logical deduction – prisoners riddle – a button, a light and a timer

Here is (yet another) prisoner riddle. I made it up (maybe it was asked before but to my best knowledge I’m the first to ask it – correct me if I’m wrong).

There are 1000 prisoners who are given a chance of freedom. They are told the rules of a game they are going to play the next day, and need to come up with a strategy (after they agree on their strategy, they may not communicate in any form anymore).

The rules of the game:

• 100 of the prisoners are randomly chosen by the warden and are placed in 100 rooms (non of the chosen prisoners know who are the other 99 that were chosen).

• Every room contains a button, a lightbulb and a timer.

• at any given time, in exactly one of the 100 cells the lightbulb is on.
  By pressing the button in the cell with the light on, the light turns off in that cell, and immediately turns on in another cell (randomly chosen with even probability – so the light might be turned on in cell 1 multiple times before being turned on in cell 100, but it will eventually reach all rooms).

• The timer shows (at all the cells) the amount of seconds passed since the game begun.

• The game ends (and the prisoners win) when one of the 100 prisoners claim the light was on at least once at every of the 100 rooms (she must be 100% certain of that – otherwise they will all be executed).

Note: Assume the prisoners are extraordinary good at arithmetic, and that they don’t mind if the game will take very long time to end. Also assume that they are not superhuman and thus cannot press the button within 0 seconds of the lighbulb turning on in their room (i.e. they have a human reaction time for pressing the button).

(My solution involves basic math – I’m interested if there is a simpler one)