348. Sakasegawa
Generalization of the VUT formula to G/G/c — multi-server queues with general arrival and service distributions.
348.1. Formula
where now:
- : number of servers
- : per-server utilization
- : expected wait time
The factor replaces the single-server .
348.2. Why the exponent
For : → , recovering Kingman VUT.
For : , and goes to rapidly → queues vanish at fixed . As you add servers, you absorb variability more efficiently than just dividing the work — a fundamental insight.
348.3. Practical impact
Two M/M/1 servers handling each vs one M/M/2 server handling :
| Setup | Comparison | ||
|---|---|---|---|
| Two separate M/M/1 | each server has own queue | ||
| Pooled M/M/2 | — lower | shared queue, better |
Pooling reduces waiting time even at the same per-server utilization. This is the queueing analog of risk pooling.
348.4. Limitations
- Approximation: tight in heavy traffic, less accurate when is small
- Identical servers assumed
- No abandonment: customers can’t leave; for that see Erlang A
348.5. 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
348.6. 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)