Question

Traffic is moving on a 6-lane dual carriageway road (3 lanes per direction). Traffic volume per direction during the peak hour (08:00 am to 09:00 am) is 6000 veh/h, distributed uniformly across the lanes in each direction. Just at 08:00 am, a truck goes out of order on the middle lane of one side, disrupting traffic on that lane. The lane capacity under normal conditions is 2000 veh/h/ln, and under queue formation it is 1600 veh/h/ln. The traffic resumes at 08:30 am on removing the truck from the middle lane. Hourly traffic volume after 09:00 am reduces to 5000 veh/h/dir. The number of vehicles in the queue at 10:00 am is ______ (in integer).

Hint:

Track cumulative arrivals vs departures in three phases; once a queue forms, the discharge rate stays capped at the reduced (queue) capacity even after the blockage clears.

Answer 1

Correct Answer: 2200

Step 1: Build cumulative arrival and departure counts from 08:00 am.
Instead of tracking the queue phase by phase, add up total vehicles that have ARRIVED and total vehicles that have DEPARTED (passed the bottleneck) from 08:00 am up to each clock time, then subtract to get the queue length at that instant.

Step 2: Cumulative arrivals.
From 08:00 to 09:00, demand is 6000 veh/h, so by 09:00, cumulative arrivals $=6000\times1=6000$.
From 09:00 to 10:00, demand drops to 5000 veh/h, so by 10:00, cumulative arrivals $=6000+5000\times1=11000$.

Step 3: Cumulative departures, using the correct discharge rate in each sub-interval.
From 08:00 to 08:30 (2 lanes working, queue forming): rate $=2\times1600=3200$ veh/h, departures in this half hour $=3200\times0.5=1600$.
From 08:30 to 09:00 (queue still present, so discharge stays capped even with 3 lanes open): rate $=3\times1600=4800$ veh/h, departures $=4800\times0.5=2400$. Cumulative departures by 09:00 $=1600+2400=4000$.
From 09:00 to 10:00 (queue still present, same capped rate): rate $=4800$ veh/h, departures in this hour $=4800\times1=4800$. Cumulative departures by 10:00 $=4000+4800=8800$.

Step 4: Queue length is cumulative arrivals minus cumulative departures.
At 09:00 am: queue $=6000-4000=2000$ vehicles.
At 10:00 am: queue $=11000-8800=2200$ vehicles.

Step 5: Sanity check - does the queue clear before 10:00 am?
After 09:00, the queue changes at a net rate of (arrival rate minus departure rate) $=5000-4800=+200$ veh/h, so it is still growing, not shrinking. It has definitely not cleared by 10:00 am, so the direct subtraction above is valid. \[ \boxed{2200 \text{ vehicles}} \]