Kingman VUT

350. Kingman VUT

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

𝑊_{𝑞} \approx \underset{𝑉}{\underset{⏟}{\frac{𝑐_{𝑎}^{2} + 𝑐_{𝑠}^{2}}{2}}} \underset{𝑈}{\underset{⏟}{\frac{𝜌}{1 - 𝜌}}} \underset{𝑇}{\underset{⏟}{\frac{1}{𝜇}}}

— $𝑉$ ariability × $𝑈$ tilization × $𝑇$ ime.

350.1. Components

$𝑐_{𝑎}$ : coefficient of variation of inter-arrival times ( $\frac{std}{mean}$ )
$𝑐_{𝑠}$ : coefficient of variation of service times
$𝜌 = \frac{𝜆}{𝜇}$ : utilization (arrival rate / service rate)
$\frac{1}{𝜇}$ : mean service time
$𝑊_{𝑞}$ : expected wait in queue (before service starts)

350.2. Three factors — three levers

Variability $𝑉 = \frac{𝑐_{𝑎}^{2} + 𝑐_{𝑠}^{2}}{2}$
- Reduce arrival variability (smooth incoming flow, level-load production)
- Reduce service variability (standardize, reduce setup-time variation)
- Halve variability → halve queue time
Utilization $𝑈 = \frac{𝜌}{1 - 𝜌}$
- Critical: $𝑈 \to \infty$ as $𝜌 \to 1$
- Going from $𝜌 = 0.8$ to $𝜌 = 0.9$ doubles the queue
- Going to $𝜌 = 0.95$ doubles again
Time $𝑇 = \frac{1}{𝜇}$
- Faster service rate (lower $\frac{1}{𝜇}$ ) shrinks queues directly
- Both reduces $𝑇$ and lowers $𝜌$ at fixed $𝜆$

350.3. Why it’s an approximation

The VUT formula is exact only for M/M/1 (Poisson arrivals + exponential service, $𝑐_{𝑎} = 𝑐_{𝑠} = 1$ ):

𝑊_{𝑞}^{M/M/1} = \frac{1 + 1}{2} \cdot \frac{𝜌}{1 - 𝜌} \cdot \frac{1}{𝜇} = \frac{𝜌}{𝜇 (1 - 𝜌)} ✓

For other distributions, VUT is a Kingman approximation — derived from upper bounds (Kingman 1962). Tight when traffic is heavy ( $𝜌 \to 1$ ); less accurate when load is light.

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

For $𝑐$ identical servers, the Sakasegawa approximation:

𝑊_{𝑞}^{G/G/c} \approx \frac{𝑐_{𝑎}^{2} + 𝑐_{𝑠}^{2}}{2} \cdot \frac{𝜌^{\sqrt{2 (𝑐 + 1)} - 1}}{𝑐 (1 - 𝜌)} \cdot \frac{1}{𝜇}

See Sakasegawa.

350.5. Variability-Utilization-Time intuition

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

Queues grow non-linearly in utilization — 80% utilization is dramatically different from 95%
Variability is half the battle — even at high utilization, low variability keeps queues small
Capacity and demand both matter — but variability multiplies their effect

350.6. Common applications

Manufacturing lines: setup variability, breakdown variability dominate
Call centers: arrival peaks + service-time spread
Hospitals: patient arrival randomness + variable treatment times
Computer systems: request bursts + response-time variance

350.7. See also

Sakasegawa — multi-server extension
Linking Equation — variance propagation
M/M/1 — exact case
Little’s Law — $𝐿 = 𝜆 𝑊$
Factory Physics