Question

Agan Interiors provides home and office decorating assistance to customers. In normal operation 2.5 customers arrive per hour. One design consultant answers problems. The consultant averages 10 minutes per customer. Arrivals follow a Poisson distribution and the service times are exponentially distributed.
Required:
a. Compute the operating characteristics of the customer waiting line, assuming Poisson arrivals and exponential service times.
b. Service goals dictate that an arriving customer should not wait for service more than an average of 7 minutes. Is this goal being met? If not, what action do you recommend?
c. If the consultant can reduce the average time spent per customer to 9 minutes, what is the mean service rate?

Answer 1

Explanation:

we find the mean service rate at 10 minutes

= 60/10 = 6 min per hour

λ = 2.5

a.

1. we find the average number that are waiting in line

Lq = 2.5²/6(6-2.5)

= 6.25/21

= 0.2976

2. we find the average customers that are in this system

= 2.5²/6(6-2.5) + 2.5/6

= 0.2976 + 0.4167

L = 0.714266

approximately 0.7143

3. we have to determine the average time that a customers stays waitong

= Lq/λ

= 0.2976/2.5

= 0.11904 hours.

we convert this to minutes

= 0.11904 x 60

Wq = 7.1424 minutes

4. we find the average time that a customer is going to stay in the system

= 7.1424 + 60/6

w = 17.14 minutes

b. this goal is not being met here. This is because the service wait time is 7.14 minutes which is greater than 7 minutes. In order for them to  meet this goal, they either have to hire other consultants or they have to raise their mean service rate.

c. mean would be =

60/9 = 6.67 per hour

Wq = 2.5/6.67(6.67-2.5)

= 2.5/27.814

= 0.0899 hour

= 0.0899*60

= 5.4 minutes