Question $2$

In studying traffic flow at a highway tool booth over a course of $60$ minutes, it is determined that the arrival and departure rates are deterministic, but not uniform. The arrival rate is found to vary according to the function of $(t)=1\mathrm{.}8+0\mathrm{.}25t-0\mathrm{.}0030t^2.$ The departure rate function is $(t)=1\mathrm{.}4+0\mathrm{.}11t.$ In both of these functions, $t$ is in minutes after the beginning of the observation and $(t)$ and $(t)$ are in vehicles per minutes.

$($i$)$ When will the queue that forms be cleared?

$($ii$)$ What time does the maximum queue length occur and what will be the corresponding queue length?

$($iii$)$ Determine the total delay

$($iv$)$ Estimate the average time delay per vehicle

$[25$ marks$]$