Marty's Barber Shop has one barber. Customers have an arrival rate of 2.2 customers per hour, and haircuts are given with a service rate of 5 per hour. Use the Poisson arrivals and exponential service times model to answer the following questions:

What is the probability that one customer is receiving a haircut and one customer is waiting?

What is the probability that one customer is receiving a haircut and two customers are waiting?

What is the probability that more than two customers are waiting?

Step-by-step explanation:

Arrival rate = ∧ = 2.2 customers per hour

Service rate = u = 5 customers per hour

1. Probability that one customer is receiving a haircut and one customer is waiting

P (2 customers) = (∧/u) ^2 * (1-∧/u) = (2.2/5) ^2 * (1-2.2/5) = 0.1936*0.56 = 0.108416

2. Probability that one customer is receiving a haircut and two customers are waiting

P (3 customers) = (∧/u) ^3 * (1-∧/u) = (2.2/5) ^3 * (1-2.2/5) = 0.085184

* 0.56 = 0.04770304

3. Probability that more than two customers are waiting

P (more than 3 customers) = 1 - P (less than 3 customers) =

1 - [P (0) + P (1) + P (2) + P (3) ]=

= 1 - [ (1-2.2/5) + 2.2/5 * (1 - 2.2/5) + 0.108416+0.04770304]=1-0.9625=0.0375

3. Probability that more than two customers are waiting =