Consider a two$-$lane freeway in one direction with a capacity of $4,000$ vehicles$/$hour and a constant

vehicle arrivals of $3,000$ vehicles$/$hour$.$ An incident occurs on this freeway at $8$:$00$ am and one of

the lanes was blocked. Consequently, the capacity instantly decreases to $2,000$ vehicles$/$hour and

a queue forms. When the incident is cleared at $8$:$45am,$ the flow instantly increases to full capacity

of $4,000$ vehicles$/$hour and the queue clears.

$($a$)$ Sketch a queueing diagram $($cumulative arrival and departure curves over time$)$ from $8$:$00$ am

and determine the time when the queue cleared.

$($b$)$ Calculate the maximum queue length $($maximum number of vehicles in the queue$)$ and the

maximum waiting time in the queue.

$($c$)$ Calculate $1)$ the total vehicle delay and $2)$ the average delay per vehicle from $8$:$00$ am until the

queue cleared. The sub$-$division indicated in Figure $4$ is comprised of five drainage areas

that drain to two ditches. The runoff from each drainage area is indicated in

Figure $4.$ Ditch #$1$ conveys the runoff from Areas $1$ and $2,$ and Ditch #$2$

conveys the runoff from Areas $1$ to $5.$ Ditch #$1($from points $A$ to $B)$ will have

a triangular cross section with $2$:$1$ side slopes, a length of $500m,$ a

longitudinal slope of $0\mathrm{.}5\%,$ and will be covered with grass $($Manning$\text{'}$s

$n=0\mathrm{.}045).$ Ditch #$2($from points $B$ to $C)$ will have a trapezoidal cross section

with $3$:$1$ side slopes, a $5m$ wide bottom, a length of $400m,$ a longitudinal

slope of $0\mathrm{.}3\%,$ and will be covered with grass $($Manning$\text{'}$s $n=0\mathrm{.}045).$

Determine the design height of Ditches #$1$ and #$2$ to convey the flows from

the five drainage areas.

Figure $4.$ Drainage areas and design details of Ditches #$1$ and #$2.$