Determine the busiest time
Brachylog, 25 bytes

⟦₂ᵐkᵐcoḅlᵒlᵍthᵐ~c{~k~⟦₂}ᵐ


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There's an obvious structure to this answer, which becomes even more obvious if you write it like this:

{⟦₂k}ᵐc oḅ lᵒlᵍ thᵐ ~c{~k~⟦₂}ᵐ


Unfortunately, Brachylog's evaluation order makes it so that the program goes into an infinite loop if you try to take advantage of the structure in question. Still, it's certainly possible to imagine a Brachylog-like language in which this is much shorter.