Analysis of a production inventory system with MAP arrivals, phase-type services and vacation to production facility

Year 2023, , 286 - 306, 23.06.2023

Şerife Özkar

https://doi.org/10.31801/cfsuasmas.1097797

Cited By: 2

Abstract

In this paper, we discuss a production inventory system with service times. Customers arrive in the system according to a Markovian arrival process. The service times follow a phase-type distribution. We assume that there is an infinite waiting space for customers. Arriving customers demand only one unit of item from the inventory. The production facility produces items according to an (s, S)-policy. Once the inventory level becomes the maximum level S, the production facility goes on a vacation of random duration. When the production facility returns from the vacation, if the inventory level depletes to the fixed level s, it is immediately switched on and starts production until the inventory level becomes S. Otherwise, if the inventory level is greater than s on return from the vacation, it takes another vacation. The vacation times are exponentially distributed. The production inventory system in the steady-state is analyzed by using the matrix-geometric method. A numerical study is performed on the system performance measures. Besides, an optimization study is discussed for the inventory policy.

Keywords

Production inventory system, vacation, phase-type distribution, Markovian arrival process, matrix geometric method

Supporting Institution

no support

Project Number

not a project

References

• Chakravarthy, S. R., Markovian arrival processes, Wiley Encyclopedia of Operations Research and Management Science, (2010). https://doi.org/10.1002/9780470400531.eorms0499
• Jeganathan, K., Abdul Reiyas, M., Padmasekaran, S., Lakshmanan, K., Two heterogeneous servers queueing-inventory system with multiple vacations and server interruptions, Adv. Model. Optim., 20(1) (2018), 113-133.
• Jeganathan, K., Padmasekaran, S., Kingsly, S. J., A perishable inventory-queueing model with delayed vacation, negative and impatient customers, Math. Model. Geom., 4(3) (2016), 11–30. https://doi.org/10.26456/MMG/2016-432
• Karthikeyan, K., Sudhesh, R., Recent review article on queueing inventory systems, Research Journal of Pharmacy and Technology, 9(11) (2016), 1451–1461. https://doi.org/10.5958/0974-360X.2016.00421.2
• Ke, J. C., Wu, C.H., Zhang, Z. G. Recent developments in vacation queueing models, Int. J. Oper. Res., 7(4) (2010), 3–8.
• Koroliuk, V. S., Melikov, A. Z., Ponomarenko, L. A., Rustamov, A. M., Models of perishable queueing-inventory systems with server vacations, Cybern. Syst. Anal., 54(1) (2018), 31–44. https://doi.org/10.1007/s10559-018-0005-4
• Krishnamoorthy, A., Lakshmy, B., Manikandan, R., A survey on inventory models with positive service time, OPSEARCH, 48(2) (2011), 153–169. https://doi.org/10.1007/s12597-010-0032-z
• Krishnamoorthy, A., Narayanan, V. C., Production inventory with service time and vacation to the server, IMA J. Manag. Math., 22(2011), 33–45.
• Manikandan, R., Nair, S. S., An M/M/1 queueing-inventory system with working vacations, vacation interruptions and lost sales, Autom Remote Control, 81(4) (2020), 746–759. https://doi.org/10.1134/S0005117920040141
• Melikov, A. Z., Rustamov, A. M., Ponomarenko, L. A, Approximate analysis of a queueinginventory system with early and delayed server vacations, Autom Remote Control, 78(11) (2017), 1991–2003. https://doi.org/10.1134/S0005117917110054
• Narayanan, V. C., Deepak, T. G., Krishnamoorthy, A., Krishnakumar, B., On an (s, S) inventory policy with service time, vacation to server and correlated lead time, Quality Technology& Quantitative Management, 5(2) (2008), 129–143. https://doi.org/10.1080/16843703.2008.11673392
• Neuts, M. F. Matrix-Geometric Solutions in Stochastic Models: An Algorithmic Approach, The Johns Hopkins University Press, Baltimore, MD. [1994 version is Dover Edition], 1981.
• Padmavathi, I., Lawrence, A. S., Sivakumar, B., A finite-source inventory system with postponed demands and modified M vacation policy, OPSEARCH, 53(1) (2016), 41–62. https://doi.org/10.1007/s12597-015-0224-7
• Padmavathi, I., Sivakumar, B., Arivarignan, G., A retrial inventory system with single and modified multiple vacation for server, Ann. Oper. Res., 233 (2015), 335–364. https://doi.org/10.1007/s10479-013-1417-1
• Sivakumar, B., An inventory system with retrial demands and multiple server vacation, Quality Technology & Quantitative Management, 8(2) (2011), 125–146. https://doi.org/10.1080/16843703.2011.11673252
• Suganya, C., Lawrence, A. S., Sivakumar, B., A finite-source inventory system with service facility, multiple vacations of two heterogeneous servers, Int. J. Inf. Manag. Sci., 29 (2018), 257–277.
• Tian, N., Zhang, Z. G., Vacation Queueing Models: Theory and Applications, Springer-Verlang, New York, 2006.
• Yue, D., Qin, Y., A production inventory system with service time and production vacations, J. Syst. Sci. Syst. Eng., 28 (2019), 168–180. https://doi.org/10.1007/s11518-018-5402-8
• Yue, D., Wang, S., Zhang, Y., A production-inventory system with a service facility and production interruptions for perishable items, In: Quan-Lin Li, Jinting Wang and Hai-Bo Yu (ed) Stochastic Models in Reliability, Network Security and System Safety, Communications in Computer and Information Science 1102, Springer Nature Singapore, (2019), 410–428.
• Zhang, Y., Yue, D., Yue, W., A queueing-inventory system with random order size policy and server vacations, Ann. Oper. Res., (2020). https://doi.org/10.1007/s10479-020-03859-3