L. Winston Martin (an allergist in Chicago) has an excellent system for handling his regular patients who come in just for allergy injections. Patients arrive for an injection and fill out a name slip, which is then placed in an open slot that passes into another room staffed by one or two nurses. The specific injections for a patient are prepared, and the patient is called through a speaker system into the room to receive the injection. At certain times during the day, patient load drops and only one nurse is needed to administer the injections.

Let’s focus on the simpler case of the two—namely, when there is one nurse. Also assume that patients arrive in a Poisson fashion and the service rate of the nurse is exponentially distributed. During this slower period, patients arrive with an interarrival time of approximately eight minutes. It takes the nurse an average of four minutes to prepare the patients’ serum and administer the injection.