Problem #7 To support National Heart Week, the Heart Association plans to install a free blood pressure testing booth in El Con Mall for the week. Previous experience indicates that on average, 10 persons per hour request a test. Assume arrivals are Poisson from an infinite population. Blood pressure measurements can be made at a constant time of 5 minutes each. Assume the queue length can be infinite with FCFS discipline

What average number in line can be expected? Lq = λ2/µ(µ- λ) Average number waiting in line= (arrival rate)2/service rate(service rate –arrival rate) 102/12(12-10) 4.17 4 customers can be expected

What average number of persons can be expected in the system? Ls = λ/µ- λ Average number in system= arrival rate/(service rate –arrival rate) 10/12-10 5 customers can be expected

What is the average amount of time that a person can expect to spend in line? Wq = Lq/ λ Average time waiting in line = average number waiting in line/arrival rate 4.17/10 0.417 hour ~25 minutes (25.02)

On the average, how much time will it take to measure a person’s blood pressure, including waiting time? Ws = Ls/λ Average total time in the system = Average number in system/arrival rate 5/10 0.5 hours 30 minutes

Using ratio and proportion: On weekends, the arrival rate can be expected to over 12 per hour. What effect will this have on the number in the waiting line? At 10 per hour: Lq = 102/12(12-10) = 4.17 Using ratio and proportion: (10/4.17)(12/x) 5 customers The number on the waiting line would increase by 1.

Problem #10 L. Winston Martin (an allergist in Tucson) 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 more 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 the 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 two minutes to prepare the patients’ serum and administer the injection

Average number in system= arrival rate/(service rate –arrival rate) What is the average number you would expect to see in Dr. Martin’s facility Ls = λ/µ-λ Average number in system= arrival rate/(service rate –arrival rate) 20/30-20 2 customers

How long would it take for a patient to arrive, get an injection, and leave? Ws = Ls/λ Average total time in the system = Average number in system/arrival rate 2/20 0.1 hour or 6 minutes

What is the probability that there will be three or more patients on the premises Pn = (1 – (λ/µ))(λ/µ)n Probability of exactly n units in the system = (1 – (arrival rate/service rate)) (arrival rate/service rate)average number of units in queuing system

Pn = (1 – (λ/µ))(λ/µ)n At n= 0, P0 At n = 1, 2 and 3 respectively (1- (20/30))(20/30)0 = 0.34 At n = 1, 2 and 3 respectively (1- (20/30))(20/30)1 = 0.22 (1- (20/30))(20/30)2 = 0.15 (1- (20/30))(20/30)3 = 0.1 TP = 0.22 + 0.15 +0.1 = 0.47 1-0.47 = 0.53 0.53 or 53 %

What is the utilization of the nurse? ρ = λ/µ Ratio of total arrival rate to service rate for a single server = arrival rate/service rate 20/30 67%

3 nurses at an average time of two minutes = 6 minutes 6-4 Assume three nurses are available. Each takes an average of two minutes to prepare the patients’ serum and administer the injection. What is the average total time of a patient in the system? Lq = 202/30(30- 20) 1.34 Wq = Lq/ λ .06 hour or 4 minutes 3 nurses at an average time of two minutes = 6 minutes 6-4 2 minutes average total time