Question

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 injection

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 three minutes. It takes the nurse an average of 2.80 minutes to prepare the patients' serum and administer the injection.

a. What is the average number you would expect to see in Dr.Martin's facilities?

b. How long would it take for a patient to arrive, get an inspection, and leave?

c. What is the probaility that there will be three or more patients on the premises?

d. What is the utilization of the nurse?

e. Assume three nurses are available. Each takes an average of two minutes to prepare the patients serum and adminsiter the injection. What is the average total time of a patient in the system?

Answer 1

Arrival rate = λ = 60/3 = 20 patients/hour

Service rate = μ = 60/2.8 = 21.42 patients/hour

a. Average Number of patients in Dr. Martins facilities = λ/(μ-λ) = 20/1.42 = 14.08

b. Average number of patients waiting in queue L_q = λ²/μ(μ-λ) = 13.15

Average time patients spend in the system = 1/(μ-λ) = 1/1.42 = 0.70 hrs = 42.25 mins

c. Probability of no patients = P₀ = 1 - (λ/μ) = 0.066

Probability of 1 patient = P₁ = (λ/μ) * P₀ = 0.061

Probability of 2 patients = P₂ = (λ/μ)² * P₀ = 0.057

Probability of 3 or more patients = 1 - P₀ - P₁ - P₂ = 1 - 0.066 - 0.061 - 0.057 = 0.816

d. Utilization = λ/μ = 93.37%