$2\mathrm{.}4.$ 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:

$1.$ Try different values of $$ and give the range of $$ such that the system is stable.

$2.$ 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 $2$ vs $,$ what is the

conclusion you can draw as for the comparison of the ratio $2$ 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 sett