When I go to a local AMP it's almost always on Saturday mornings, as early as I can.  In my town most AMPs open at 9:30, but they don't like taking customers before 10.  I can think of only once in the past few years when there was anyone else there at that time--customers I mean.  

Also, where I go the front doors have some kind of ringer and the rooms have doors.  I always make sure a towel is within reach should I need to throw it over me quickly.

Maybe they get busy later in the day on weekends, but I don't have much experience there.  

...the odds of getting the 60yo mamasan out front giving you the "massage" increases since she will usually be the only one there. I have run into this many times. Unfortunately for me the late morning is the only time that I have available, so now I pre-book with known (cute) masseuses to ensure that I get who I want, when I want.

I have noticed girls are more appreciative on a Sunday, especially if there is a big game for your region.

My observations are that most busts happen during a weekday afternoon.  I think this is for a couple of reasons:

1. You don't have to pay cops overtime because the bust is during their normal shift;
2. You don't have to get a judge out of bed in the middle of the night for an arraignment or search warrant;  
3. You can tip off the media and you are sure to get local TV news coverage.

Also, they don't want to go through the hassle of busting a MP when it's raining or during other bad weather.  I know there are exceptions to every rule, but I think, categorically, if you stay away from normal business hours or go when it's pouring rain, you'll be OK.