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I believe the 317s are only really used at peak times. That being said, with the repaint/interior refresh for the 315s going on at the moment, there may be slightly more chance of getting a 317.
Before the Lea Valley Lines passed to TFL in May last year, the Chingford line and the Hertford East line had a joint diagram where the 317s used on Hertfords would work a Hertford East service and back to Liverpool Street, and then work a Chingford service and back to Liverpool Street, ad nausum. The 315s would also do this, meaning 2tph out of the 4tph Chingford service was guaranteed 317s. Good pickings for someone like me, who rarely had the chance to catch 317s. :P
I'd guess they have sufficient 315s to cover the weekend timetables and formations. The 317s are far from ideal on that nature of work so if they can avoid using them, especially the 317/6s they hire from AGA then I'm sure they will. The 317/7s were a lot more expensive to lease when they were leased by NXEA and AGA (hence why they got rid of them!) so I'd guess they are costing LOROL similar which means they wont use them unless they have to.
A book I have, "Network Southeast from Sectorisation to Privatisation" says the 305s allocated to the West Anglia sector of NSE were only really used during peaks by the early 1990s. Given the book was published in 1994, and there doesn't seem to be any indication that WAGN had 305s by the time the franchise started, I guess the 305s were finally withdrawn in 1995, when the 365s were delivered to the Great Northern sector, resulting in a cascade of 317s. Presumably the 305s were sent over to the LTS line for a few extra years of service?