I don't know if it makes sense, but I know what you are getting at.
You want the user to know that he has been logged off, without the user doing anything, just being idle.
If you want to rely on JavaScript, I'd use a timer. If the timer reaches e.g. 240 seconds (4 minutes), you can redirect the user or make a popup saying "You are logged out" (and of course killing the session in the back).
However I think it's far too complicated. I'd still say, do a check up whenever the user requests a page, meaning, if he had been idle before (and got logged off), just reroute to a page telling him.
That's easy to achieve and user friendly as well, 'cause he will be watching the screen when it happens, he won't return and "something" has happened to his browser and the page it sat on, and he doesn't know why.