Introduction: This case study analyzes the check-out processes at a large retail store with the objective of determining the most efficient queue system. The two queue models under consideration are:

A single queue system where the first person in line goes to the first available cashier (e.g., Marshalls) and that has five cashiers available;

Five separate queues with one cashier assigned to each line (e.g., Walmart).

Store Background: The store has a total of five cashiers on duty. The rate of customer arrivals for check-out is 990 people per hour, while each cashier can serve an average of 200 people per hour. Both interarrival times and service times are assumed to follow an exponential distribution.

Methodology: To compare the performance of the two queue models, utilize the Excel file provided in the Week 15 folder. The file contains relevant data and calculations needed to evaluate the efficiency of each queue system.

1. At first, let us assume a system with one single line leading to the five cashiers. What type of queue is this? (10 Points)

M/M/1

M/M/5

M/G/1

M/G/5

2. What is the expected number of people in the system (including people waiting in line and being served)? (10 Points) The value must be a number

3. What is the expected time people need to wait between the time they join the line and the time they leave the store in minutes? (10 Points) The value must be a number

4. What proportion of the time are the cashiers idle? (provide your answer as a number between 0 and 1) (10 Points) The value must be a number

5. Now what if we have a different layout. We now have 5 lines, each with one cashier. Each line is of what type? (10 Points)

M/M/1

M/M/5

M/G/5

M/G/1

6. In this case, the rate of arrivals for each queue will be different than before. We shall assume that a person arriving to checkout will choose one of the five queues randomly. What should be the rate of arrival (lambda) for each one of the five queues? (10 Points)

990

198

445

7. In this particular case, what is the expected number of people in the whole system (remember that there are five queues in total!) (10 Points) The value must be a number

8. For each of the five queues, what is the probability of having at least someone checking out? (10 Points) The value must be a number

9. In this new system, what is the expected time people need to wait between the time they join the line and the time they leave the store in minutes? (10 Points) The value must be a number

10. Naturally, we would like for people to wait as little as possible to checkout. Which system would you implement then?

5 queues with one cashier

One queue with 5 cashiers