In my experience, time on site isn't going to be a useful metric however you go about it. At least with the news sites my company runs, checking if a page is still open would be hopeless because it's very common for users to just open the 2, 5, 20 pages they want to read in different tabs, then work through them slowly. And for clicking outbound links, well nearly all exits for us are window/tab closing, not links elsewhere.
Maybe this isn't so much the case for other industries, or even for the same type of site targeting a less techy audience? (e.g. My mother wouldn't ever open multiple tabs as a queue of pages - but then, she might well open a page and then go do some cooking, or whatever, before reading it.)
