Question

Problem 15-9 (Algorithmic)

Marty's Barber Shop has one barber. Customers have an arrival rate of 2.3 customers per hour, and haircuts are given with a service rate of 4 per hour. Use the Poisson arrivals and exponential service times model to answer the following questions:

1. What is the probability that no units are in the system? If required, round your answer to four decimal places.
   
   P₀ =
   
2. What is the probability that one customer is receiving a haircut and no one is waiting? If required, round your answer to four decimal places.
   
   P₁ =
   
3. What is the probability that one customer is receiving a haircut and one customer is waiting? If required, round your answer to four decimal places.
   
   P₂ =
   
4. What is the probability that one customer is receiving a haircut and two customers are waiting? If required, round your answer to four decimal places.
   
   P₃ =
   
5. What is the probability that more than two customers are waiting? If required, round your answer to four decimal places.
   
   P(More than 2 waiting) =
   
6. What is the average time a customer waits for service? If required, round your answer to four decimal places.
   
   W_q =  hours

Answer 1

arrival rate λ=		2.3
service period=μ=		4.00

a)

probability of 0 people waiting in system =(1-λ/μ)*(λ/μ)0 =

0.4250

b)

probability that one customer is receiving a haircut and no one is waiting =P(1 unit in the system):

probability of 1 people waiting in system =(1-λ/μ)*(λ/μ)1 =

0.2444

c)

probability of 2 people in system =(1-λ/μ)*(λ/μ)2 =

0.1405

d)

probability of 3 people in system =(1-λ/μ)*(λ/μ)3 =

0.0808

e)

probability that more than two customers are waiting

probability that number of customer in system is 4 or greater=Pn=>k=(λ/μ)4 =

0.1093

f)

average time in queue Wq =λ/(μ(μ-λ))=

0.3382 hours

(please try 0.5882   if this comes wrong)