Modem/Network connection time

Does the amount of time it takes a modem to power up and connect to a cellular network depend mostly on:
The modem itself?
The network carrier?
The sim card and service you’re using?

History behind the query…
The application is a remote area and prone to high temp. As such, the idea was to only boot up the raspberry pi when needed and then only long enough to send data, then power back down. I’m trying to get a grasp on how long I need to wait after powering up before sending data and then powering down. Obviously the less time the better, so I was wondering if I should look at certain carriers, modem models, or whatever to try and cut the connection time down, or if it’s just a matter of circumstances at the time as to the time required.

Thanks for any info you can send my way.