Customers arrive at a full-service gas station, with two pumps, at rate of 10 cars per hour. However, customers will go to another station if there are at least four cars in the station: i. e., two being served and two waiting. Suppose that the service time for customers is exponential with mean 10 minutes. Denote by Xt the number of cars at the gas station at time t (both waiting and being served) (a) Write down the generator matrix for (Xt).
(b) In the long-run, how many cars per hour are served at the station?
(c) If the gas station is currently empty, what is the expected time until it becomes ull
(d) Assume that there is currently one car at the gas station. Find a quadratic approximation of the probability that there are exactly 2 cars at the station at time t. In other words, find the constants a b and c, such that p (1,2)+ bt +ct2 +o(t2), where (pi(i, j) are the transition probabilitics of this Markov process, and the residual term is such that it vanishes faster than t: o(t)/t20, as t0