What does this equation mean?

The reason this matters more for collectives than for ordinary traffic is synchronisation. A collective is a barrier. Every participant waits for the operation to complete, so the operation completes when the last contribution arrives. Tail latency therefore stops being a tail statistic and becomes the mean cost of every step. Dean and Barroso’s analysis of latency variability in large fan-out services is the canonical statement of the mechanism: background activity, queueing at multiple layers including network switches, and shared-resource contention produce occasional slow responses, and a request that must wait on many servers is very likely to encounter one [ 18 ] . The arithmetic is unforgiving. If each participant independently exceeds its budget with probability q , the probability that at least one of p participants does so is $P_{\mathrm{slow}} = 1 - (1-q)^{p}$. so at a one-percent per-participant rate and 100 participants, roughly 63 percent of collectives are affected — and at 1,000 participants, essentially all of them. That is my own arithmetic on an independence assumption that real systems violate in both directions, but the direction of the effect is not in doubt.