Meet with your boss (or bosses) and say "I'm taking on a lot of additional responsibilities and I want to be certain that I have them prioritized correctly. I want to work as efficiently as possible."
Present them with a detailed list of the job tasks and make them put the tasks in order of importance. Don't allow them to cop out and say they are all equally important. You must get a list in order.
Then take on the tasks in exactly that order. When people start complaining about unfinished business, point out that all the tasks at the top of the list are completed and that your boss put the list in that order. Then the complainers need to go to your boss to complain about his priorities instead of blaming you. Eventually the boss will realize that the list is too long and hire someone or shift tasks to someone else.
Repeat this process as often as needed. Every new task must be given a priority number.
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