Constant Curve for Cumulative Binomial

I am simulating traffic flow and have come across a rather interesting statistics problem that I'm having trouble wrapping my head around.

I am modeling a stretch of road comprised of 2 lanes. Lets say for simplicity they head North for 500 feet. The lanes then diverge with the left lane moving West and the right lane moving East. I am trying to model when the drivers decide to switch lanes for the upcoming diverge.

My method of modeling it is having a binomial distribution for lane switching decision run every foot to where by the end of the stretch of road their cumulative probability of having switched approaches 1.

The problem is that entities may enter the road at multiple points during the 500' stretch. The decision making for the drivers would be that the urgency to switch lanes increases as they approach the diverge, but not based on absolute distance from diverge but from relative distance. For example, for someone who entered 100' from the diverge, they will have the same instantaneous probability to switch at 50' as someone who enters at 200' would at 100' from the diverge. The problem I'm having trouble with is assuring the cumulative probability reaches 1 near the diverge.

I have played around with this in Excel and arrived at the following equation which works with several instances:

$P = 1-\prod_{i=1}^{n}(1-\frac{i^{4.6}}{n^5})$

Where $i$ is how long they have been traveling on the road and $n$ is the total distance traveled until the diverge.

This shows the following curves for cumulative probability.

Entering the road 494' before diverge:

Entering the road 187' before diverge:

Although these will work for these 2 instances, they do not work for smaller values of $n$ and they are not as similar as I would like. I believe at $n=50$ the curve cuts short at a final cumulative probability of ~.86

Is there anyway to derive an equation that will keep the same curve regardless of the n value? I am having trouble thinking of the maths for this scenario. My equation was derived through intuition and trial and error; so I am open to completely different approaches to this.

• (+1) Your immediate problem is that the expression scales like $n^{-0.4}$ and therefore depends on that arbitrary number $n$. The approach is nevertheless appealing: start with a simple assumption about the second-by-second decision-making process as a function of the entry point and the distance to the divergence. Let that determine the distribution of the decision points. That suggests you still need to supply another piece of information: exactly how should the instantaneous decision probability depend on where the driver enters? Tell us that and we can give you a solution.
– whuber
Sep 8, 2017 at 18:54
• @whuber The decision making would be that the urgency to switch lanes increases as they approach the diverge, but not based on absolute distance from diverge but from relative distance. For example, for someone who entered 100' from the diverge, they will have the same instantaneous probability to switch at 50' as someone who enters at 200' would at 100' from the diverge. The problem I'm having trouble with is assuring the cumulative probability reaches 1 near the diverge. Hope this clarifies. Sep 8, 2017 at 19:24
• Nice! Please incorporate that comment into your question: it's clear enough that I think it may lead to a simple, elegant solution.
– whuber
Sep 8, 2017 at 22:11