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.

$1.$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$)$