$2\mathrm{.}4.$ 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 $3$ 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 $3$ to determine

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

screening stations $3$ that minimizes the below cost function

$=+\%$

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.

Tasks:

$1.$ Find the range of $3$ such that the system is stable. Note that when the system is unstable,

the cost is infinity, since L is infinite.

$2.$ For different values of $3$ in the range you gave in part $1,$ how does S compare to $$ and

why?

$3.$ What is the best $3$ in the range $\{$ in $$: $1<=<=20\}$ that attains the minimum cost?