Suppose you are at coordinate $(m,n)$, and wish to go to $(0,0)$ using unit steps, either down or left. However, at each point, there is a traffic light, which switches from allowing horizontal to vertical crossing every 10 seconds. All crossings are independent, you you can assume that when you reach a crossing, there is a uniform distribution of which state it is in when you reach it.
As long as we are at $(m,n)$ with $m,n\geq 1$, we can choose if we wish to go vertically or horizontally, so we of course cross the street in the direction which is green. The tricky part is what happens after one of the coordinates become $0$.
Since the Manhattan distance from $(0,0)$ to $(m,n)$ is constant for every path, we don't really need to consider walking speed.
So, what is the expected total waiting time?
EDIT: So, the expected time to go from $(1,0)$ to $(0,0)$ would be $10/4$ seconds, since with 50% probability, there is no need to wait at all, and in the other case, the average waiting time is $5$ seconds.
Thus, the expected time from $(k,0)$ to $(0,0)$ is $10k/4$.
Define $$ f(m,n) = \sum_{k=0}^m \frac{10k}{4}\frac{\text{number of paths from (m,n) to (k,0)}}{\text{total number of paths leading to an edge}} $$ Hence, expected time from $(m,n)$ to $(0,0)$ is then $f(m,n)+f(n,m)$.