## Early or Late, Never On-Time

Say that you live in a metropolitan area and that everyone works downtown. If you leave early, then you get to work WAY early. If you leave late, then you get to work WAY late. What’s up with that? Let’s say that the closer people live to downtown, the proportionally deeper you work in downtown.

Odds are that you live somewhere in between super far away and somewhere super close. That means that when you arrive at work, there are people closer to and further from the city center also arriving at their jobs. They are your competition. Their mere existence adds congestion to the roads and slows your velocity. As you all make your way closer to the downtown area, the congestion increases and the velocity falls still further such that your slowest speed occurs as you approach work.

If you leave home early, then you get to enjoy a nice low density ride for the first portion of your journey. The last portion of your journey is still slower than the first, but you’ll spend less time on the road and face less congestion. The result is that you get to work not only sooner, but also faster.

Conversely, if you leave home late, then you are already in a more congested area due to those people who live further from downtown than you. The last portion of your journey is still slower as the congestion increases nearer to downtown, but the mile-for-mile speed is slower than if you had left on time. You not only get to work later, you get to work more slowly.

The relationship between arrival time and departure time illustrated above can be approximated by an upward sloping line with a normal distribution function whose mean is just greater than 9 AM (redline is uniform traffic conditions over time). This mean>9 is sensible certainly if we consider that some jobs include driving and adding to congestion. The slope increases until the mean is reached. If you leave super early, then your travel time doesn’t change much regardless of whether or not you enjoy a 5 minute coffee. Same for leaving super late. I know people who don’t do their makeup if they’re running late… Unless their super late. Then they know that the impact on travel time is negligible.

But, if you leave at the time that will get you to work exactly at 9 AM, then it’s you and everyone’s brother out on the road. A small change in your departure time affects your average velocity and travel time a lot. That quick coffee might have a 5 minute impact on your departure time. But the impact on your arrival time is huge. That’s because 9 AM or just before is when the most people are on the road. Any time interval that includes travelling at or near 9 AM will take more time. If you’re trying to minimize travel time, it’s better to leave far earlier or far later. For many of us, far later isn’t an option.

### One thought on “Early or Late, Never On-Time”

1. Joy May 26, 2023 / 11:49 am

I used to leave my office in grad school at 6:30pm. I picked “late”.

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