I must say, days off are what I live for, so I sympathise utterly with the comments about time spent at the face.
How much sick leave are you guys entitled to and when are you "allowed" to take it off? It would seem that there's a valid fatigue problem that any doctor would be happy to sign off on if the standbys are worked as indicated.
What does management think about calling in sick for a standby after a 10 minute pause between shifts?