Question

Please do it by type not pic.

1.You’re working at a local hospital and have been given the task of planning to provide flu shots for the upcoming flu season. You anticipate very heavy demand because of the nature of this season’s flu. You anticipate that patients will arrive at the rate of 24 per hour and it will take 3 minutes to prep each patient, provide the vaccine and take care of the paperwork.

a.Your management wants people to wait in line for no more than one minute. How many nurses will you require to meet this requirement (round to nearest minute)?

b.If each nurse costs you $20 per hour and you’ve calculated the cost of a customer waiting to be $40, how much will the number of nurses arrived at in part a. cost the hospital?

Answer 1

Arrival rate, a = 24 patients per hour

Service rate, s = 60 minutes per hour / 3 minutes per patient = 20 patients per hour

Arrival rate is more than service rate. Therefore, multiple servers (nurses) are needed

Minimum number of nurses required = a/s = 24/20 = 1.2 ~ 2

Starting with 2 nurses, evaluate the waiting time in line (Wq)

For k number of nurses, k=0, (a/s)^k/k! = (24/20)^0/0! = 1

k=1, (a/s)^k/k! = (24/20)^1/1! = 1.2

k=2, (a/s)^k/k! = (24/20)^2/2! = 0.72

Cumulative sum of this expression (for k=0,1) = 1+1.2 = 2.2

term 2 = ((a/s)^k/k!)/(1-a/ks) (where k=2)  = ((24/20)^2/2!)/(1-24/(2*20)) = 1.8

P0 = 1/(2.2+1.8) = 0.25

Average time a caller is put on hold, Wq = P0*((a/s)^k/k!)*(a/ks)/(1-(a/ks))^2/a (where k=3)

= 0.25*((24/20)^2/2!)*(24/(2*20))/(1-24/(2*20))^2/24

= 0.028125 hour

= 1.7 minutes

This is more than 1 minutes.

Therefore, we evaluate with 3 nurses

_____________________________

3 nurses

For k number of nurses, k=0, (a/s)^k/k! = (24/20)^0/0! = 1

k=1, (a/s)^k/k! = (24/20)^1/1! = 1.2

k=2, (a/s)^k/k! = (24/20)^2/2! = 0.72

k=3, (a/s)^k/k! = (24/20)^3/3! = 0.288

Cumulative sum of this expression (for k=0,1,2) = 1+1.2+0.72 = 2.92

term 2 = ((a/s)^k/k!)/(1-a/ks) (where k=3)  = ((24/20)^3/3!)/(1-24/(3*20)) = 0.48

P0 = 1/(2.92+.48) = 0.2941

Average time a caller is put on hold, Wq = P0*((a/s)^k/k!)*(a/ks)/(1-(a/ks))^2/a

= 0.2941*((24/20)^3/3!)*(24/(3*20))/(1-24/(3*20))^2/24

= 0.003921 hours

= 0.2353 minutes

This is less than 1 minutes. Therefore, 3 nurses are required

b) Nurse cost per hour, Cs = $ 20

Cost of customer waiting, Cw = $ 40

Average number of patients waiting in queue (Lq) = a*Wq = 24*0.003921 = 0.0941

Total hourly cost = Cost of nurses + Waiting cost of patients

= Cs*k + Cw*Lq

= 20*3 + 40*0.0941

= $ 63.76 per hour