Consider a model for inventory where inventory is depleted and replenished according to Poisson processes. Thus times between depletions are iid exponential with mean 1/u and times between arrivals of new items are also iid exponential with mean 1/a, and the two processes are independent. No backlogging is allowed, so unsatisfied demand just disappears. Suppose that every unit of time that the inventory is out of stock a penalty C2 is incurred. Howere, there is also a holding cost C1n for every unit of time that there are n units of stock on hand, with C2 > C1. assue a
(1) what is the long run average cost in the system ?
(2) What is the optimal value of p= a/u ?