Overnight induction-what to expect?

I’m being induced on Sunday afternoon (the day before my due date) if this little one doesn’t make her appearance before then. What should I expect? My OB has prepared me for a long labor, unfortunately. Her plan is Cervidil overnight, then foley catheter and pitocin Monday morning. Last Monday at 39w I was only half a cm dilated. Since then I have lost my mucus plug, so I’m hoping I’ve dilated more! Will my doctor meet me there on Sunday when I’m admitted or do they typically wait and come the next morning? I didn’t think to ask at my appointment.