Consider an inventory demand process for some item that can be purchased or rented. That is, the item might or might not be returned after some period. Suppose that the demands follow a Poisson process with rate m and the returns form another Poisson process with rate l. Of course, the demand rate is greater than the return rate. Suppose that the replenishment policy is to order a quantity of Q (for example, 100) units as soon as the stock level reaches r (for example, 10). Backorders are not taken, so that demands that occur when the stock is empty are simply rejected. The lead time for the replenishment order to arrive is L time units. Using a continuous time Markov model with the state defined as the number of units on hand,

Draw the state-transition diagram, and write the associated differential equations for P(t) and for (t).