How do you numerically respond to "Is the request issued to all servers equally?"

Actually, I left the distribution of requests to the load balancer.Now that I've found some requests with longer response times, I've guessed that some servers are more loaded than others.

It's questionable to think mathematically about distributed situations in order to see the load.
First of all, in order to confirm the reality, why not check the resource status?
　→ If the processing is delayed due to heavy load, of course there should be a change in the resource situation.
　In the first place, if it's in production, it's probably monitoring resources.

Also, the server collects logs indicating acceptance and completion of the process and analyzes them. It's the ruins of the castle.

In order to test whether the observed frequency distribution and theoretical frequency distribution are the same or not, I think it is appropriate to use the chi-square test (fit test).
If you search on the web, you will find many examples to test whether the dice appear equally or not using Kai square test, so I think it can be applied directly to the test of the number of requests this time.

Just in case, there was an R tag attached.
If you write in R,

result<-chisq.test(x=c(2054, 1989, 1957), p=c(1/3, 1/3, 1/3)))

will look like this