2. The director of the study abroad program at a college advises one, two, or three students at a time depending on how many students are waiting outside his office. The time for each advisement session, regardless of the number of participants, is exponential with mean 1/μ, independent of other advisement sessions and the arrival process. Students arrive at a Poisson rate of λ and wait to be advised only if two or fewer other students are waiting to be advised. Otherwise, they leave. Upon the completion of an advisement session, the director will begin a new session if there are students waiting outside his office to be advised. Otherwise, he begins a new session when the next student arrives.

Let X(t) = f if the director of the study abroad program is free and, for i = 0, 1, 2, 3, let X(t) = i if an advisement session is in progress and there are i students waiting outside to be advised. Show that



X(t) : t ≥ 0

is a continuous-time Markov chain and find πf , π0, π1, π2, and π3, the steady-state probabilities of this process.