The chat times, wait times and time stamps shown in the Agent Console are controlled by the individual agent workstation clock. Therefore, the displayed chat times may differ between workstations.
During the initial login to the Agent Console, the local station times are synchronized with the LivePerson servers. However, if the agent doesn't log out from the Agent Console for long periods of time, the console will start displaying slightly different times from the other workstations.
In order to synchronize the Agent Console, log out from the console and then log in again. This will synchronize the local time of the Agent Console with the time of the LivePerson servers.
NOTE: The times that appear in the chat transcripts are governed by the server and are always consistent.