Thursday, October 25, 2012

Low-Key Hillclimbs week 4 passing prediction

The 2012 Low-Key Hillclimbs have reached their fourth week. After two weeks of mass-start, and the third week being a small group start, week four will be an individual time trial due to the relative narrowness and relative flatness of the west side of Highway 9 as it leaves Boulder Creek. We're a hill climb series, not a road race series.

There's three ways to do the start: fast riders first, fast riders last, or intermixing the fast and slow riders. We've done all three of these at different points in the past, and this week coordinator Rich Brown has chosen to pre-assign start times going from slower to faster riders.

A few riders expressed concern over this, expecting the extended mass of riders to begin to converge, the faster riders gaining on the slower, until a singularity is reached similar to that of a black hole's mass converging on a singular point causing a cataclysm in the very fabric of space-time. Okay, I'm overstating the case, but at least there have been worries about excessive pass events.

As I've said, we've done slow-to-fast before, because the opposite takes too much time for volunteers who then have to wait for the slowest riders to not only complete the course but additionally to wait for their late starts. So it's worth calculating how many pass events each rider can expect to experience.

Looking at the week 1 results, there was around a 1/2% per rider difference in speed. That means if you line up riders in order of score, and each rider is perfectly repeatable, then if there's a 1/2% difference between riders and a typical rider takes around 1 hour (3600 seconds), then each rider would be 18 seconds faster than the rider ahead, 18 seconds slower than the rider behind. The finish line would thus have riders finishing with gaps 18 second less than they started with.

If riders started with 18 second gaps between riders, we'd thus have issues. But Rich is a smart guy, and realized such short gaps would be a problem, so instead set the start gap to 60 seconds. Over the course of the hour, on average, riders will therefore reduce the gap to the rider ahead by 18 seconds, down to 42 seconds. Additionally, the rider behind will reduce the gap by 18 seconds, to 42 seconds.

Now of course this is terribly idealistic. On top of this systematic gap reduction, there will be essentially a random perturbation. There's multiple reasons for this. One is that riders aren't uniformly distributed in ability. There might be a 0.5% gap, or the gap might be 2%, or it might be 0.01%. If it's 2%, the rider behind over the course of an hour would gain 72 seconds, catching the rider ahead.

But additionally, riders vary week-to-week in how they ride. Some riders do better on flatter roads, some on steeper, some on shorter, some on longer, some when it's hotter, some when it's cooler. Things are never the same. Even on the same course, rider performance chances.

I'm going to assume on average rider performance varies by about 2%. That's around 72 seconds, plus or minus.

So what we have is the initial gap of 60 seconds is systematically reduced to 48 seconds, but on top of that systematic gap reduction we add a random perturbation. Imagine I assign each rider a time 48 seconds different from the riders adjacent to them, then I randomize their times with a normally distributed random number of sigma 48 seconds.

The result is there's a good chance the rider will catch the rider ahead or be caught by the rider behind. Further, the gap to the rider two ahead is 96 seconds: that rider might be caught, or similarly the rider 2 behind might catch the rider. By three ahead, the gap is 144 seconds, so perturbing positions by around 48 seconds is going to have a hard time closing that gap.

Without doing a numerical simulation, I seems clear riders will be passed or will pass around two total riders during the ride. Some won't experience any passes. Others might experience four. But on average it will be around two.

This is a super-crude analysis, of course, better to do a simulation. But it's clear that the concern of some sort of collapsing singularity of riders is unfounded. A few passings over the course of an hour ride is really no big deal. The far greater concern, as it always is, is vehicular traffic. Yet that's something we deal with every time we go onto the roads.

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