We have much the same issue here in AOL Platforms. We have Feature Teams that work both on new features and also get dragged into fixing bugs. We did a number of things to fix this, the first starting with education.
1- Secrets of Multi-Tasking Exercise: This is great as it can be done in five minutes with a piece of paper and a pen. I'll detail it at the bottom of this post. The outcome is people understand in real terms the cost of multi-tasking.
2- Leverage Severity/ Priority: In AOL Platforms we have Minor, Major, Critical and Blocker. Looking at our SLAs we knew that Blockers would always be sprint interrupters are they require follow the sun attention. However our Critical defects have an SLA of 10 days and because our functional areas all have at least four teams, there is always a team starting a sprint within a couple of days. So we determined that only Blockers could be sprint interrupts. Everything else had to be scheduled into a sprint.
Tracking Cycle Time: We started tracking cycle time on the work done by the teams. From when something moved to "In Progress" to when it was marked as "Done". By having this data, we could then look at teams whose cycle times spiked. Most often there was a direct correlation to interrupt work, showing that teams that were interrupted ended up going slower on all their work, including the interrupt work.
Have everyone put their paper in landscape position. On the paper divide the paper vertically with a line 1/3 down the page. Then divide the page horizontally into three sections. You will end up with six boxes on the paper, three small squares and three long columns.
In the small squares (column headers), write an "A", a "1" and a "I" (the last being a Roman Numeral one.
Round 1: Explain that the exercise is to create three features, a letter feature, a number feature and a Roman numeral feature. A complete feature is ten characters (A-J, 1-10, I-X) Management is very keen to see work being done on all the features. So when they write down your characters you must start with the letter A, then move to the number 1 and then the Roman Numeral I. Only then can they start on the next row with the letter B.
Ask that when they are done, they raise their hands. Get a stop watch and say begin.
As soon as the first person puts their arm up, hit the lap timer. When the majority of the room has their arm up, stop the timer. Review the times. Generally you'll have a range between about 30 and 45 seconds.
Then ask them, if each column is a feature, when did they see the first value. The answer should be something like "a couple of seconds before the first person got done."
Round 2: In this round tell them they are now to do each column completely before moving on, starting with the Letter column.
Start the timer and pay attention. When you notice someone has finished the Letter column hit your lap timer. When the majority of the room is done, stop the timer. Review the numbers. Generally you're first person will have finished between 15-20 seconds and everyone between 30-35 seconds. The first column done was less than 10 seconds.
When I teach this I generally then ask the room, "So the secret to multi-tasking is what?"
Someone always answers with "Don't do it"