Question $2(60$ marks$)$ you may submit a single Excel file with organized layout and clearly labelled

answers to different parts of the question

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.

Table $1$: Inter$-$arrival time distribution $($all vehicles$)$

The service start time $($end time, respectively$)$ of a vehicle is assumed to be the time when the

vehicle front $($back$,$ respectively$)$ passes a fixed location $($e$.$g$.,$ camera scanning the car plate$).$

The length of private cars and goods vehicles are estimated to be $5$ metres and $11$ metres,

respectively, based on the local standard car park sizes. The vehicle speed passing through the

fixed location is assumed normally distributed with an average of $35km$ per hour and a standard

deviation of $5km$ per hour for any vehicle. The service time can be treated as the time taken

for the entire vehicle $($from its front to end$)$ to pass the fixed location. $($Note: Time taken $=$

Length$/$speed$.)$ To ensure enough stopping time in an emergency, each vehicle in the lane

should maintain a safe time gap of at least $2$ seconds with the vehicle in front.

Assume the simulation starts at $7$:$00$:$00($or $7am)$ with no car at the entrance to the single lane.

When the front car plate of a vehicle is detected at the fixed location $($e$.$g$.,$ camera scanning car

plate$),$ the time is recorded and the charge is based on Table $2.$ For example, if a private car is

detected at $7$:$29$:$59,$ the charge will be $$20.$ If a private car is detected at $7$:$30$:$00,$ it will be

charged $$22.$ A goods vehicle pays a fixed charge of $$50$ at any time.

Table $2$: Time$-$varying toll

$($a$)$ Simulate $5000$ vehicle arrivals using spreadsheet starting at $7$:$00$:$00.$ For each vehicle, show

its arrival order, vehicle type, time between arrival of previous vehicle and itself, arrival

time, service start time, service time, service end time, waiting time and charge.

$($For the first vehicle, the arrival of the previous vehicle is assumed to be the simulation

starting time.$)$

$(30$ marks$)$

$($b$)$ Based on $1$ replication, find the following performance measures:

$($i$)$ Revenue collected during the period $7$:$15$:$00-10$:$30$:$00$

$($ii$)$ Number of vehicles arriving between $7$:$15$:$00$ and $10$:$30$:$00$

$($iii$)$ Mean waiting time for vehicles arriving between $7$:$15$:$00$ and $10$:$30$:$00(5$ marks$)$

$($iv$){90}^{th}$ percentile waiting time for vehicles arriving between $7$:$15$:$00$ and $10$:$30$:$00$

$(5$ marks$)$

$($c$)$ Replicate the above simulation for $200$ times using Data Table. For each replication, show

the average performance measures in $($b$)($i$),($ii$),($iii$)$ and $($iv$).$

$(10$ marks$)$