Question

. Peter International Barbershop is a popular haircutting and styling saloon near the

campus of the B College. One barber is available to work full time and spend an

average of 3 minutes on each customer. Customers arrive all day long at an average rate

of 4 minutes Arrivals tend to follow the Poisson distribution, and service times are

exponentially distributed. Explain your results.

1. What is the probability that the shop is empty?

(b) What is the average number of customers in the barbershop?

(c) What is the average time spent in the shop?

4. What percentage of time the barber is busy?
5. What is the average number of customers waiting to be served?
6. What is the probability of more than 3 customers at the Barbershop?

Answer 1

Arrival rate (lambda) = 60/4 = 15 customer per hour

Service rate (mu) = 60/3 = 20 customers per hour

Utilization (rho) = 15/20 = 0.75 or 75%

a.

Probability of empty shop (P0) = 1 – rho = 1 – 0.75 = 0.25

b.

Average number of customers in the shop (L) = lambda/(mu-lambda) = 15/(20-15) = 3

c.

Average time spent in the shop (W) = 1/(mu-lambda) = 1/(20-15) = 0.2 or 12 minutes

d.

The barber is busy for the utilization percentage of 75% of the time

e.

Average number of customers waiting (Lq) = lambda^2/(mu*(mu-lambda)) = 15*15/(20*(20-15)) = 2.25

f.

Probability of no customer = 0.25

Probability of n customer = [(lambda/mu)^n]*P0

Probability of 1 customer = (15/20)*0.25 = 0.1875

Probability of 2 customers = (15/20)^2 *0.25 = 0.1406

Probability of 3 customers = (15/20)^3 *0.25 = 0.1054

Probability of 3 customers or less = 0.1054 + 0.1406 + 0.1875 + 0.25 = 0.6835

Probability of more than 3 customers = 1 - 0.7835 = 0.3165