Hi,
The thing to understand is that a server side session is not the same thing as a browser session. By default the IIS session timeout is 20 minutes, so 20 minutes of no activity and the session expires and the next activity from the same user/browser will start a new session. A browser session is determined by when the browser is closed or restarted, so if the browser is left open the server side session may timeout but its still an active browser session. Note that you can close a web site but the browser session is still active if you visit the site again without closing the browser, only closing the browser expires a session cookie.
For something like this my advice would be to not use server session but use a cookie set from javascript, that cookie should be a session cookie meaning that it will expire when the browser is closed, ie when the browser session is over.
This article Add a Notification Bar to Your Web Site Using jQuery by Bipin Joshi should help you.
Hope that helps,
Joe