Cars arrive at a parking lot according to a Poisson process with rate λ. There are only four parking spaces, and any car that arrives when all the spaces are occupied is lost. The parking duration of...

Cars arrive at a parking lot according to a Poisson process with rate λ. There are only four parking spaces, and any car that arrives when all the spaces are occupied is lost. The parking duration of a car is exponentially distributed with mean 1/μ. Let p_k(t) denote the probability that k cars are parked in the lot at time t, k = 0, 1,..., 4.

a. Give the differential equation governing p_k(t).

b. What are the steady-state values of these probabilities?

c. What is m₁₄, the mean first passage time to state 4 given that the process started in state 1?