347. Kingman VUT

A foundational approximation for the G/G/1 queue (general arrival distribution, general service distribution, single server). The VUT equation:

— ariability × tilization × ime.

347.1. Components

347.2. Three factors — three levers

  1. Variability

    • Reduce arrival variability (smooth incoming flow, level-load production)
    • Reduce service variability (standardize, reduce setup-time variation)
    • Halve variability → halve queue time

  2. Utilization

    • Critical: as
    • Going from to doubles the queue
    • Going to doubles again

  3. Time

    • Faster service rate (lower ) shrinks queues directly
    • Both reduces and lowers at fixed

347.3. Why it’s an approximation

The VUT formula is exact only for M/M/1 (Poisson arrivals + exponential service, ):

For other distributions, VUT is a Kingman approximation — derived from upper bounds (Kingman 1962). Tight when traffic is heavy (); less accurate when load is light.

347.4. Generalization: G/G/c (multi-server)

For identical servers, the Sakasegawa approximation:

See Sakasegawa.

347.5. Variability-Utilization-Time intuition

This is the single most important formula in factory physics. It says:

347.6. Common applications

347.7. See also