the simulation for $N_{\text{r}\text{e}\text{p}}$ replications and output the average of the performance measures $($over

replications$)$ and the confidence intervals. You are free to try different values of $N_{\text{r}\text{e}\text{p}}$ such that the

expectation is close enough to empirical average with large confidence, i$.$e$.,$ randomness is

reasonably reduced, while your program runs within reasonable time.

$2\mathrm{.}4.$ Task $1-$ understanding impact of arrival rate on performance measures

Our first task is to explore the relationship between the first two performance measures and the

arrival rate $.$ In particular, throughout the course of the day, there may be busy periods in which

$$ is high, and calm periods in which $$ is low. For very high values of $$ the system may even be

unstable, i$.$e$.$ the as the average number of people in the system $L$ explodes as time increases $($i$.$e$.$

the rate that people arrive is higher than what the service capacity can support$).$

Tasks:

Try different values of $$ and give the range of $$ such that the system is stable.

For different values of $$ in the range you gave in part $1,$ estimate the average number of

people in the system $L$ and the average waiting time $W.$ Plot $\frac{L}{W}$ vs $,$ what is the

conclusion you can draw as for the comparison of the ratio $\frac{L}{W}$ with $?$

Suggestion: You should simulate the system long enough so that the estimates for $L$ and $W$ stabilize.

In order to show the system is unstable, you may need to run for a long enough time to show that

$L$ blows up for large time horizons.

You should discover the famous "Little's Law" in queueing theory; you may even have realized

this relationship when setting up the calculations for $L$ and $W$ in the simulation, which gives the

key idea of the proof!

$2\mathrm{.}4.$ Task $1-$ Optimizing number of luggage screening stations

For this section use the default arrival rate of $=10.$ Our goal in this section is to investigate

how the performance measures are affected by the number of screening stations opened for security

check. An important policy change that is being considered is to optimize the number of screening

stations in the security line $n_3$ to achieve the best trade$-$off between cost and overall system

efficiency. Opening a new luggage screening station $($servers$)$ is costly as it requires paying another

attendant to staff the station, but it would help decrease the system queue length $L($and hence

waiting time $W$ by our previous observation$).$ In addition, if the number of servers is too low the

system will also be unstable. Run the simulation with different number of servers $n_3$ to determine

what is the best trade$-$off. In particular our goal will be to determine the optimal number of

screening stations $n_3$ that minimizes the below cost function

$cost=h*L-R*S+c*n_3$

where $h=1$ is the per person cost that penalizes the total number of people in the system, $R=1$ is

the reward per minute per person for completing service, and $c=10$ denotes the cost per minute of

hiring one attendant.

Task:

Find the range of $n_3$ such that the system is stable. Note that when the system is unstable,

the cost is infinity, since $L$ is infinite.

For different values of $n_3$ in the range you gave in part $1,$ how does $S$ compare to $$ and

why?

What is the best $n_3$ in the range $\{ninN$:$1n20\}$ that attains the minimum cost?