Sakasegawa

361. Sakasegawa

Generalization of the VUT formula to G/G/c — multi-server queues with general arrival and service distributions.

361.1. Formula

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

where now:

$𝑐$ : number of servers
$𝜌 = \frac{𝜆}{𝑐 𝜇}$ : per-server utilization
$𝑊_{𝑞}$ : expected wait time

The factor $\frac{𝜌^{\sqrt{2 (𝑐 + 1)} - 1}}{𝑐 (1 - 𝜌)}$ replaces the single-server $\frac{𝜌}{1 - 𝜌}$ .

361.2. Why the exponent

For $𝑐 = 1$ : $\sqrt{2 \cdot 2} - 1 = \sqrt{4} - 1 = 1$ → $𝜌^{1} = 𝜌$ , recovering Kingman VUT.

For $𝑐 \to \infty$ : $\sqrt{2 (𝑐 + 1)} - 1 \to \infty$ , and $𝜌^{large}$ goes to $0$ rapidly → queues vanish at fixed $𝜌$ . As you add servers, you absorb variability more efficiently than just dividing the work — a fundamental insight.

361.3. Practical impact

Two M/M/1 servers handling $\frac{𝜆}{2}$ each vs one M/M/2 server handling $𝜆$ :

Setup	$𝜌$	$\frac{𝑊_{𝑞}}{\frac{1}{𝜇}}$	Comparison
Two separate M/M/1	$𝜌_{1} = \frac{𝜆}{2 𝜇}$	$\frac{𝜌_{1}}{1 - 𝜌_{1}}$	each server has own queue
Pooled M/M/2	$𝜌_{2} = \frac{𝜆}{2 𝜇}$	$\frac{𝜌_{2}^{\sqrt{6} - 1}}{2 (1 - 𝜌_{2})}$ — lower	shared queue, better

Pooling reduces waiting time even at the same per-server utilization. This is the queueing analog of risk pooling.

361.4. Limitations

Approximation: tight in heavy traffic, less accurate when $𝜌$ is small
Identical servers assumed
No abandonment: customers can’t leave; for that see Erlang A

361.5. Where used

Call center sizing: how many agents for given call volume and service-time distribution
Web service capacity planning: request handlers
Bank teller / DMV staffing
Manufacturing: multiple parallel machines

361.6. See also

Kingman VUT — single-server case
Linking Equation — variance propagation across stations
Erlang C — exact M/M/c waiting probability
Erlang B — exact M/M/c/c (loss)