Question

Steph Curry’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. The shop charges $3.50 per cup. The service time is a constant 2.8 minutes, and the arrival rate is 15 per hour (Poisson distributed).

a) What is the average waiting time in line?

b) What is the average number of people in line?

c) The shop decides to raise 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. Now what are the average wait time and the average number of people in the queue (waiting)?

Answer 1

Steph Curry’s coffee shop:

Given:

Arrival rate λ = 15 per hour

Service rate µ = (1/2.8)*60 = 21.429 customers per hour

Service rate = 1/service time. The service time is in minutes, so multiplied by 60 to get the rate in customers per hour.

It is mentioned in the scenario that the arrival is Poisson distributed and the service rate is constant. So we will use M/D/1 model.

(a) Average wait time in the line

(b) Average number of people in line

(c) 60 seconds off means 60 seconds reduced from total service time. Service time = 2.8 minutes = 168 seconds

New service time = 168 – 60 = 108 seconds = 1.8 minute

Service rate µ= (1/1.8)*60 = 0.555*60 = 33.333 customers per hour

Arrival rate = λ = 10 per hour

Average wait time in the line

Average number of people in line

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