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Chapter 6: Problem 13

A small barbershop, operated by a single barber, has room for at most two customers. Potential customers arrive at a Poisson rate of three per hour, and the successive service times are independent exponential random variables with mean \(\frac{1}{4}\) hour. (a) What is the average number of customers in the shop? (b) What is the proportion of potential customers that enter the shop? (c) If the barber could work twice as fast, how much more business would he do?

Short Answer

Expert verified
The average number of customers in the shop is 3. The proportion of potential customers that enter the shop is 75%. If the barber could work twice as fast, he would do 12.5% less business.

Step by step solution

01

Calculate arrival rate, service rate, utilization factor, average number of customers in the system and probability that the system is empty

We have the following parameters: Arrival rate (λ) = 3 customers per hour Service rate (μ) = 4 customers per hour Using these values, we can calculate the utilization factor (ρ), the average number of customers in the system (L), and the probability that the system is empty (P0). Utilization factor (ρ) = λ/μ = 3/4 Average number of customers in the system (L) = ρ / (1 - ρ) = (3/4) / (1 - 3/4) Probability that the system is empty (P0) = 1 - ρ = 1 - 3/4
02

Answer to part (a): Average number of customers in the shop

Average number of customers in the system (L) = (3/4) / (1 - 3/4) = 3. So, there are on average 3 customers in the shop.
03

Answer to part (b): Proportion of potential customers that enter the shop

Probability that a potential customer enters the shop (1 - P0) = 1 - (1 - 3/4) = 3/4. So, 3 out of 4 potential customers (75%) enter the shop.
04

Calculate business increase if the barber works twice as fast

We are given that the barber could work twice as fast, so the new service rate (μ') would be: New service rate (μ') = 2 * μ = 2 * 4 = 8 customers per hour Now, calculate the new utilization factor (ρ'), average number of customers in the new system (L'), and probability that the new system is empty (P0'). New utilization factor (ρ') = λ/μ' = 3/8 Average number of customers in the new system (L') = ρ' / (1 - ρ') = (3/8) / (1 - 3/8) Probability that the new system is empty (P0') = 1 - ρ' = 1 - 3/8 The business increase would be the difference in proportion between the old and new systems: Business increase = (1 - P0') - (1 - P0) = P0 - P0' = 1/4 - 5/8 = -1/8
05

Answer to part (c): How much more business would the barber do?

Business increase = -1/8. However, since the negative sign implies a decrease, we can say that the barber would do 1/8 or 12.5% less business if he worked twice as fast. This may seem counterintuitive, but working twice as fast would make potential customers less likely to enter the shop when there is already one customer inside, as they will expect the service to be very fast. The barber would finish his current task faster, but less customers will enter the shop on average.

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Most popular questions from this chapter

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