Dave's Photography Store has the following inventory problem. The store stocks a particular model camera that can be ordered weekly. Let D1,D2, represent the demand for this camera during the first week, second week, , respectively. So the random variable Dt is the number of cameras that would be sold in week t if the inventory is not depleted. This number includes lost sales when the inventory is depleted. It is assumed that the Dt are independent and identically distributed random variables having a discrete uniform distribution over {0,1,2,3}. Let X0 represent the number of cameras on hand at the outset, X1 the number of cameras on hand at the end of week 1 , and Xt is the number of cameras on hand at the end of week t. Assume X0=3, so that week 1 begins with three cameras on hand. Then {Xt}={X0,X1,X2,} is a stochastic process where the random variable Xt represents the state of the system at time t, namely, state at time t is the number of cameras on hand at the end of week t. As the owner of the store, Dave would like to learn more about how the status of this stochastic process evolves over time while using the current ordering policy described below. Whenever the number of cameras on hand at the end of the week is 0 or 1 , an order is placed that will bring this number up to 3 . Otherwise, no order is placed. a) Construct the state space of Xt. (10 points) b) Construct the one-step transition probability matrix P. (20 points) c) Find the steady state probability of Xt. (15 points) d) Suppose the camera store finds that a storage charge is being allocated for each camera remaining on the shelf at the end of the week. The cost is charged as follows: C(xt)=02818ififififxt=0xt=1xt=2xt=3 What is the expected average cost per unit time? (10 points) e) The order of z>0 units of cameras incurs ordering cost 10+25z. If no cameras are ordered, no ordering cost is incurred. For each unit of unsatisfied demand, there is a penalty of $50. Including the inventory holding cost in d ), what is the expected average cost per unit time? (25 points) f) Suppose that the current inventory level is 3 . What is the expected first time to deplete the stock? (Hint: the expected first passage time from state 3 to state 0). (20 points)