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 2.0 per hour. It will take Benny an average of 25 minutes to give a haircut. Based on these figures, find the following:

The average number of customers waiting.

Answer 1

Using queueing theory, the average number of customers waiting at Benny the Barber's one-chair shop, with an arrival rate of 2 per hour and service rate of 2.4 per hour, is calculated to be 9 customers.

Step-by-step explanation:

To determine the average number of customers waiting at Benny the Barber's shop, we can utilize queueing theory concepts. With customers arriving at a rate of 2 per hour and Benny taking an average of 25 minutes to give a haircut, we can calculate the traffic intensity (often represented by λ/μ, where λ is the arrival rate and μ is the service rate).

Since Benny can service 60/25 = 2.4 customers per hour, the traffic intensity is 2/2.4 = 0.8333.

To find the average number of customers in the system (L), we can use the formula L = λ/(μ - λ) when 0 < λ < μ.

Plugging the values into the formula, we get L = 2/(2.4 - 2) = 10 customers in the system.

This average includes both the customer being serviced and those waiting.

Therefore, the average number of customers waiting (Lq) is L - 1/customer being serviced = 10 - 1 = 9 customers.