AMCAS is actually ahead of schedule compared to last year. Last year apps could be submitted June 1st. So it took 15 calendar days/11 business days (June 15th) to get to the second day of apps (June 2nd). This year, we started on May 31, and got to the second day of apps (June 1st) on June 13th. Only 14 calendar/10 business days later. It took AMCAS 19 calendar/13 business days to get to the 4th day of apps. So if AMCAS gets to the 4th day (June 3rd apps) by Monday, June 18th, we're right on schedule with last year or better.
Bottom line is: don't stress. If you submitted on or before June 11th and things go as they did last year (which so far they are actually faster), you'll be in with the first batch. And even if you submitted after June 11th, you're pretty darn early and will be fine!