It depends on how you go "offline." Sometimes if you simply end your Viewer program operation by closing a window or terminating via WIndows Task Manager, SL will look at it like a temporary fault/connection glitch and keep you logged in to the Sim server. If that happens you will have to try twice, at least, to get logged in again. You often recieve a message, "the region is currently rebooting" or somethign like that. Then second time through you will be able to log in again, In between the faulty log off and the new log in, your avatar will continue to rezz in the Sim server at the same location as you logged out, but will no move, talk, etc. This is literally an after-image of your avatar, stuck in the Sim server's cache.