Yes, there is a delay time.
With our current version live chat, there will be a delay time for a visitor’s status to change with following three scenarios:
From Chat ended to Out of site (approximately 3 minutes).
From In site to Out of site (approximately 3 minutes).
From Out of site to disappearing from Agent Console (approximately 1 minute).
Please note that the delay time becomes a little longer if visitors have stayed on your site for more than half an hour.