You might wonder if customers can open a chat when all support agents are busy and just add that chat to the queue so that agents can answer it as soon as they are available.
When all support agents are busy, customers can still open a chat that goes into an open queue. There's a maximum time (set in the Admin CP > Live Support > Settings > Maximum wait time) that a chat request can wait in the open queue before the user is invited to leave an offline message instead.
Yes, end-users can leave offline chat messages. These offline messages can be controlled through the 'Message Routing' feature. The Message Routing feature allows clients or end-users to leave offline messages via the live chat window when no staff user is available for a chat, which can be routed to the Staff Control Panel as tickets.
To enable this feature, check out the information from the Routing Incoming Messages When the Live Chat Is Offline article.