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.

At first, let us assume a system with one single line leading to the five cashiers.

What type of queue is this? $(10$ Points$)$

$\frac{M}{M}\frac{?}{1}$

$\frac{M}{M}\frac{?}{5}$

M$/$G$/1$

M$/$G$/5$

$2.$ What is the expected number of people in the system $($including people waiting in line and beirtg 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$)$

What proportion of the time are the cashiers idle? $($provide your answer as a number between and $1)(10$ Points$).{?}_w$

The value must be a number

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

$\frac{M}{M}\frac{?}{1}$

$\frac{M}{M}\frac{?}{5}$

M$/$G$/5$

M$/$G$/1$

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$)$

$990$

$198$

$445$

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$)$

For each of the five queues, what is the probability of having at least someone checking out? $(10$ Points$)$

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

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

$5$ queues with one cashier

One queue with $5$ cashiers