>apptQ.push(apptQ.front + apptList.pop); but that would only work with a stack right?
No, that wouldn't ever work because front is a member function, pop is a member function, and apptList isn't even an object, it's an array.
Assuming time24 has a single argument constructor, you could do something similar to this:
for (int i = 0; i < 10; i++)
apptQ.push(apptList[i]);
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Oh, I see what you're trying to do. Now it really does matter how time24 is defined. Could you please elaborate on that type? This has a good chance of working if you have a way of extracting the first part of a time24 value:
for (int i = 0; i < 10; i++)
apptQ.push(time24(apptQ.top().first(), apptList[i]));
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