So first thing to state is that assuming your are in coverage then the unit can wake up and send data at any point.
With regards receiving downlink connectivity and trying to keep it as simple as possible.
- Sleep - Unit is on the network, not employing any techniques to specifically turn the radio off.
- eDRX - Unit is on the network and it will listen to incoming paging requests up to every 80 seconds (theoretically longer but not in practice), so there will be a maximum latency of 80 seconds between trying to connect and getting a response from the unit.
- PSM - So the unit is in power save mode, when it is in PSM it has an IP address assigned to it and can go to sleep for minutes/hours/days (depending on network setup), the network typically determines when it needs to ping into the network and this is typically around every 4 hours which is the only time you would be able to send it a message (obviously not good for IP based messaging.
The above is a really simple explanation of the modes, there is a lot more detail and it does not touch on the IP set up.