Aren't people who close their browser automatically logged out on "timeout"?
If the "logout event" does not fire in that case, maybe the following is helpful in getting creative to get stats:
page 129 user guide
The Properties window will show other properties of the business space version:
Timeout page – a page that the system will display when a browser session times
out. If not specified the default one will be used
Description: Returns the number of time a particular user is logged in the system,
0 is returned if the user is not logged in
Else, piwik (now matomo) is able to give you all stats just like google analytics. Easy, but no idea how/if easy to integrate with aware.