For general purpose web development response times, I like to refer to p99 and p95 for general healthiness. P50 doesn't do much more than make me feel better. Max has its uses, it can be a good timeout gauge as the commenter above me pointed out. If the RATE at which max values occur changes, you may have something worth investigating.
To my point though, you mention that you noticed a "consistent" max of 2hr. Meaning the consistency was the interesting part. You probably consider this issue to be more important than some other hypothetical issue that caused, say, literally a single request to have a 2hr latency on Tuesday.
In other words, if you monitored over a period that was long enough to establish a statistically significant p99.9 or even p99.99 and then measured those percentiles, you still would have not only caught this issue, but also distinguished it from the other hypothetical issue. Monitoring your maximum and worrying about consistently high maximums might be a more effective mechanism, but in principle, you still care more about top percentiles.
Just like monitoring the arithmetic average ain't completely useless either. It's just far from the most important metric.
Comparisons like this are what is going to lead us into the next 100 years.
To confirm though, I am indeed not that or any other mathematician and I can’t claim any credit for any contribution to category theory. I’m simply a programmer who knows a little bit about a few things, no one special :)