., Retrial Inventory Model with Negative Customers and Multiple Working Vacations, Int. J. Manage. Sci. Eng. Manage., 2017. https://doi.org/10.1080/17509653.2016.1233837
Latouche, G. and Ramaswami, V., A Logarithmic Reduction Algorithm for Quasi-Birth-and-Death Processes, J. Appl. Probability, 1993, vol. 30, no. 3, pp. 650–674.
Li, J. and Tian, N., The M/M/1 Queue with Working Vacations and Vacation Interruptions, J. Syst. Sci. Syst. Eng., 2007, vol. 16, no. 1, pp. 121–127. https://doi.org/10.1007/s11518-006-5030-6
Melikov, A.Z. and Molchanov, A.A., Stock Optimization in Transport/Storage, Cybern. Syst. Anal., 1992, vol. 28, no. 3, pp. 484–487.
Melikov, A.Z., Rustamov, A.M., and Ponomarenko, L.A., Approximate Analysis of a Queueing-Inventory System with Early and Delayed Server Vacations, Autom. Remote Control, 2017, vol. 78, no. 11, pp. 1991–2003.
Melikov, A.Z. and Shahmaliyev, M.O., Queueing System M/M/1/∞ with Perishable Inventory and Repeated Customers, Autom. Remote Control, 2019, vol. 80, no. 1, pp. 53–65.
Viswanath C. Narayanan, Deepak, T.G., Krishnamoorthy, A., and Krishnakumar, B., On an (s, S) Inventory Policy with Service Time, Vacation to Server and Correlated Lead Time, Quality Techn. Quantitative Manage., 2008, vol. 5:2, pp. 129–143. https://doi.org/10.1080/16843703.2008.11673392
Padmavathi, I. and Sivakumar, B., Comparison of Three Vacation Policies in Discrete Time Inventory System with Postponed Demands, Int. J. Syst. Sci.: Operat. Logistics, 2019. https://doi.org/10.1080/23302674.2019.1587027
Padmavathi, I., Sivakumar, B., and Arivarignan, G., A Retrial Inventory System with Single and Modified multiple Vacation for Server, Ann. Oper. Res., 2015, vol. 233, no. 1, pp. 335–364.
Saffari, M.S., Asmussen and R. Haji, The M/M/1 Queue with Inventory, Lost Sale, and General Lead Times, Queueing Syst., 2013. https://doi.org/10.1007/s11134-012-9337-3
Schwarz, M., Sauer, C., Daduna, H., Kulik, R., and R. Szekli, R., M/M/1 Queueing Systems with Inventory, Queueing Syst., 2006, vol. 54, pp. 55–78.
Schwarz, M., Wichelhaus, C., and Daduna, H., Product Form Models for Queueing Networks with an Inventory, Stochas. Models, 2007, vol. 23, no. 4, pp. 627–663.
Shajin, D., Benny, B., Razumchik, R.V., and Krishnamoorthy, A., Discrete Product Inventory Control System with Positive Service Time and Two Operation Modes, Autom. Remote Control, 2018, vol. 79, no. 9, pp. 1593–1608.
Shajin, D. and Krishnamoorthy, A., Stochastic Decomposition in Retrial Queueing Inventory System, RAIRO Oper. Res., 2018. https://doi.org/10.1051/ro/2018118
Sigman, K. and Simchi-Levi, D., Light Traffic Heuristic for an M/G/1 Queue with Limited Inventory, Ann. Oper. Res., 1992, vol. 40, pp. 371–380.
Sivakumar, B., An Inventory System with Retrial Demands and Multiple Server Vacation, Quality Techno. Quantitative Manage., 2011, vol. 8:2, pp. 125–146. https://doi.org/10.1080/16843703.2011.11673252
Neuts, M.F., Matrix-Geometric Solutions in Stochastic Models-An Algorithmic Approach, New York: Dover, 1994, 2nd ed.
Acknowledgments
The authors wish to thank Prof. A. Krishnamoorthy for helpful suggestions. We also wish to thank two anonymous referees for their comments and constructive criticisms. These helped in improving the presentation of the paper.
Author information
Authors and Affiliations
Corresponding authors
Additional information
Russian Text © The Author(s), 2019, published in Matematicheskaya Teoriya Igr i Ee Prilozheniya, 2019, No. 3, pp. 53–76.
Rights and permissions
About this article
Cite this article
Manikandan, R., Nair, S.S. An M/M/1 Queueing-Inventory System with Working Vacations, Vacation Interruptions and Lost Sales. Autom Remote Control 81, 746–759 (2020). https://doi.org/10.1134/S0005117920040141
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1134/S0005117920040141