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 1.50 minutes, and the arrival rate is 30 per hour (Poisson distributed).

a. The average wait time in the line _____minutes (round your response to two decimal places).
b. The average number of customers waiting in the line _____ customers (round your response to three decimal places)
c. Mehmet raises the price to? $5 per cup and takes 30 seconds off the service time?

However, because the coffee is now so expensive, the arrival rate drops to 24 per hour. The average wait time in the line _____minutes (round your response to two decimal places). The average number of customers waiting in the line _____ customers ?(round your response to three decimal places.

Answer 1

a. Mean arrival rate = λ = 30

Service rate = μ= 60/1.50 = 40

Average wait time = λ / 2μ(μ - λ)

Average wait time = 30 / 2*40*(40-30)

Average wait time = 0.0375

The average wait time in minutes = 0.0375 hrs * 60 minutes = 2.25 minutes

b. The average number of customers = Lq = λ² / 2μ(μ - λ)

= 30² / 2*40*(40-30)

= 1.125

= 1.13

c. μ = 60 / (1.50−0.5) ​ = 60

Average wait time = λ / 2μ(μ - λ)

Average wait time = 24 / 2*60*(60-24)

Average wait time = 0.0069

The average wait time in minutes = 0.0069 hrs * 60 minutes = 0.414 minutes

The average number of customers = Lq = λ² / 2μ(μ - λ)

= 24² / 2×60×(60−24)

= 0.1333

= 0.133