I'm pretty sure what the author is saying is:
E(X) =:= \sum_t(t * P(X = t)) is the definition
another important note is P(X^2 = t^2) = P(X = t) - because it's the same distribution.
E_a(X) is a bit sloppy, but consider X_a aka Alice's latency "experience" distribution. The argument is:
P(X_a = t) = t * P(X = t) / \sum_u(u * P(X = u)) - i.e. scale the probability up by t but make it sum to 1.
Then
E(X_a) = \sum_t(t * P(X_a = t)) = \sum_t(t * t * P(X = t) / \sum_u(u * P(X = u))
aka
E(X^2) / E(X)
Then (from wikipedia)
Var(X) = E(X^2) - (E(X))^2
And we get
E(X_a) = (Var(X) + (E(X))^2) / E(X) = E(X) + Var(X) / E(X)
I find it much more inquisitive and visceral, to the extent that p99 now boggles my mind. 2N would be dreadful as an availability figure, yet for UX it's treated very different. So much so that my measurements corroborate exactly that; good UX requires the same many-nines reliability, not one or two.
I wonder if it's p90 and p99 to blame for the shoddy services we have in a way. It's pretty hard to argue for fixing something when it's presented as only going wrong 0.5% or less of the time after all. Even if at scale that means most of your users are experiencing it weekly.
trb•1h ago
By focusing on the tail and optimizing worst cases you help users more than by improving your median latency.