An exam is being written by students who are sitting in an $m \times n$ grid in a classroom. The proctor, who is both extremely bored and extremely well acquainted with his students, has assigned each student a probability $P_{i,j}$ of asking a question during the exam. The proctor also knows that once a student has asked a question, their probability of asking a question is reduced to $\alpha_{i,j}P_{i,j}$, where $\alpha_{i,j} < 1$.
Once a student asks a question, the proctor must travel from his current position to the student's location to answer the question.
Let us now assume that the proctor is "tired" - he wants to travel the least Manhattan Distance possible over the course of the exam. Now, I suspect but cannot prove several things, and I'd like to know if my suspicions are correct (or at least valid) and how to go about investigating further:
I suspect that:
1) The optimal thing for the proctor to do, once he has answered a question, is to travel to the location that is nearest the highest average probability. I envision something like a "probability center", similar to the center of mass, and this is where the proctor should move to.
2) Not moving to this center of probability is sub-optimal, since it is more likely students near this center of probability are going to ask questions right after the other.
Is there a more formal, possibly well-studied, version of this problem that I can start taking a look at?