Automation and Remote Control

, Volume 80, Issue 1, pp 53–65 | Cite as

Queueing System M/M/1/∞ with Perishable Inventory and Repeated Customers

  • A. Z. MelikovEmail author
  • M. O. Shahmaliyev
Stochastic Systems


We propose a model of a queueing system with a single server, perishable inventory and repeated customers that can form an orbit of infinite size. In the absence of inventory in the system, primary customers according to the Bernoulli scheme either enter the queue or go into the orbit. The system uses the (s, S)-policy of replenishing the inventory. We develop a method for calculating system characteristics and solve the problem of minimizing total costs by choosing the critical level of inventory.


queueing-inventory systems perishable inventory repeated customers computation algorithm optimization 


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  1. 1.
    Artalejo, J.R., Krishnamoorthy, A., and Lopez-Herrero, M.J., Numerical Analysis of (s, S) Inventory System with Repeated Attempts, Ann. Oper. Res., 2006, vol. 141, pp. 67–83.zbMATHGoogle Scholar
  2. 2.
    Ushakumari, P.V., On (s, S) Inventory System with Random Lead Time and Repeated Demands, J. Appl. Math. Stoch. Anal., 2006, Article ID 81508.zbMATHGoogle Scholar
  3. 3.
    Lopez-Herrero, M.J., Waiting Time and Other First-Pasage Time Measures in an (s, S) Inventory System with Repeated Attempts and Finite Retrial Group, Comput. Oper. Res., 2010, vol. 37, pp. 1256–1261.MathSciNetzbMATHGoogle Scholar
  4. 4.
    Anbazhagan, N., Wang, J., and Gomathi, D., Base Stock Policy with Retrial Demands, Appl. Math. Model., 2013, vol. 37, pp. 4464–4473.MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Krishnamoorthy, A. and Jose, K.P., Comparision of Inventory Systems with Service, Positive Lead-Time, Loss, and Retrial of Customers, J. Appl. Math. Stoch. Anal. (Hindawi Publishing Corparation), vol. 2007, Article ID 37848.Google Scholar
  6. 6.
    Nair, A.N. and Jacob, M.J., (s, S) Inventory System with Positive Service Time and Retrial of Demands: An Approach Through Multi-Server Queues, ISRN Oper. Res. (Hindawi Publishing Corparation), vol. 2014, Article ID 596031.Google Scholar
  7. 7.
    Yadavalli, V.S.S., Anbazhagan, N., and Jeganathan, K., A Retrial Inventory System with Impatient Customers, Appl. Math. Inform. Sci., 2015, vol. 9, no. 2, pp. 637–650.Google Scholar
  8. 8.
    Amirthakodi, M. and Sivakumar, B., An Inventory System with Service Facility and Finite Orbit for Feedback Customers, OPSEARCH, 2015, vol. 52, no. 2, pp. 225–255.MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Manikandan, R. and Nair, S.S., M/M/1/1Queuing-Inventory System with Retrial of Unsatisfied Customers, Comm. Appl. Anal., 2017, vol. 21, no. 2, pp. 217–236.Google Scholar
  10. 10.
    Neuts, M.F., Matrix-Geometric Solutions in Stochastic Models: An Algorithmic Approach, Baltimore: John Hopkins Univ. Press, 1981.zbMATHGoogle Scholar
  11. 11.
    Manuel, P., Sivakumar, B., and Arivarignan, G., A Perishable Inventory System with Service Facilities and Retrial Customers, Comput. Ind. Eng., 2008, vol. 54, pp. 484–501.CrossRefGoogle Scholar
  12. 12.
    Sivakumar, B., An Inventory System with Retrial Demands andMultiple Server Vacation, Qual. Technol. Quantit. Manage., 2011, vol. 8, no. 2, pp. 125–146.CrossRefGoogle Scholar
  13. 13.
    Melikov, A.Z., Ponomarenko, L.A., and Shahmaliyev, M.O., Models of Perishable Queuing-Inventory Systems with Repeated Customers, J. Autom. Inform. Sci., 2016, vol. 48, no. 6, pp. 22–38.CrossRefGoogle Scholar
  14. 14.
    Novitskaya, E.V. and Terpugov, A.F., Optimizatsiya roznichnoi prodazhy skoroportyashcheisya produktsii (Optimizing Retail Sales of Perishable Products), Tomsk: Tomsk Univ., 2004.Google Scholar
  15. 15.
    Livshits, K.I. and Ul’yanova, E.S., Diffusion Approximation for the Process of Production and Sales of Perishable Products, Izv. Vyssh. Uchebn. Zaved., Fiz., 2015, vol. 58, no. 11/2, pp. 281–285.Google Scholar
  16. 16.
    Krishnamoorthy, A., Manikandan, R., and Lakshmy, B., Revisit to Queuing–Inventory System with Positive Service Time, Ann. Oper. Res., 2015, vol. 233, pp. 221–236.MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Melikov, A.Z., Ponomarenko, L.A., and Shahmaliyev, M.O., Analysis of Perishable Queuing–Inventory Systems with Different Types of Requests, J. Autom. Inform. Sci., 2017, vol. 49, no. 9, pp. 42–60.CrossRefGoogle Scholar
  18. 18.
    Melikov, A.Z., Ponomarenko, L.A., and Rustamov, A.M., Approximate Analysis of a Queueing–Inventory System with Early and Delayed Server Vacations, Autom. Remote Control, 2017, vol. 78, no. 11, pp. 1991–2003.MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Mitrani, I. and Chakka, R., Spectral Expansion Solution for a Class of Markov Models: Application and Comparison with the Matrix-Geometric Method, Perform. Evaluat., 1995, vol. 23, pp. 241–260.CrossRefzbMATHGoogle Scholar
  20. 20.
    Gillespie, D.T., A General Method for Numerically Simulating the Stochastic Time Evolution of Coupled Chemical Reactions, J. Comput. Phys., 1976, vol. 22, pp. 403–434.MathSciNetCrossRefGoogle Scholar

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© Pleiades Publishing, Ltd. 2019

Authors and Affiliations

  1. 1.Institute of Control SystemsAzerbaijan National Academy of SciencesBakuAzerbaijan
  2. 2.National Aviation AcademyBakuAzerbaijan

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