Hellobee Boards

I'm not sure there's much you can do. They're probably required to do a certain number of on call shifts per month, that's how it was at my hospital in the midwife group. You can ask for a copy of the on call calendar when you get closer to your due date so when you go into labor you will know who will be there.
If it makes you feel any better, during my labor I went through 4 different midwife shift changes and I spent WAY more time with my nurses. The only midwife who spent much time in my room was the one who was on during the middle of the night (I'm assuming she was bored!)

My practice rotates patients through them various midwives and doctors for the weekly and monthly appointments and there is one doctor I always request not to see. He delivered my first and I just didn't like him. It was easy to keep from seeing him for scheduled care.

There was nothing to be done about delivery though - whoever was on call was who I got. I never doubted this doctor's competence, so I was hoping not to have him deliver but not overly concerned. Can you ask for the call schedule around the time you're due? I'd also find out how long their shifts are-- the doctors at my practice do 24 hour call but the midwives do 12. So there was a lot more changeover there.

I asked my nurse about this at my last appointment and she said that her advice is to see who is on call that day when I go into labor. (I was debating between seeing an OB versus a midwife). She basically said if I get there and it is my doctor then I can go with OB or if I don't like the doctor I can ask for a midwife. I think you can basically be like nope, I want the OB please.