I usually vacuum sinter by pulling the unit down to 50 microns. For heating, backfill with N2 and hold the partial pressure around 300 microns. All of your cooling will be done at that partial pressure using forced N2 as the cooling medium.
One observation I'll make is that you need to burn-off / delube your samples outside of your sintering furnace unless it's a newer one with some sort of specialized lube removal system. You don't want the wax coating your heating elements or destroying the internals on your rougher pump. Either a belt burn-off or a specialized vacuum burn-off is fine. For the belt burn-off, run the parts under straight N2 with no air additions to control your oxygen content. If you oxidize the parts, they will react with the carbon and sinter differently. Those are both methods we use.
To Ed's point, you have to make sure the carbon is out of the part. However, you may have to run a slightly higher carbon content in order to react with the oxygen that exists as oxides on the powder particles. That's what will assist in the sintering mechanism under vacuum.
That being said, can I inguire why you're vacuum sintering these? Most of my vacuum sintering of 316L is only done to take the stuff to full density where strict temperature control is required (7.65 g/cc min). 7.00 g/cc is somewhat of a hybrid density. I'm assuming you're molding around a 6.6 g/cc density. Low temp sintering on a mesh belt at 2100°F will put you around a 6.8 g/cc density. If you move to a pusher plate furnace around 2200°F you should be able to hit your density spec. The pusher furnaces are usually more economical to run for production. A ceramic belt may give you similar results if your parts are light.
If you want to discuss pusher conditions, feel free to post and we can go down that route.