Problem 1: Waiting Line Problem (12 points)

A small grocery store has one checkout line. Customers arrivals at the checkout line exhibit a Poisson arrival distribution with an average of 12 per hour. The cashier checks out customers at an average rate of 15 per hour. The service time is exponentially distributed.

a) What is the utilization of the cashier? (2 points)

Utilization = 12/15 = 80%

b) How many minutes, on average, does the cashier take to check out a customer? (2 points make sure your answer is in minutes)

Service Time = 60 mins per hour/ 15 hours = 4 minutes per customer

c) What is the average number of people in the checkout line? (2 points you can leave this as a decimal)

Average number in line = 144/15(3) = 3.2 people

d) How many minutes on average does a customer wait in the line? (2 points make sure your answer is in minutes) 12/15(3) x 60 = 16 minutes

e) What is the probability there will be no customers in the system? (2 points)

f) What is the probability there will be exactly 5 people arrive in the next hour? (2 points)

Could you please answer part E and F only.