After setting it up out of the box I never turn it off.I have this system and it does indeed go to sleep in about a minute after I get off the bike. Bumping it or closing the top box is enough to wake the system up if it has gone to sleep.
The numbers you see when it wakes up (or you power it on) are the last readings it had before it went to sleep or got shut off. You should be able to tell by the temperature readings next to the pressure readings.
Mine wakes up and I get front tire readings just backing the bike out of the garage. The back tire takes about 100' of riding to show up. That is if went to sleep.
If I shut if off (usually for charging the display) then turn it back on it will take 5 to 10 minutes to get both tires to show up. My front shows up within a few minutes but my back will take 5 minutes at least before it does.
I shut mine off and take the display into work to charge it. It turns on during charge but won't show any pressures or alarms as long as it's charging. Once back on the bike I have to turn it on manually and it will beep at me until both pressures show up again.
I do like that it's loud enough that I can just hear it in my helmet with ear plugs while I'm riding. In the sunlight the flashing isn't very noticeable but at night it's really clear.

