Consider the following blood inventory problem facing a hospital. There is need for a rare blood type, namely, AB Rh-negative blood. The demand D (in pints) over any 3-day period is given by P[D=0]=0.4,P[D=1]=0.3,P[D=2]=0.2, and P[D=3]=0.1.
Note that the expected demand is 1 pint, since E[D]=0.3(1)+0.2(2)+0.1(3)=1. Suppose that there are three days between deliveries. The hospital proposes a policy of receiving 1 pintat each delivery and using the oldest blood first. If more blood is required than is on hand,then an expensive emergency delivery is made. Blood is discarded if it is still on hand after 21days. Denote the state of the system as the number of pints on hand just after a delivery. Thus, because of the discarding policy, the largest state is 7 .
(a) Construct the transition matrix for this Markov chain.