Consider a small clinic staffed by one nurse who injects patients with an antifl u vaccine. The clinic has space for only four chairs. One chair is reserved for the patient being vaccinated. The other three chairs are reserved for patients who wait. The arrival process is deterministic. One new patient arrives every 20 min. A single patient arrives at the clinic at precisely on the hour, at precisely 20 min past the hour, and at precisely 40 min past the hour. Any patient who arrives to fi nd all four chairs occupied does not enter the clinic. If the clinic is empty, the nurse is idle. If the clinic is not empty, the number of patients that she vaccinates during a 20 min interval is an independent, identically distributed random variable. If V_n is the number of vaccinations that the nurse provides during the nth 20 min interval, then V_n has the following conditional probability distribution, which is stationary in time. 

The quantity pk = P(V_n = k) is a conditional probability, conditioned on the presence of at least k patients inside the clinic available to be vaccinated. To model this problem as a regular Markov chain, let the state X_n represent the number of patients in the clinic at the end of the nth 20 min interval, immediately before the arrival of the nth patient.

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4 5 or more Ps 123 Po P1 P2 P3 P4 Number V, of vaccinations in period n, V = k 0 Probability, P = P(V =k)