What happens to session when browser is closed

I am working on concurrent login in asp.net for that m maintaining one table with a flag. as soon as the user logs in it gets inserted in table with flag as '1' .Till this flag remains '1' ,no other user with same credentials can logs in and at the time of log out and session expiry m updating the table n making it '0' so that other user can log.
but there are two situation arising where the flag will remain 1.
1.When user closes the browser without log out.
2.when user navigates/changes the url.
 

 
so i just want to know will the session end at the specific time if the browser is closed before timeout or what will happen actually to session when browser is closed..?
 

plz help....thanx in advance