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Equation 30 · Part 1 · The Network Is the Computer Again

Symbol P_slow

Pslow=1−(1−q)p,P_{\mathrm{slow}} = 1 - (1-q)^{p},
PslowP_{\mathrm{slow}}

What this part means

the probability that at least one of p participants does so.

Its job in the formula

PsP_slow is part of the quantity the equation computes from the expression on the right.

Where the article explains it

If each participant independently exceeds its budget with probability q , the probability that at least one of p participants does so is

The passage around this formula

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…

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A subscript is a label attached below a symbol. It often selects a time step, component, category, or member of a sequence.

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Sources cited in the surrounding passage

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