Undershoot and order quantity probability distributions in periodic review, reorder point, order-up-to-level inventory systems with continuous demand Articles uri icon

publication date

  • March 2021

start page

  • 791

end page

  • 814


  • 91

International Standard Serial Number (ISSN)

  • 0307-904X


  • The undershoot of the reorder point in the periodic review, order-up-to-level (R, s, S) inventory system is known to follow a complex probability distribution which depends on the value of S-s (Δ) and the distribution of the demand during the review interval (R). We focus on the continuous demand case with full backlogging and variable lead-time. For this case, a generic formulation of the undershoot probability density function (p.d.f.) is developed. The order quantity probability distribution in (R, s, S) systems is the same as the undershoot probability distribution with a shift of Δ in the random variable. Therefore, the latter opens the possibility of calculating valuable managerial information such as the expected average order quantity, its standard deviation, and the probability that the order quantity is lower than or exceeds a predetermined value. Based on the proposed formulation, we derive an analytical expression of the undershoot p.d.f. (and hence the order quantity p.d.f.) for the case of gamma distributed demand, as well as a tractable approximation for the normal distributed demand. Both expressions are shown to be dependent upon two nondimensional parameters, Δ/μR and the coefficient of variation, with the mean demand during the review interval (μR) acting as a scale parameter. We thus define a nondimensional undershoot p.d.f. (NUPDF). The relevance of full nondimensionalization stems from the fact that gamma and normal NUPDF analyses can be scaled to any case of gamma and normal distributed demands. Although we focus on the inventory management viewpoint, the results for the gamma distributed case can be directly adapted for use in any renewal process.


  • inventory; order quantity; renewal theory; reorder point; undershoot