Using FRP to model road network with jams

Question

I am currently trying to understand arrows and FRP, and I came upon a question, which I cannot seem to map to FRP, namely how to model a road network.

I thought I could model a road network as Arrows, where each Arrow represents a road segment. It accepts streams of cars at locations and times and produces the same type, albeit with different locations and times.

So far so good. But this model does not take into account, that segments may get jammed. While each segment could well respond to heavy traffic and delay cars more and more, the more congested it gets, there would be no backwater effect, i.e. the jam would not propagate backwards to other road segments.

I suspect I am applying too much OO thinking here, instead of focusing on what needs to be computed, but I cannot get it right in my head.

• How can I model a road network with Arrows such that backwater effects are taken into account?

Answer 1

The problem is that in arrows and in FRP the flow of information is in general unidirectional. Think of a FRP arrow like a piece of digital circuit. The output of a circuit element doesn't depend on what's connected to it - it just "offers" the output to whoever is interested. This is also described visually in Primitive signal functions in the Yampa overview:

Your situation is different. The state of a segment of a road depends on both the next and previous segments - cars are comming from the previous one, but if cars can't leave to the next one, they have to stay. It's just like a pipe with running water. If you close the pipe at its end, water stops, and the information about that propagates backwards through the pipe at the speed of sound in water.

So each road segment will need to have 2 inputs: One saying let's say how many cars can the following segment accept, and how many cars are coming from the previous segment (which should always be less or equal to the number of cars the segment can accept at the moment). This means that the FRP signal flow will be actually circular. For this you'll need loops, shown in the last image in the above diagram, which are captured by ArrowLoop type-class. Most likely you'll have a custom binding function for road segments that'll be internally creating the required loops. Note that there must be a time delay in a loop, to prevent it from diverging, which makes sense as it takes some time for cars to go from one segment to another.

(I'll perhaps expand the answer with an example, if I'll have more time.)