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Good question Ross as it may not always be clear.
Persistent means "stickiness", so there's a persistence entry on the BIG-IP, or cookie persistence on the client. Active connections means there's a TCP connection table entry that has not expired.
With Disabled, returning traffic may not match an existing, open connection, but if there's a persistence entry then the BIG-IP will return the traffic to the pool member. This can lead to pool member connections not draining, or draining very slowly.
With Forced Offline, as soon as the connection table expires that entry then any returning traffic from the client will be load balanced to a new pool member. This will drain connections a lot quicker - check the TCP Timeout session for an idea of how long this may take for stale connections.
Hope this helps,
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