A problem came up in a course of a conversation with my friend. Suppose we have a street with several traffic lights, placed equidistantly one from another. Time of them being green $t_g$ and time of them being red $t_r$ is equal on all of them and much lower than time that is required to travel between lights. Your task as pedestrian is to go from one end of a street to another while crossing it somewhere on green light. After some thinking we came up with this more strict formulation:
Suppose we have $n$ non-negative independent random variables $\xi_i$ with mixed probability distribution, specifically: $$ P(\xi_i = 0) = \frac{t_g}{t_g + t_r},\\ P(\xi_i < x) = \frac{t_g}{t_g + t_r} + \frac{x}{t_g + t_r}, \hspace{2mm}x < t_r,\\ P(\xi_i < x) = 1, \hspace{2mm} x \geq t_r. $$ Simply speaking we have a probablity of light being green ($\xi_i = 0$) and a uniform distribution on time that is left until that event if light is red. Independence simulates the fact that no one really wants to check how lights are synchronized so they are basically independent for relevant purposes. Now the task is to come up with a strategy that gives the lowest expected time of going to the end of a street.
That is a point that moved me to ask this question here. One can suggest that after each light you can look at a time that is left to wait on this particular light and compute a probability of encountering a time that is lower than present if you move forward. If this probability is greater than 1/2 you move forward and repeat a process on next light until satisfaction: either you are more likely to wait less not moving, so you don't move, or light is green and you cross the street and go to the end of it.
Problem is that while this seems reasonable is this really the strategy that gives the lowest expeted time? Is there a framework for those kinds of problems that allows to rigorously establish extremum properties of strategies or find those that possess them?