Train intersection problem with unequal speeds As shown in this question, it is trivial to schedule trains running either north-south or east-west in a square city along randomly placed (vertical and horizontal) tracks and ensure that two trains always meet at track intersections--so long as all trains have the same speed.
What are the conditions on the number and placement of vertical and horizontal tracks and train speeds such that such intersections remain even though the trains have different speeds?  Must all the train speeds be related by rational numbers?
 A: Likely one can represent this system by a set of linear equations.  However, the speeds are a function of distance over time, and do not have to be related by rational numbers but the times do have to be related.
Here is a geometric model which is a little different from what you present,  but is easy both to visualize and analyze, and you can adapt it to your situation.  Imagine a train running on a (topologically) circular track at constant speed in 3d-space.  Now pick two points on this track, and add a second circular track and train that "comes sufficiently close" to the first train at the appropriate times.  Both of them have to cover the same fraction of track in the same time, but their speeds can be any ratio you desire , especially if you bend the first track.  You can now pick two other points on this combination, and under mild restrictions add a third train and track so that there are meetings between each pair of trains.  It is an interesting exercise to determine how to place a new circular track and a train to satisfy intersection conditions.  Perhaps using a Minkowski space-time model will help visualize the conditions needed for intersections.
There was a linear version of this at Alexander Bolgomony's website cut-the-knot, but involving constant velocities, not just constant speeds.  You might find it useful to consider the Four Travelers problem there.
Gerhard "Is Going Back In Time" Paseman, 2017.09.25.
