If the role is primarily support this might not apply as much.
I consult in a lot of non-startup environments.
People rarely can wait an hour or two. Hell, they can't even wait 10 minutes. They over trivialize matters, wave their hands at developers to go away and fix it right away.
Building the process to teach them to prioritize those things that "can't wait" -- teaches them to understand everything isn't a priority and can't be. Resources, time, and attention are limited and ultimately you have to pick what to do first.
Each request is fundamentally different. Each request does not need immediate processing, or there is likely something that is reactive that should be pro-active.
I've worked in the IT food chain from the front lines, to hardware, software, networking, sales, management, architecture. It really comes down to a culture of reactiveness vs. proactiveness.
What do I do? We use a case manager heavily, with different priorities meaning different response times. There's no room for misinterpretation. An emergency is a fire alarm. Urgent client impacting issues need to be resolved under 24-48 hours, if not the same business day. But, every request does not get looked at immediately. I've handled hundreds of thousands of cases in 15 years of getting requests.
Immediate action doesn't meant constant interruption. Raging fires of an emergency should get our attention. If other requests rae coming into a queue we're already working on, we don't need to see more, we just keep working through the queue in the order we need.
On a side note, I can't wait for ICS letting me dismiss individual notifications, long over due. Also, I hope I can turn off the audio notifications that are happening in Gingerbread even though I turn off all notifications in the apps.