A butcher buys chickens for his shop, and the sequence of events during each month for his shopping is as follows.

(1) He observes the number of chickens in the freezer (call it a) at the end of a month. 

(2) If a ≤ 9, 15 – a chickens are ordered. If a≥ 10, there is no order. Delivery of all ordered chickens is immediate, i.e., they are available at the beginning of the next month. 

(3) During the next month, the probability distribution for customers’ chicken demand is P{D=2}=1/8, P{D=3}=1/4, P{D=4}=1/8, P{D=5}=1/2. 

(4) He observes the number of chickens in the freezer at the end of the next month. 

Define a month’s state to be the number of chickens in the freezer at the beginning of the month (right after receiving the order given at the end of the previous month but before satisfying the demand of the month). Determine the transition matrix that could be used to model this problem as a Markov chain.