Q:

The Two Roads Problem: How many cars take Roads 1 and 2?

Suppose the points A and B are connected by two roads of the length $S_1$ and $S_2$. Cars can drive from A to B on either of the two roads. They start at point A and must decide, one after the other which road to take. They know how many cars already chose to drive on each road.
The speed of all cars on a road is equal to $\frac{1}{\sqrt{(N)}}$, where $N$ is the number of cars driving on that road at any time. 
What is the limit of the ratios of cars on road $1$ to road $2$ as the number of cars that arrive per unit of time tends to infinity?
EDIT: The drivers can make decisions instantly and are $100\%$ logical.

A:

I think you're better off looking for the long-run equilibrium result in the limit as time goes to infinity (assuming that interarrival times are small compared to the travel time), rather than as the arrival rate goes to infinity.
With that interpretation, at equilibrium, the travel times on the two roads are equal.  That is, the speeds on the two roads are proportional to their lengths.  Symbolically,
$$
\frac{S_2}{S_1} = \frac{1/\sqrt{N_2}}{1/\sqrt{N_1}} = \sqrt\frac{N_1}{N_2}
$$
This can be simplified to obtain
$$
\frac{N_2}{N_1} = \left(\frac{S_1}{S_2}\right)^2
$$

