Kingman's Formula

Kingman's approximation for the mean wait in a single-server queue (G/G/1) is E(Wq) ≈ (ρ/(1−ρ)) · ((ca² + cs²)/2) · τ, where ρ is utilization, ca and cs are the coefficients of variation of arrival and service times, and τ is the mean service time; it is also called the VUT equation, after its three factors. The utilization term does the damage: ρ/(1−ρ) is 4 at 80%, 9 at 90% and 19 at 95%, so going from 90% to 95% busy roughly doubles the wait with nothing else changed. Those multipliers, like the formula, are for one server; a pool of several servers sharing a queue keeps the wait short up to a higher utilization before it starts to climb. Variability multiplies it, so bursty arrivals and uneven job sizes lengthen the wait at any load. The approximation is generally accurate, and most accurate for a system close to saturation. Little's Law says L = λW but not why W grows; Kingman's formula shows where the growth comes from. The same arithmetic plausibly applies to people: a team whose every hour is booked tends to make every new request wait. That is an inference from the model, not something Kingman measured.

In February 2026 DNSimple described version 10 of erldns, its authoritative DNS server: when scheduler utilization goes above 90%, the server reduces the rate at which it accepts work. The post asks why not push for 100% utilization and answers with Kingman's formula, saying queueing latency grows hyperbolically with utilization, so the 90% ceiling is a deliberate trade-off to stay away from the latency cliff.

Headroom Costs Money: spare capacity is paid for whether or not anyone uses it, so whoever owns the budget pushes utilization up, and often rightly; the Google SRE book says "Efficient use of resources is important any time a service cares about money." and treats provisioning as a large lever on a service's total cost, so where a slower answer costs little, running hotter frees money for work that matters more.