Session Expiry when browser window is closed

Ever wondered what occurs when a user closes his browser window. For the time the user was interacting with a site, his session was active and everything was going smooth. But how does our player ("Server" , here) come to know that a user has closed his window and the session is no longer active? Read further to unfold the mystery...
 
Whenever a browser requests a server for a page/resource, that request is served with a particular SessionID and the Session timer starts. It is this ID which identifies the future requests coming from the same user. Now when the user remains idle for a duration equalling the Session timeout (default 20 minutes), the Session expires at the server-side and if a request is made after this, a new SessionID is generated.
 
However, if the user abruptly or mistakenly closes his window, the SessionID remains active at the server-side, thinking that the user is idle. Until the Session expires for this particular SessionID, it cannot be used for another request. Once the Session is time-out, the SessionID is free again to be used for serving another new web-request.
 
Thanks for reading.
 
Any suggestions/different views are always welcome.