Second Time Around sells popular used items on consignment. Its operation can be viewed as an inventory problem in which the stock is replenished and depleted randomly according to Poisson distributions with rates l and m items per day. Every time unit the item is out of stock, Second Time loses $C1 because of lost opportunities, and every time unit an item is held in stock, a holding cost $C2 is incurred.

(a) Develop an expression for the expected total cost per unit time.

(b) Determine the optimal value of r = lm

. What condition must be imposed on the relative values of C1 and C2 in order for the solution to be consistent with the assumptions of the 1M/M/12: 1GD//2 model?