Question

In a gas station there is one gas pump. Cars arrive at the gas station according to a Poisson proces. The arrival rate is 20 cars per hour. An arriving car finding n cars at the station immediately leaves with probability qn = n/4, and joins the queue with probability 1−qn, n = 0,1,2,3,4. Cars are served in order of arrival. The service time (i.e. the time needed for pumping and paying) is exponential. The mean service time is 3 minutes.
(i) Determine the stationary distribution of the number of cars at the gas station.
(ii) Determine the mean number of cars at the gas station.
(iii) Determine the mean sojourn time (waiting time plus service time) of cars deciding to take gas at the station.
(iv) Determine the mean sojourn time and the mean waiting time of all cars arriving at the gas station.

Answer 1