A dental clinic at which only one dentist works is open only two days a week. Du

Question

A dental clinic at which only one dentist works is open only two days a week. During those two days, patients are arriving at the rate of three per hour. The doctor serves patients at the rate of one every 10 minutes.

a) What is the probability that the clinic is empty (except for the dentist)? (1 point)

b) What is the probability that there are five or more patients in the system? (1 point)

c) What is the probability that there are fewer than three patients in the system? (2 points)

I got 7/8 for this.

d) What is the probability that there are six patients in the system? (2 points)

Explanation / Answer

Arrival rate, = 3 per hour

Service rate, = 60/10 = 6 per hour

Utilization, = / = 3/6 = 0.5

a) Probability that the clinic is empty (except for the dentist), P(n=0) =(1-)*n = (1-0.5)*0.50  = 1/2 = 0.5

b) Probability that there are five or more patients in the system, P(n>=5) = 1-(P0+P1+P2+P3+P4) = 1-(0.5+(1-0.5)0.51+(1-0.5)*0.52+(1-0.5)*0.53+(1-0.5)*0.54) = 1/32 = 0.03125

c) Probability that there are fewer than three patients in the system P(n<3) = (1-)*0 + (1-)*1 + (1-)*2 = 0.5 + (1-0.5)*0.51 + (1-0.5)*0.52 = 7/8 = 0.875

d) Probability that there are six patients in the system, P(n=6) = (1-)*6 = (1-0.5)*0.56 = 1/128 = 0.0078125

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