A floating phone number that can be handed to, say, 24-hour support staff, that will automatically direct incoming calls to the on-duty engineer.

The value is that when support staff has real phone numbers available to them, they tend to dial historically responsive individuals directly in order to get a problem resolved, thus creating a negative feedback loop -- if you ignore notifications and phone calls, you get called less by support staff in the future.

Having a floating number -- especially if we could get statistics on who answers and who always ignores them, and if the number could "call up the chain" automatically when nobody answers, would be a useful tool to solve this issue.

Of course, the preference is that human staff doesn't need to call anyone, but it still happens.