Vehicles arrive in a single approach for two consecutive cycles at a signalized intersection. The

signal has a $80sec.$ cycle time with a $40sec.$ green interval and a $40sec.$ red interval $($ignore

yellow interval$).$ The arrival rates are assumed to be constant in both cycles. The arrival rate was

$1,170$ vehicles$/$hour in the first cycle and it decreased to $630$ vehicles$/$hour in the second cycle.

Assume that the vehicles in the queue formed on red pass through the intersection during the

subsequent green interval at the saturation flow rate of $1,800$ vehicles$/$hour immediately after the

start of the green interval.

$($a$)$ Sketch a queueing diagram $($cumulative arrival and departure curves over time$)$ for the

approach during the two cycles.

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

$($c$)$ Calculate $1)$ the total vehicle delay and $2)$ the average delay per vehicle during the two

cycles.