I admit -- I have my phone set to "do not disturb" each evening because I also have push notifications set for all my sites. Often, there are sales in the middle of the night, and before I set my phone to "do not disturb" I woke up to loud cha-ching's or "dings" in the middle of the night.
It's an odd situation though, not seeing anything until the next morning, so invariably, I feel like I wake up behind -- not that I would be able to honestly ship in the middle of the night. It doesn't matter, I still feel behind, because I often feel like many of my items are gifts and needed quickly.
So I prepare items for shipping at the crack of dawn, as my kids are dressing for school, so that I can mail quickly in the morning (knowing full well that item's won't really move at the post office until specific times are met -- 1:30 pm is the earliest on weekdays, 5:30 pm is the latest). Still, I like to have items there because in my mind they are still "moving" along their trajectory to their final, happy destination.