So our first daily Scrum started on Monday and it went pretty well. In meeting with the developers, they confirmed what I originally suspected. Our throughput is low, not because the problems we have to solve are too complex, but because the developers are constantly interrupted to handle the next emergency.
So a prioritized product backlog is the right answer--with each item in the backlog being a defect (or more specifically, the tasks necessary to fix a defect). But we can't follow the traditional scrum methodology and create a 30-day-long backlog. We have to adjust our backlog on a daily basis to ensure that we are handling problems in the right order. Since we don't have a prioritized list, I started the team out by just having them pick defects from our defect tracking system and get working on them. I also committed to minimizing interruptions. That might be easier said than done, but we'll try. It takes a long time to break old habits.
Today (Thursday), I distributed a first cut at a prioritized list of defects. I'm also trying to recruit someone to serve as our "product manager", and maintain that prioritized list.
In the next couple of days, I'll regroup with the team to measure our progress in the first Scrum week. Stay tuned for more details.