Imagine a long, skinny prison, composed of a single row of cells, oriented East-West.
Each cell has two doors, each door opens into the adjoining cell.
The prisoners know that Westward lies the exit, Eastward lies the dungeon.
Not every cell is occupied, but some randomly-distributed fraction of them are.
In a cruel twist, each prisoner is also the warden for the person East of him.
There's our setup. Now, the rules:
Every minute, ALL of the doors open instantly for ONE SECOND. There is JUST time (if you choose) to slip into the Western cell, IF it is unoccupied.
There are now two variants of the game, escape-oriented or capture-oriented.
If we're playing escape-oriented, then IF the Western cell is occupied, AND that occupant is NOT CURRENTLY TRYING TO ESCAPE, then you cannot move, and in fact, have to trade cells with the person behind you (no matter how far back they may be). If the cell is unoccupied, you get to move up.
If we're playing capture-oriented, then IF the Western cell is
occupied, REGARDLESS of what that occupant is doing, then you
cannot move, and in fact, have to trade cells with the person behind you
(no matter how far back they may be). If the cell is unoccupied, you
get to move up. In the event of a stacked capture (ie, many contiguous people try to simultaneously move up), then the one first in line swaps with the nearest un-caught person behind him (ie- propels some lucky soul way far forward).
We may assign a "riskiness level" to the prisoners, either randomly distributed among them (as in a general population), or uniform across them all (as might happen in a soul-crushing Gulag).
How long does it take for "n" people to escape from a prison of "N" cells if they look "F" fraction of the times available to them?
I suppose a variant question is "What is the optimum level of riskiness factor "F" for someone to escape as fast as possible?"