Stationary analysis of an (R, Q) inventory model with normal and emergency orders

Onno Boxma (Corresponding author), David Perry (Corresponding author), Wolfgang Stadje (Corresponding author)

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We consider an (R, Q) inventory model with two types of orders, normal orders and emergency orders, which are issued at different inventory levels. These orders are delivered after exponentially distributed lead times. In between deliveries, the inventory level decreases in a state-dependent way, according to a release rate function. This function represents the fluid demand rate; it could be controlled by a system manager via price adaptations. We determine the mean number of downcrossings of any level x in one regenerative cycle, and use it to obtain the steady-state density f (x) of the inventory level. We also derive the rates of occurrence of normal deliveries and of emergency deliveries, and the steady-state probability of having zero inventory.

Original languageEnglish
Pages (from-to)106-126
Number of pages21
JournalJournal of Applied Probability
Issue number1
Publication statusPublished - 4 Mar 2023


  • (R, Q) inventory
  • level crossings
  • steady-state analysis
  • stochastic lead time


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