1. In an inventory system for a single product the depletion of stock is due to demand and deterioration. The demand process for the product is a Poisson process with rate A. The lifetime of each unit product is exponentially distributed with mean 1/. The stock control is exercised as follows. Each time the stock drops to zero an order for Q units is placed. The lead time of each order is negligible. Determine the average stock and the average number of orders placed per time unit. 2. An electronic system uses one operating unit but has built-in redundancy in the form of R standby units. The standby units are not switched on (cold standby). The operating unit has an exponentially distributed lifetime with mean 1/A. If the operating unit fails, it is immediately replaced by a standby unit if available. Each failed unit enters repair immediately and is again available after an exponentially distributed repair time with mean 1/. It is assumed that the mean repair time is much smaller than the mean lifetime. There are ample repair facilities (infinite number of repairmen). The system is down when all R + 1 units are in repair. (a) Let pj = limt Pj operating units are present at time t) for 0 js N. Give a recursion scheme to compute the pj. (b) What is the long-run average number of busy repairmen? (c) Proportion of the time the system is down (equivalent to the probability)?