Health Check that lessens the down time for failover

You could use a iRule to look for response codes or other information returned from a server, and potentially select another server, see: https://devcentral.f5.com/wiki/iRules.LB__reselect.ashx

The LB::down method in iRules can mark a node as down: https://devcentral.f5.com/wiki/iRules.LB__down.ashx

There are a few examples in those pages that may fit your needs. For example, from the LB::down page: when HTTP_RESPONSE { if { [HTTP::status] == 500 } { LB::down } }

you can also use LB::down to mark selected pool member in LB_FAILED event.

LB_FAILED wiki

LB::down wiki