The live chat build into all our solutions does check every x seconds if the client has left or not. That is important otherwise the operator is talking to a client that does not exist anymore. It works the same way the other way around as well.
Now some server, clients, internet connections or whatever magic interrupt the connections from either the client or operator side will mark the client or operator offline.
Important are your settings in your operator panel. Login to your operator panel and go to Settings - General settings.
Important here: Mark client as left after x seconds should that be set too low your clients can be marked as left because they loose internet connection, instable wifi connection, server did not response or any other weird stuff within the time set.
Should you encounter any of this, change it to a higher number means wait for the system longer to mark the client as left.