Question

A barber owns a one-chair shop. At barber college, he was told that his customers would exhibit a Poission arrival distribution and that he would provide an exponential service distribution. His market survey data indicate that customers arrive at the rate of two per hour and that it takes him an average of 20 minutes to give a haircut. Based on these figures, find the following:

The average number of customers waiting in line. (2 pts)

The average time a customer waits in line (2 pts)

The average time a customer is in the shop. (2 pts)

The average number of customers in the shop. (2 pts)

The barber would like to increase his utilization to .85. Given he cannot speed up giving haircuts, how often would customers have to arrive for him to achieve his goal? (round to two decimal places). (2 pts)

Answer 1

Given are following data :

Arrival rate = a = 2 per hour

Service rate ( to give haircut ) = S = 60/20 = 3 per hour ( @ 20 minutes to give one haircut )

Average number of customers waiting in the line = a^2/ S x ( S – a ) = 2 x 2 / 3 x ( 3 – 2 ) = 4/3 = 1.33

Average time customer waits in line= a / S x ( S – a ) = 2 / 3 x ( 3 -2 ) hour = 2/3 hour = 40 minutes

Average time customer is in the shop = a/ S x ( s – a ) + 1/S = 2/ 3 x ( 3-2) + 1/3 hour = 1 hour

Average number of customers in the shop

= a^2/S x ( S – a ) + a/S

= 2 x 2 / 3 x ( 3 -2 ) + 2/3

= 4/3 + 2/3

= 2

AVERAGE NUMBER OF CUSTOMERS IN THE SHOP = 2

Let the required arrival rate = a1 per hour for untilization of 0.85

Therefore . a1/S = a1/3 = 0.85

Hence a1 = 3 x 0.85 = 2.55

CUSTOMERS TO ARRIVE 2.55 TIMES IN AN HOUR TO ACHIEVE HIS GOAL OF 0.85 UTLIZATION LEVEL