Question

​Altug's Coffee Shop decides to install an automatic coffee vending machine outside one of its stores to reduce the number of people standing in line inside. Mehmet Altug charges​ $3.50 per cup. ​ However, it takes too long for people to make change. The service time is a constant 4.00 ​minutes, and the arrival rate is 12 per hour​ (Poisson distributed). ​

a. The average wait time in the line ​
b. Mehmet raises the price to $5 per cup and takes 60 seconds off the service time. However, because the coffee is now so expensive, the arrival rate drops to 10 per hour.
c. What is theaverage number of customers waiting in the line.

Answer 1

The average time between two successive arrivals at Altug's Coffee Shop is 5 minutes before the price change and 6 minutes after the price change and arrival rate reduction. The average time for three customers to arrive is 15 minutes before the price change and 18 minutes after the price change.

Step-by-step explanation:

To calculate the average wait time in the line for Altug's Coffee Shop before adjusting the price, we use the data that the service time is a constant 4 minutes and the arrival rate is 12 customers per hour, which is Poisson distributed.

a. The average wait time in the line

Since there are 60 minutes in an hour and customers arrive at a rate of 12 per hour, on average, the time between successive arrivals is 5 minutes. Therefore, it is 5 minutes on average between two successive arrivals.

b. When the arrival rate is reduced

After Mehmet raises the price to $5 per cup and shortens the service time by 60 seconds, the arrival rate drops to 10 per hour. Thus, on average, one customer arrives every 6 minutes, and for three customers to arrive, it would take 18 minutes on average.

c. The average number of customers waiting in the line

Since we're not given a specific formula or additional data to calculate the actual average number of customers waiting, we would need either the average queue length formula or additional information on the system's capacity and utilization rate to provide an accurate number.

Answer 2

a. Wₙ = 0.1333 hours or 8 minutes

b. The average number of people in line is 1,6

c. average wait time is 0.0083 hours or 0.5 minutes

Step-by-step explanation:

The arival rate is λ and μ service rate.

μ= 60/4 = 15 cars/hour

Arrival rate is 12 per hour

a. Using the given formula Wₙ =λ / 2μ(μ-λ)

Wₙ =12 / 2*15*(15-12) = 12/90 = 0.1333 hours or 8 minutes

b. Determine Lₙ = Wₙ =λ
`λ^(2)` / 2μ(μ-λ)

Lₙ=12^2/2*15*(15-12) = 1,6

The average number of people in line is 1,6

c. μ= 60/2 = 30 cars/hour

Wₙ =10 / 2*30*(30-10) = 10/(60*20) = 0.0083 hours or 0.5 minutes

Hence average wait time is 0.0083 hours or 0.5 minutes