Potential customers arrive at a full-service, two-pump gas station according to a Poisson process at a rate of 40 cars per hour. There are two service attendants to help customers, one for each pump. If the two pumps are busy, then arriving customers wait in a single queue, to be served in the order of arrival by the first available pump. However, customers will not enter the station to wait if there are already two customers waiting, in addition to the two in service. Suppose that the amount of time required to service a car is exponentially distributed with a mean of three minutes.
I wonder how to find Markov chain model, which equivalently asking what is q(i,j) for state spaces for this problem. This is textbook question, and the textbook question is asking for long run fraction, and the solutions are easily found on google, but I wonder how to find the generator matrix q(i,j). Thanks!