Here are my thoughts based on what I have seen in real life.

But one thing I have seen in real life that we don't have here is: traffic lights at adjacent intersections working together.

In general there are 4 states that the intersection can be in, and we can cycle through them in order.

The states are:

A) vertical (top to bottom or bottom to top) traffic, including pedestrian traffic

B) left turn from horizontal to vertical (also allows horizontal U-turns)

C) horizontal traffic, including pedestrian traffic

D) left turn from vertical to horizontal (also allows vertical U-turns)

There are also intermediate states if we want to get more complicated. In state B we are allowing two left turns: left-to-up and right-to-down. We might decide to stop the left-to-up traffic (in order to allow right-to-left traffic) but keep the right-to-down traffic going. So this is a state between B and C.

How long to stay in each state? We need all of these criteria to be met in order to be done with the state:

We can leave state A when all the cars have gone through, and X seconds have passed*, and the pedestrian time has passed (only applies if a pedestrian pressed the button)

We can leave state B when all the cars have gone through

We can leave state C when all the cars have gone through, and Y seconds have passed*, and the pedestrian time has passed (only applies if a pedestrian pressed the button)

We can leave state D when all the cars have gone through

*Example X and Y calculations:

X = 60 seconds during rush hour, 20 seconds otherwise

Y = 10 seconds

We could also put a time limit on each state.

In my drawing I show the 4 states. Black = the main flow. Blue = right turn traffic that is always allowed. Brown lines = pedestrian traffic.

https://github.com/Skullbash27/trafficemgmt/tree/master/Traffic%20Management