Drivers arrive according to a Poisson process with rate λ to fill up their cars at a service station where there are two employees who serve at the exponential rates μ₁ and 𝜇₂, respectively. However, only one employee works at a time serving gasoline. Moreover, there is space for only one waiting car. We suppose that

• when the system is empty and a customer arrives, employee no. 1 fills up the car,
• when employee no. 1 (respectively, no. 2) finishes filling up a car and another car is waiting, there is a probability equal to pi (resp.,P2)that this employee will service the customer waiting to be served, independently from one time to another.

Finally, we suppose that the service times are independent random variables.

(b) Write the balance equations of the process.

(c) Calculate, in terms of the limiting probabilities, the probability that

(i) an arbitrary customer entering the system will be served by employee no. 2,

(ii) two customers arriving consecutively will be served by different employees, given that the first of these customers arrived while there was exactly one car, being filled up by employee no. 1, in the system.