Every so often on my way home, I find myself walking North on Stone Way, attempting to catch a Westbound bus #44 on 45th St. at Stone Wy. I can see the buses go past from afar on Stone Way, and when I see a bus go past, I wonder what the implications are for my wait time at the stop. When Iʼm far enough away, I want a bus to go past already, so that the next bus might show up just as I get to the stop. But as I get nearer the stop, I donʼt want a bus to go past, as that would mean a longer wait.
The worst scenario is a bus whizzing past just before I get to the stop, while the best is a bus showing up just as I do. So thereʼs a discontinuity there, and that made me want to model the problem, and help figure out how I should adjust my walking speed to synchronise my arrival and a bus being there.
- $l$ be the distance from the stop when I sight a bus go past,
- $nu$ the frequency of Westbound buses along 45th St.,
- $v_0$ my current speed, and
- $v$ my new (for simplicity) constant speed once I see the bus.
So it takes me $l/v$ time to get to the stop, and $l/v mod (1/nu)$ time since the last bus passed by. Thus, itʼs $(-l/v) mod (1/nu)$ time for the next bus. I thus want to minimise (the non-negative quantity) $(-l/v) mod (1/nu)$.
Let: $-l/v mod (1/nu) = epsilon_0$, for some $epsilon_0 ≥ 0$ $⇒ -l/v = -k/nu + epsilon_0$, for some $k in ZZ_(≥0)$ $⇒ v = -l/(-k/nu + epsilon_0) = l/(k/nu - epsilon_0) = (l nu)/k + epsilon_1$, for some $epsilon_1 ≥ 0$
Thus, I should pick speeds just greater than $(l nu)/k$, for some (positive) integral $k$. Letʼs say I donʼt want to go any faster than $v_0$, but I want to get home as soon as possible. Iʼd then pick $k$ such that $(l nu)/k$ is just less than $v_0$, i.e. minimising the positive quantity $v_0 - (l nu)/k$.
Letʼs let: $v_0 - (l nu)/k = v_delta$, with $v_delta > 0$ $⇒ k = (l nu)/(v_0 - v_delta) = ceil((l nu)/v_0)$.
Therefore, the speed I should target, that would
- minimise my wait time at the stop
- minimise the overall time
- not exceed my current walking speed
is $(l nu)/ceil((l nu)/v_0$.