Context switching an engineer
Every time you ask an engineer to postpone what he/she was working on and immediately work on some new task, you've done the equivalent of grabbing at least $500 from your company's piggy bank and flushing it down the toilet. You'd better make sure that new task is worth it.
Any progress the engineer has made on the previous task, you can now kiss goodbye. You should consider that previous task as completely unbegun.
Context switching is incredibly expensive for engineers. It can take upwards of an hour for an engineer to build state on a problem. Building this state is a fixed cost of entry for the engineer to enter a state of flow that enables productivity. Breaking that flow and wasting all that state is expensive.