, Volume 47, Issue 4, pp 266–283 | Cite as

An inventory system with Markovian demands, phase type distributions for perishability and replenishment

  • Srinivas R. ChakravarthyEmail author
Theory and Methodology


In this paper we consider a version of (s, S)-type inventory system in which the demands occur according to a Markovian arrival process (MAP). The shelf life times of the items in the inventory as well as the replenishment times are assumed to be of (possibly different) phase type. Demands that are not met immediately are stored in an unmet buffer of finite capacity. Any arriving demand finding the inventory level at zero and the unmet buffer to be full is considered lost. Demands in the unmet buffer compete for the inventory after waiting for a random amount of time that is exponentially distributed. The steady state analysis of the inventory model is performed using the well-known matrix analytic methods. An optimization along with a couple of illustrative numerical examples are presented.


Markovian arrival process Phase type distribution Inventory Shelf life Algorithmic probability 


  1. 1.
    Bellman, R.E.: Introduction to Matrix Analysis. McGraw Hill, New York, NY (1960)Google Scholar
  2. 2.
    Chakravarthy, S.R.: The batch Markovian arrival process: a review and future work. In: Krishnamoorthy, A., et al. (eds.) Advances in Probability Theory and Stochastic Processes, pp. 21–39. Notable Publications Inc., NJ (2001)Google Scholar
  3. 3.
    Chakravarthy, S.R., Daniel, J.K.: A Markovian inventory system with random shelf time and back orders. Comput. Ind. Eng. 47, 315–337 (2004)CrossRefGoogle Scholar
  4. 4.
    Goyal, S.K., Giri, B.C.: Recent trends in modeling of deteriorating inventory. Eur. J. Oper. Res. 134, 1–16 (2001)CrossRefGoogle Scholar
  5. 5.
    Lucantoni, D.M.: New results on the single server queue with a batch Markovian arrival process. Stoch. Models 7, 1–46 (1991)CrossRefGoogle Scholar
  6. 6.
    Marcus, M., Minc, H.: A Survey of Matrix Theory and Matrix Inequalities. Allyn and Bacon, Boston, MA (1964)Google Scholar
  7. 7.
    Moinzadeh, K.: An improved ordering policy for continuous review inventory systems with arbitrary inter-demand time distributions. IIE Trans. 33, 111–118 (2001)Google Scholar
  8. 8.
    Nahimas, S.: Perishable inventory theory: a review. Oper. Res. 30, 680–708 (1982)CrossRefGoogle Scholar
  9. 9.
    Neuts, M.F.: Matrix-Geometric Solutions in Stochastic Models: An Algorithmic Approach. The Johns Hopkins University Press, Baltimore, MD (1981)Google Scholar
  10. 10.
    Neuts, M.F.: Structured Stochastic Matrices of M/G/1 Type and their Applications. Marcel Dekker, NY (1989)Google Scholar
  11. 11.
    Neuts, M.F.:. Models based on the Markovian arrival process. IEICE Trans. Commun. E75B, 1255–1265 (1992)Google Scholar
  12. 12.
    Raafat, F.: Survey of literature on continuously deteriorating inventory models. J. Oper. Res. Soc. 42, 27–37 (1991)Google Scholar
  13. 13.
    Sahin, I.: Regenerative Inventory Systems. Springer, New York (1990)Google Scholar

Copyright information

© Operational Research Society of India 2010

Authors and Affiliations

  1. 1.Department of Industrial and Manufacturing EngineeringKettering UniversityFlintUSA

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