A small company maintains a fleet of four cars to be driven by its workers on business trips. Requests to use cars are a Poisson process with rate 1.5 per day.

A car is used for an exponentially distributed time with mean 2 days. Forgetting about weekends, we arrive at the following Markov chain for the number of cars in service.

0 1 2 3 4 0 1:5 1:5 0 0 0 1 0:5 2:0 1:5 0 0 2 0 1:0 2:5 1:5 0 3 0 0 1:5 3 1:5 4 0 0 0 2 2

(a) Find the stationary distribution.

(b) At what rate do unfulfilled requests come in?

How would this change if there were only three cars?

(c) Let g.i / D EiT4. Write and solve equations to find the g.i /.

(d) Use the stationary distribution to compute E3T4.