Issue 026 - Wildfire evacuation - Traffic flow
How long does it take 60,000 people to evacuate?
Reuters and AP reported that fires around Spokane forced about 60,000 people to evacuate. AP reported about 8.2 square miles burned, roughly 5,200 acres, with hundreds of structures destroyed and fires still uncontained.
The problem
Estimate how many vehicles were needed to evacuate 60,000 people from the Spokane-area wildfires.
Then estimate how long those vehicles would take to pass through a limited number of major outbound roads.
Convert your result into lane-hours of road capacity, bumper-to-bumper traffic length, and the minimum number of usable outbound lanes needed to complete the evacuation within a reasonable warning period.
Because Fermi problems target an order of magnitude, I normally use no more than two significant digits and write most calculations in scientific notation; the Fermi reference explains both conventions.
Before checking sources
Matt's first pass
I assumed 95% of the people being evacuated would take their own vehicles instead of some form of mass transit or bus evacuation option.
people by car ~= 6.0 x 10^4 x 0.95
~= 5.7 x 10^4 people
people by bus ~= 3.0 x 10^3 peopleFor the most part, the bus evacuees are negligible for the rest of the problem, but I added them in anyway.
I assumed the average vehicle contains 3 people, is about 4 meters long, and takes up about 6 meters in slow-moving traffic from the front bumper of one car to the front bumper of the next. I also assumed most evacuees can maintain an average travel speed of about half highway speed: roughly 25 mph, or 12 m/s.
I assumed there are about 4 major roadways out of the area that account for the vast majority of exiting traffic, with a combined 10 lanes available.
cars ~= 5.7 x 10^4 people / 3 people/car
~= 1.9 x 10^4 carsIt would only take about 60 buses to hold the other 3,000 people.
If we are counting the time it would take for all vehicles to pass a specific checkpoint, at the assumed speeds and intervals, with no accidents or lane closures:
passage rate per lane ~= 12 m/s / 6 m/car
~= 2 cars/s
~= 7.2 x 10^3 cars/hourAt that rate, 19,000 cars would take about 2.5 hours on a single lane, so 1/10 that time across 10 equal and free-flowing lanes, or about 15 minutes total.
Of course, this volume of vehicles would not all flow at that rate continuously, and there are bound to be accidents and other blockages that halt and slow traffic. I bet in a real-world scenario, the passage of vehicles through a checkpoint like a freeway on-ramp on a crowded freeway might be more like 5 m/s.
traffic length ~= 1.9 x 10^4 cars x 6 m/car
~= 1.1 x 10^5 m
~= 110 km
single-lane time at 5 m/s ~= 1.1 x 10^5 m / 5 m/s
~= 2.2 x 10^4 s
~= 6 hours
10-lane ideal time ~= 6 hours / 10
~= 0.6 hoursI wrote down about 52 hours for a single lane and just over 5 hours across 10 lanes, which appears to mix up the division by seconds per hour. Still, with the rate of movement of wildfires, officials would need a large time buffer before the expected arrival of the fire front. In the real world, I would not expect the circumstances to be ideal.
Calibration Score
Matt's Calibration Score: 60 / 100
Higher is better: earn points for accurate pegs, sound models, correct math, and a result close to the sourced answer. The image shows percent full of it: 100 minus the Calibration Score.
Pegs: 10/30. Vehicle count was close, but realistic lane-flow pegs were too optimistic.
Model: 30/30. Vehicles divided by lane throughput and queue length was the right model.
Math: 0/10. The lane-time calculation had a substantial rate/units slip.
Result: 20/30. The final warning-buffer conclusion stayed right even though the idealized time was too short.
Grounding facts
A useful evacuation intuition is that lane-hours are the scarce resource. If 20,000 vehicles need to leave and each lane can realistically process about 1,000 vehicles per hour under degraded conditions, that is 20 lane-hours before accounting for uneven timing and bottlenecks.
Road closures matter disproportionately. Losing half the outbound lanes can roughly double the clearance time, and losing one key merge point can make the whole road network behave like a much narrower pipe.
After checking sources
Check and recalibrate
Matt's vehicle count is probably close. If 60,000 people evacuate and the average vehicle carries 2.5 to 3 people, then:
vehicles ~= 6.0 x 10^4 people / (2.5 to 3 people/vehicle)
~= 2.0 x 10^4 to 2.4 x 10^4 vehiclesCall it 20,000 to 25,000 vehicles, plus buses, emergency vehicles, trailers, and people moving supplies or animals.
Ideal freeway capacity can be around 2,000 to 2,400 vehicles per lane-hour, but wildfire evacuation is not ideal freeway flow. Use 1,000 to 1,500 vehicles per lane-hour:
lane-hours required ~= vehicles / flow per lane-hour
low traffic case ~= 2.0 x 10^4 / 1.5 x 10^3
~= 13 lane-hours
conservative case ~= 2.4 x 10^4 / 1.0 x 10^3
~= 24 lane-hoursA central estimate is about 20 lane-hours. That means the clearance time depends sharply on the number of open outbound lanes:
with 10 useful lanes: time ~= 20 lane-hours / 10 ~= 2 hours
with 5 useful lanes: time ~= 20 lane-hours / 5 ~= 4 hours
with 3 useful lanes: time ~= 20 lane-hours / 3 ~= 7 hoursThe bumper-to-bumper length is also large. If vehicles average 6 to 8 meters of road space in slow/stopped traffic:
traffic length ~= (2.0 x 10^4 to 2.4 x 10^4 vehicles) x (6 to 8 m)
~= 1.2 x 10^5 to 1.9 x 10^5 m
~= 120 to 190 km
~= 75 to 120 milesThat does not mean one single line of traffic runs 100 miles out of Spokane. It means the evacuation volume contains enough vehicles to fill that much lane length if compressed into slow traffic.
For a reasonable 4-hour warning window, the minimum lane count is:
lanes needed ~= lane-hours / hours
~= 20 lane-hours / 4 hours
~= 5 usable outbound lanesThat is the clean Fermi answer: evacuating 60,000 people likely means something like 20,000 to 25,000 vehicles, about 20 lane-hours, and several hours even with multiple usable outbound lanes. If smoke, closures, merging, crashes, or panic reduce capacity, the warning time must grow. That is why officials may order evacuations hours before a neighborhood is directly threatened.
Post-check reflection
Matt's reflection
Vehicle count was probably close, maybe a bit low. My expected rate of car passage per lane was probably quite high. That gave me an unrealistically low time to get everyone out of town.
Not much changes in my overall opinion of the news piece.
Recommended memory peg
Remember freeway capacity is roughly 2,000 vehicles per lane-hour in good conditions, but emergency evacuation may be closer to 1,000 vehicles per lane-hour. Traffic length is roughly vehicles x 6 to 8 meters in slow or stopped traffic.
Reader results
Bars show how submitted estimates sort into the answer choices from the gut-check prompt.