C++11 lockfree single producer single consumer: how to avoid busy wait

what if the worker thread completes and before executing working_ = false, another enqueue comes?

Then the value will be pushed to the queue but will not be processed until another value is enqueued after the flag is set. You (or your users) may decide whether that is acceptable. This can be avoided using locks, but they're against your requirements.

The code may fail if the running thread is about to finish and sets working_ = false; but hasn't stopped running before next value is enqueued. In that case your code will call operator= on the running thread which results in a call to std::terminate according to the linked documentation.

Adding worker_.join() before assigning the worker to a new thread should prevent that.

Another problem is that queue_.push may fail if the queue is full because it has a fixed size. Currently you just ignore the case and the value will not be added to the full queue. If you wait for queue to have space, you don't get fast enqueue (in the edge case). You could take the bool returned by push (which tells if it was successful) and return it from enqueue. That way the caller may decide whether it wants to wait or discard the value.

Or use non-fixed size queue. Boost has this to say about that choice:

Can be used to completely disable dynamic memory allocations during push in order to ensure lockfree behavior. If the data structure is configured as fixed-sized, the internal nodes are stored inside an array and they are addressed by array indexing. This limits the possible size of the queue to the number of elements that can be addressed by the index type (usually 2**16-2), but on platforms that lack double-width compare-and-exchange instructions, this is the best way to achieve lock-freedom.