A mutex is used in multi-threaded environments. I don't see mention of threads in your question, so I don't see a need for a mutex.
However, if we assume by reader and writer you mean you have two threads, you need to protect mutual data with a mutex (or other multi-threaded protection scheme.)
What happens when the queue has items, and the reader thread pops something off while the writer thread puts something on? Disaster! With a mutex, you'll be sure only one thread is operating on the queue at a time.
Another method is a lock-free thread-safe queue. It would use atomic operations to ensure the data isn't manipulated incorrectly.
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