A barber who has six chairs to accommodate people waiting for hair cut. Assume that customers
who arrive when all the six chairs are full leave without entering the shop. Customers arrive at the
per hour and spend an average of 15 minutes for service. Find: (a) The probabilit
that a customer can get directly into the barber chair upon arrival. (b) Expected number of
customers waiting for a haircut. (c) Effective arrival rate. (d) The time a customer can expect to
spend in the barber shop
Answers
Step-by-step explanation:
Given A barber who has six chairs to accommodate people waiting for hair cut. Assume that customers who arrive when all the six chairs are full leave without entering the shop. Customers arrive at the rate of 3 per hour and spend an average of 15 minutes for service. Find: (a) The probabilit y that a customer can get directly into the barber chair upon arrival. (b) Expected number of customers waiting for a haircut. (c) Effective arrival rate. (d) The time a customer can expect to spend in the barber shop
Now M is maximum number in the system = 7
Given λ = 3 / hr and μ = 4 / hr
1. Now probability a customer can get directly into chair upon arrival is the probability of no one in the shop.
So P = 1 – λ / μ / 1 – (λ / μ )^k + 1
= 1 – 3 / 4 / 1 – (3/4)^7 + 1
= 1/4 / 1 – 0.100112915
= 0.25 / 0.899887085
P = 0.27781
2. Expected numbers waiting for a haircut is
L = 3/4 / 1 – 3/4 - 8 x (3/4 )^8 / 1 – (3/4 )^8
= 3 – 0.1112505298 / 0.899887085
L = 2.88
Now L q = L – (1 – P)
= 2.88 – (1 – 0.27781)
= 1.3878
3. Now effective arrival rate will be
So λ = μ (1 – P)
= 4 (1 – 0.27781)
= 4 x 0.72219
= 2.89 / hr
4. We know that W = 1/ λ effective
= 2.88 / 2.89
= 0.99 hr
= 59.4 min