I don't think this is possible, as no OnBrowserClosed event exists. The only thing that gets close is the non-standard window.onunload
DOM event, which you could catch, but that would mean having custom logic run whenever you navigate from one page to another, and even then I don't think you can access the URL the user is going to next, which would be required to know that the user is actually going away from your site (something like null
meaning that he goes to no site at all).
One think you could do is to set the timeout even shorter and have a JavaScript on every page that pings the server in regular short intervals using AJAX, but that's a dirty hack, and it would lock out anyone who has JS turned off. A clean solution is not possible IMHO.