A Production Inventory System with Service Time and Production Vacations

  • Dequan Yue
  • Yaling QinEmail author


In this paper, we consider a production inventory system with service time and production vacations. Customers arrive in the system according to a Poisson process requiring service from a single server. The single production facility produces items according to an (s, S) policy, and it takes a vacation of random duration once the inventory level becomes (S. It is assumed that all arriving customers are lost during the stock out period. We first derive the stationary joint distribution of the queue length and the on-hand inventory level in product form. Then, we compute explicitly some performance measures, and develop a cost function based on these performance measures. Finally, some numerical results are presented.


Production inventory system service time lost sales queue length cost function 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.



The authors sincerely thank the anonymous reviewers for their helpful comments that improved the quality of the paper. This work was supported in part by the Natural Science Foundation of Hebei Province, China (No. A2017203078).


  1. Baek JW, Moon SK (2014). The M/M/1 queue with a production-inventory system and lost sales. Applied Mathematics and Computation 233: 534–544.MathSciNetCrossRefzbMATHGoogle Scholar
  2. Baek JW, Moon SK (2016). A production-inventory system with Markovian service queue and lost sales. Journal of the Korean Statistical Society 45(1): 14–24.MathSciNetCrossRefzbMATHGoogle Scholar
  3. Berman O, Kaplan EH, Shimshak DG (1993). Deterministic approximations for inventory management at service facilities. IIE Transactions 25(5): 98–104.CrossRefGoogle Scholar
  4. Berman O, Kim E (1999). Stochastic models for inventory management at service facilities. Stochastic Models 15(4): 695–718.MathSciNetCrossRefzbMATHGoogle Scholar
  5. 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(3): 497–521.MathSciNetCrossRefzbMATHGoogle Scholar
  6. Doshi BT (1986). Queueing systems with vacations: a survey. Queueing Systems 1: 29–66.MathSciNetCrossRefzbMATHGoogle Scholar
  7. Ke JC, Wu CH, Zhang ZG (2010). Recent developments in vacation queuing models: a short survey. International Journal of Operantion Research 7: 3–8.Google Scholar
  8. Krenzler R, Daduna H (2015). Loss systems in a random environment steady-state analysis. Queueing Systems 80(1): 127–153.MathSciNetCrossRefzbMATHGoogle Scholar
  9. Krishnamoorthy A, Viswanath NC (2011). Production inventory with service time and vacation to the server. IMA Journal of Management Mathematics 22(1): 33–45.MathSciNetCrossRefzbMATHGoogle Scholar
  10. Krishnamoorthy A, Viswanath NC (2013). Stochastic decomposition in production inventory with service time. European Journal of Operational Research 228(2): 358–366.MathSciNetCrossRefzbMATHGoogle Scholar
  11. Li N, Jiang Z (2013). Modeling and optimization of a product-service system with additional service capacity and impatient customers. Computer & Operations Research 40(8): 1923–1937.MathSciNetCrossRefzbMATHGoogle Scholar
  12. Neuts MF (1981). Matrix-Geometric Solutions in Stochastic Models: an Algorithmic Approach. John Hopkins Press, Baltimore.zbMATHGoogle Scholar
  13. Padmavathi I, Lawrence AS, Sivakumar B (2016). A finitesource inventory system with postponed demands and modified M vacation policy. OPSEARCH 53(1): 41–62.MathSciNetCrossRefzbMATHGoogle Scholar
  14. Saffari M, Asmussen S, Haji R (2013). The M/M/1 queue with inventory, lost sale, and general lead times. Queueing Systems 75(1): 65–77.MathSciNetCrossRefzbMATHGoogle Scholar
  15. Schwarz M, Sauer C, Daduna H, Kulik R, Szekli R (2006). M/M/1 Queueing systems with inventory. Queueing Systems 54(1): 55–78.MathSciNetCrossRefzbMATHGoogle Scholar
  16. Sigman K, Simchi-Levi D (1992). Light traffc heuristic for an M/G/1 queue with limited inventory. Annals of Operations Research 40(1): 371–380.CrossRefzbMATHGoogle Scholar
  17. Sivakumar B (2011). An inventory system with retrial demands and multiple server vacation. Quality Technology and Quantitative Management 8(2): 125–146.CrossRefGoogle Scholar
  18. Takagi H (1991). Queueing Analysis-A Foundation of Performance Evaluation. Elsvier, Amusterdam.Google Scholar
  19. Tian N, Zhang ZG (2006). Vacation Queuing Models: Theory and Applications. Springer-Verlag, New York.CrossRefGoogle Scholar
  20. Viswanath CN, Deepak TG, Krishnamoorthy A, Krishkumar B (2008). On (s, S) inventory policy with service time, vacation to server and correlated lead time. Quality Technology and Quantitative Management 5(2): 129–144.MathSciNetCrossRefGoogle Scholar

Copyright information

© Systems Engineering Society of China and Springer-Verlag GmbH Germany 2019

Authors and Affiliations

  1. 1.School of ScienceYanshan UniversityQinhuangdaoChina
  2. 2.School of Economics and ManagementYanshan UniversityQinhuangdaoChina

Personalised recommendations