Question $2(60$ marks$)$

A cross$-$harbour tunnel has implemented a time$-$varying toll for arriving vehicles. Simulating

vehicle arrivals to the automatic toll collection system can help estimate the revenue and

vehicle waiting time during a morning peak period, given the past distribution of time between

vehicle arrivals $($or inter$-$arrival time$)$ to a single lane at the entrance of the tunnel in Table $1.$

For simplicity, consider only two major types of vehicles arriving at the single lane. The

distribution of private cars and goods vehicles are $70\%$ and $30\%,$ respectively.