Question

Benny the Barber owns a one-chair shop. At barber college, they told Benny that his customers would exhibit a Poisson arrival distribution and that he would provide an exponential service distribution. His market survey data indicate that customers arrive at a rate of 3.0 per hour. It will take Benny an average of 17 minutes to give a haircut. Based on these figures, find the following:a. What is the average number of customers waiting? (Round your answer to 4 decimal places.)b. What is the average time a customer waits? (Round your answer to nearest whole number.)c. What is the average time a customer is in the shop?d. What is the average utilization of Benny's time? (Round your answer to the nearest whole percent.)

Answer 1

Answer:

Below is the solution to the given problem

Explanation:

a. What is the average number of customers waiting?

With one barber and exponential service, this system fits Model 1 in the text. λ = 3.000 per hour (given), μ= 60/17 = 3.529 per hour.

You are looking for Lq here.

Lq = λ$^{2}$ / μ(μ – λ) = $3.00^{2}$/3.529 (3.529 – 3.000) =  = 4.82 customers

b. What is the average time a customer waits?

Wq = Lq/λ = 4.82/3.00 = 1.607 hours  = 96.40 minutes  

c. What is the average time a customer is in the shop?

You are looking for Ws here, and need to calculate Ls first.

Ls =    λ / μ(μ – λ) = 3.00/3.529 – 3.000= 5.671

Ws =  Ls/λ = 5.671/3.00 = 1.890 hours  = 113.4 minutes

d. What is the average utilization of Benny's time?

ρ =  λ/μ = 3.00/3.529 = 0.85 = 85.0%