Sadly the browser does not tell the server when it closes. I don't think you can tell the difference between a)user closed the browser b)user went out for dinner & dancing or c)user is busily surfing other sites.
If you really, really gotta know (really!) you can put an
applet in the browser and have it send "I'm still here!" messages to the server or have the server send "Are you still there?" messages to the applet. We're buying a vendor product that does this so it can synchronize telephony events and workflow availability with the browser. We're buying it because it ain't easy to build!