In a small train station, there are two counters where the travelers can buy their tickets, but the customers form a single waiting line. During the slack hours, only one counter is manned continuously by a clerk. When there is at least one customer waiting to be served, the second clerk opens his counter. When this second clerk finishes serving a customer and there is nobody waiting, he goes back to attending to other tasks. We suppose that the clerks both serve in a random time having an exponential distribution with parameter 𝜇 and that, during the slack hours, the customers arrive according to a Poisson process with rate λ. We also suppose that the slack period lasts long enough for the process to reach a stationary regime. The state X(t) of the system is defined as being the total number of customers present in the system at time t.
(b) Write the balance equation for the state I2 corresponding to the case when only the second clerk is busy (serving a customer).
(c) What fraction of time is the second counter open?