Parti-gyle brewing question

I've searched the threads on here that discuss parti-gyle brewing, but I never found a direct answer to the question I had:

What do you do with the mash while you're boiling the first batch? Do you add the new hot liquor immediately after running off the high gravity beer and then leave it until you've got the first batch out of the kettle, or do you just leave the grain in the tun and add the hot liquor to it when you're almost ready to boil it?

I imagine if you're capping the mash, you'd want to give it a good 30-45 minutes to convert. I've only got the one kettle and would have nowhere to run it off to, so in any case the grain would be sitting for at least an hour and half for me to boil/chill/dump into the fermenter/wipe down the kettle.

I usually have my second runnings mashing while my first beer is boiling. I've evn left it go for the entire length of time from boil to chilling of the first beer.