A queueing model with randomized depletion of inventory

In this paper we study an M/M/1 queue, where the server continues to work during idle periods and builds up inventory. This inventory is used for new arriving service requirements, but it is completely emptied at random epochs of a Poisson process, whose rate depends on the current level of the acquired inventory.
For several shapes of depletion rates, we derive differential equations for the stationary density of the workload and the inventory level and solve them explicitly. Finally numerical illustrations are given for some particular examples, and the effects of this depletion mechanism are discussed.