Question

Benny the Barber gives a haircut in an average of 8 minutes.
HIs market survey data indicates that customers arrive at a rate of 4 per hour.
Benny knows that customers exhibit a Poisson arrival distribution and that he provides an exponential service distribution.

(Round intermediate calculations to 3 decimals & final answers to 2 decimals)
[a] What is the average number of customers waiting?
[b] What is the average wait time, per customer?
[c] What is the average time a customer is in the shop?
[d] What is the average utilization of Benny's time

Answer 1

Let ? = Arrival Rate = 4 per hour

Benny gives a haircut in an average of 8 minutes. Thus per hour, he gives 60/8= 7.5

? = Service Rate = 7.5

[a] What is the average number of customers waiting?

Lq = average number of customers waiting

      = ?^2 / ? (?- ?) = 4^2 / (7.5 * (7.5-4))=16/26.25= 0.60952 = 0.61

[b] What is the average wait time, per customer?

Wq= average wait time per customer = Lq/ ? = 0.61/ 4= 0.1525= 0.153hr = 0.153*60mins = 9.18 mins

[c] What is the average time a customer is in the shop?

W= average time a customer is in the shop

    = Wq + 1/ ?

    = 0.153 + 1/7.5 = 0.153 + 0.133= 0.286hr = 17.16 min

[d] What is the average utilization of Benny's time

P = average utilization of Benny’s time = ?/ ?= 4/7.5= 0.53= 53%