In short, you cannot (and shouldn't) do this. If a user closes the browser, it's closing...no unload style events are guaranteed to finish, and something doing AJAX with involves latency is more unlikely to finish.
You should look at firing your events at another point, or change the approach altogether, but making an AJAX call in an unload event is going to unreliable, at best.
As an addendum to the above on the shouldn't part, think about it this way, how many tabs do you usually have open on any given window? I typically have 4-6 chrome windows open with 5-12 tabs each...should my browser window hang open because 1 of those tabs wants to make some AJAX request I don't care about? I wouldn't want it to as a user, so I wouldn't try and do it as a developer. This is just an opinion of course, but food for thought.
