Question

A small barbershop, operated by one barber, has room for only one waiting customer. Potential customers arrive at a rate of 6 people per hour, and it takes an average of 15 minutes for the barber to serve a customer.

a) Find the steady state probabilities.

b) Find the probability that an arriving customer will be turned away.

c) Find the expected number of people in the barbershop.

d) Find the expected number of people in the barbershop.

e) Find the expected time spent by a customer (who enters the shop).

f) Find the expected waiting time of a customer (who enters the shop).

g) Give the pdf of the time spent by a customer who enters the barbershop.

h) Find the probability that a customer does not have to wait given that he enters.

i) Give the pdf of the waiting time of a customer who enters the barbershop

Answer 1

Arrival rate per minute, λ = 6/60=0.1

Service rate, µ = 60/15 = 4 customers per hour

ρ = λ/µ = 0.1/4 = 0.025

Lq = ρ2/(1-ρ) = (0.025^2)/(1-0.025) = 0.00641025

Wq = Lq/λ = 0.00641025/0.1=0.006410256

Ws = Wq + 1/µ = 0.006410256 + 1/4 = 0.256410256

Ls = λWs = 0.1*0.2564 = 0.025641025

a) Steady state probabillity: ρ = λ/µ = 0.1/4 = 0.025

b) Probability that an arriving customer will be turned away = (λ/µ)nP0 = 0.6191

n: No. of customers in the system = No. of customers being served + No. of customers waiting = 0.25641025

P0: Probablity of zero customers in the system = (1-ρ)/(1-ρn+1) = 1.59417

c) d) Expected number of people in the barbershop: N = λT = 0.1*0.256410256 = 0.25641025

e) Expected time spent by a customer (who enters the shop): T = 1(µ-λ) = 1/(4-0.1) = 0.256410256

f) Expected waiting time of a customer (who enters the shop): Ws = 0.256410256

g) pdf of the time spent by a customer who enters the barbershop = 1 - e-λt

h) Probability that a customer does not have to wait given that he enters = 1 – ρ = 1-0.025 = 0.975

i) pdf of the waiting time of a customer who enters the barbershop = λe-λt