In Springfield, Massachusetts, people drive their cars to a state inspection center for annual safety and emission certification at a Poisson rate of λ. For n ≥ 1, if there are n cars at the center...

In Springfield, Massachusetts, people drive their cars to a state inspection center for annual safety and emission certification at a Poisson rate of λ. For n ≥ 1, if there are n cars at the center either being inspected or waiting to be inspected, the probability is 1−α_n
that an additional driver will not join the queue and will leave. A driver who joins the queue has a patience time that is exponentially distributed with mean 1/γ . That is, if the car’s inspection turn does not occur within the patience time, the driver will leave. Suppose that cars are inspected one at a time, inspection times are independent and identically distributed exponential random variables with mean 1/µ, and they are independent of the arrival process and patience times. Let X(t) be the number of cars being or waiting to be inspected at t. Find the birth and death rates of the birth and death process $ X(t): t ≥ 0 % .