If the Live Agent Status Dashboard is used, the presence state logout check should be enabled. This will detect if a user logged out out from Salesforce and it will set the presence state of this user to Logout (only for the Dashboard, not for the phone system).
In Step 1 you need to enable the Logout Event. Just click on the link on the screen and select: Enable Streaming.
In Step 2 you activate the check if a user is logged out in Salesforce.