Every day, a number of automobiles arrive in the morning at Casey’s Auto Body Shop. Casey has room for one car in the shop and up to two on the lot.

Every day, a number of automobiles arrive in the morning at Casey’s Auto Body Shop. Casey has room for one car in the shop and up to two on the lot. If there is no room for a car, Casey is forced to send it to his competitor, Bob’s Body Boutique. Once a car has been dropped off at Casey’s it remains there until it is repaired. It takes a whole day to repair a car, but once repaired, it is gone by the following morning. Assume the number of autos that arrive each morning follows a Poisson(µ) distribution, and that the numbers on different days are independent. The situation at Casey’s can be viewed as a five-state Markov chain, where the states are 0, 1, 2, 3, B, where the numeric states refer to the total number of cars at Casey’s, and the state B is that there are 3 cars at Casey’s and he had to send some overflow to Bob. 

1. Draw the graph corresponding to this Markov chain. 

2. Write down the transition matrix for this in terms of the Poisson pmf pµ(k) 

3. For µ = 0.2 and µ = 1.5, write down the numeric transition matrix. 

4. Today, Casey has no cars. For µ = 0.2 and µ = 1.5, what is the numeric probability of each state in 30 days. 

5. For µ = 0.2 and µ = 1.5, numerically find the steady-state (invariant) distribution of the Markov chain. 6. For µ = 0.2 and µ = 1.5, what is the steady-state probability that Casey will send Bob some customers on April 1. (Hint: It’s the probability of state B under the invariant distribution.)

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