Question

Please do it by type not pic.

1.       The Heart Association plans t install a free blood pressure testing booth at the Palouse mall for a week. Previous experience indicates that, on average, 10 persons per hour request a test and the arrivals are Poisson distributed from an infinite population. Blood pressure measurements are exponential with an average time of 5 minutes per exam.

a.      What is the average number of people in line?

b.      What is the average number of people in the system?

c.      What is the average amount of time a person can expect to spend in line?

d.      What is the average amount of time it will take to measure a person's blood pressure, including waiting in line?

e.      What is the probability of having 3 people in line ?

Answer 1

Arrival rate, = 10 customers per hour

Service time= 5 minutes for a customer

--> Service rate, µ = 12 customers per hour

First, we find the classification:

Arrival rate-Poisson: Service rate- Exponential: 1 server – hence MM1 system-

a.      What is the average number of people in line?

Utilization factor, = percentage of time that the worker (server) is busy=

=/µ =10/12 = 0.83

Average number of people in line

= length of queue

=/(1-)

=0.83² /(1-0.83) =4.05 customer

b.      What is the average number of people in the system?

The number of people in the system is (waiting line + being served)

=/(µ-) =10/(12-10) =5 customers

c.      What is the average amount of time a person can expect to spend in line?

Average time spent in the line, Wq

=/ ( µ(1- ) ) =0.83/(12(1-0.83)) = 0.41 hour

d.      What is the average amount of time it will take to measure a person's blood pressure, including waiting in line?

Average time spent in the system (waiting line or drive through+ being served),W

=1/ µ( 1-) =1/12(1-0.83) =0.49 hour

e.      What is the probability of having 3 people in line?

Probability that there are zero customers in the system,

P₀= 1-

= 1-0.83

=0.17

Probability that there are n customers in the system     

= ⁿ .P₀

= ( utilization factor) ⁿ   * (Probability that there are zero customers in the system)

= (0.83)³ * (0.17)

=0.097 or 9.7%