In my team there are 4 of us that rotate a oncall roster. Starts on a Monday and finishes the following Monday. It's 24x7 support. We also use an external monitoring company that monitors all our routers and calls us if there are issues. The service desk rarely calls the oncall support person, it would have to be urgent if they do.
It works pretty well but I used to get calls in the middle of the night from the monitoring company saying a particular site was down. Then they'd ask, does the site have power. As the site is only 9x5 obviously I have no idea of power availability. So I created some guidelines on how to confirm power. This has stopped almost all the calls. If the link goes down, they log it with the carrier, if the power goes down they wait for it to come back up.
I don't mind the oncall as you make a little bit of extra money. But it took some work to streamline it and minimise the chances of getting called.