USL: throughput C(N) = λN / (1 + σ(N−1) + κN(N−1)). The second term (coherency κ) explains why adding nodes yields negative returns: crosstalk grows quadratically.
The model is Neil Gunther’s: he set it out in Guerrilla Capacity Planning (2007) and in “A General Theory of Computational Scalability Based on Rational Functions” (arXiv, 2008). σ is contention — the share of work queued on a shared resource; κ is coherency — the cost of keeping copies of state consistent, which grows with the number of pairs of workers and therefore roughly as N². Setting the derivative to zero gives the peak at N* = √((1 − σ) / κ): with σ = 0.05 and κ = 0.001 throughput is highest at about 31 workers and falls after that.
The relation to Amdahl’s law (89) is direct: with κ = 0 USL reduces to it, and σ is that same serial fraction. The difference is that Amdahl’s law follows from arithmetic and gives a ceiling — the curve levels off into a plateau — while USL adds a term that makes the curve go down past its peak: coordination between nodes becomes work in its own right. The status differs too: Amdahl leaves nothing to fit, whereas USL’s σ and κ exist only as the output of a regression on your measurements.
Where κ comes from in data systems. Anything every worker must agree on: a consumer group rebalance that stops all members, checkpoint barriers that wait for the slowest subtask, a distributed lock, cache invalidation broadcast to every node, a shuffle in which each of N tasks talks to each of the others (93). Kafka adds a hard step on top: consumers beyond the number of partitions sit idle, which no smooth curve predicts. Hence the practical rule: measure σ and κ on a load-test rig before scaling out, or you pay for nodes that lower throughput.
Limits of the fit. USL is a tool for extrapolation and a vocabulary, not a formula to fill with numbers from documentation. Fit on measurements at many load levels across the range you care about; through three points almost any curve fits. Extrapolate a little past the last measurement, not by a factor of ten: a new bottleneck (network, disk, a downstream service) changes the parameters. A fitted κ that comes out negative or a σ above 1 means the model does not describe this system, not that the system is superlinear by magic — although genuine superlinear effects exist, for example when added nodes bring enough cache that the working set suddenly fits in memory.