Question

The owner of a barbershop has two barber chairs and one employee. The owner will cut a customer’s hair only if her employee is already busy cutting someone else’s hair. Service times are exponentially distributed with mean 1 hour, and the time between arrivals is exponentially distributed with mean 1.5 hours. When the barbershop is full(i.e., two customers are getting haircuts), people must wait outside in line.

a)What percentage of the time is the owner cutting hair?

b)If the owner wants to spend at most 10% of her time working, what is the maximum arrival rate that can be tolerated?

Answer 1

As in the question given there are two tables and one waiter and also when the second table becomes occupied the owner has to wait himself so in this case there is one server.when there are 2 customers then only the owner waits.

As given in question inter-arrival rate = 1.5 hours

so to calculate the arrival rate of customer i.e in 1 hr how many customers come to the restaurant we can do it by this way:-

arrival rate of customer($\lambda$) = $1/1.5=0.667$ customers per hour

Similarly, service rate($\mu$ ) as given will be 1 customer per hour.

so we can calculate the probability of waiting($\rho$) =$\lambda /\mu$=0.667

Now as per the first question we have to calculate when the second server i.e the owner has to wait.This case will arise when there are 2 customers in the system/restaurant.

so this will be a case we have to find the probability that there will be two customers in the system

we know the P_n= probability of having n customers in the system

P₀= probability of having zero customers in the system= 1-$\rho$=1-0.67=0.33

and P_n = $\rho ^{n-1}$ * P₀

so, the probability of having two customers P₂ = =0.22

a) so 22% of the time the owner has to wait in the tables

b) In the 2nd part, it is given that owner can maximumly devote 10% of the time i.e. maximum 10% of the time there should be 2 customers in the system.So.P₂ = 0.1(given)

putting this in the above equation and solving

P₂ = $\rho ^{2-1}$ * P₀ = $\rho$ * (1- $\rho$ )

=> 0.1 =  $\rho$ - $\rho ^{2}$ => $\rho ^{2}$- $\rho$ +0.1 = 0

solving the above quadratic equation we get two values of $\rho$ i.e.0.887 and 0.112 obviously 0.887 will not be possible as our service rate is fixed i.e 1 so we consider second one i.e 0.112

so our arrival rate should be 0.112 customers per hour