Model

Kingman's approximation

Utilization sets the multiplier, variability sets what is multiplied.

W_{q} \approx \frac{ρ}{1 - ρ} \cdot \frac{c _{a}^{2} + c _{s}^{2}}{2} \cdot S

$W_{q}$: Waiting time; the average waiting time
$ρ$: Utilization; fraction of time the server is busy
$c_{a}^{2}$: Arrival variability; how variable arrivals are
$c_{s}^{2}$: Service variability; how variable service times are
$S$: Service time; the average service time

Recurring principle: Variance costs more than volume

How it behaves

One input on the horizontal axis, the others held fixed.

cₛ² = 1 (steady)cₛ² = 4cₛ² = 19 (a few very slow requests)

Data table

Waiting time by utilization, for three service-time variabilities (cₐ² = 1, S = 1)
Utilization ρ	series	Waiting time Wq (× S)
0.012	cₛ² = 1 (steady)	0.012×
0.024	cₛ² = 1 (steady)	0.024×
0.036	cₛ² = 1 (steady)	0.037×
0.048	cₛ² = 1 (steady)	0.05×
0.059	cₛ² = 1 (steady)	0.063×
0.071	cₛ² = 1 (steady)	0.077×
0.083	cₛ² = 1 (steady)	0.091×
0.095	cₛ² = 1 (steady)	0.105×
0.107	cₛ² = 1 (steady)	0.12×
0.119	cₛ² = 1 (steady)	0.135×
0.131	cₛ² = 1 (steady)	0.15×
0.142	cₛ² = 1 (steady)	0.166×
0.154	cₛ² = 1 (steady)	0.183×
0.166	cₛ² = 1 (steady)	0.199×
0.178	cₛ² = 1 (steady)	0.217×
0.19	cₛ² = 1 (steady)	0.235×
0.202	cₛ² = 1 (steady)	0.253×
0.214	cₛ² = 1 (steady)	0.272×
0.226	cₛ² = 1 (steady)	0.291×
0.237	cₛ² = 1 (steady)	0.311×
0.249	cₛ² = 1 (steady)	0.332×
0.261	cₛ² = 1 (steady)	0.354×
0.273	cₛ² = 1 (steady)	0.376×
0.285	cₛ² = 1 (steady)	0.399×
0.297	cₛ² = 1 (steady)	0.422×
0.309	cₛ² = 1 (steady)	0.447×
0.321	cₛ² = 1 (steady)	0.472×
0.332	cₛ² = 1 (steady)	0.498×
0.344	cₛ² = 1 (steady)	0.525×
0.356	cₛ² = 1 (steady)	0.553×
0.368	cₛ² = 1 (steady)	0.583×
0.38	cₛ² = 1 (steady)	0.613×
0.392	cₛ² = 1 (steady)	0.644×
0.404	cₛ² = 1 (steady)	0.677×
0.416	cₛ² = 1 (steady)	0.711×
0.427	cₛ² = 1 (steady)	0.747×
0.439	cₛ² = 1 (steady)	0.784×
0.451	cₛ² = 1 (steady)	0.822×
0.463	cₛ² = 1 (steady)	0.863×
0.475	cₛ² = 1 (steady)	0.905×
0.487	cₛ² = 1 (steady)	0.949×
0.499	cₛ² = 1 (steady)	0.995×
0.511	cₛ² = 1 (steady)	1.04×
0.522	cₛ² = 1 (steady)	1.09×
0.534	cₛ² = 1 (steady)	1.15×
0.546	cₛ² = 1 (steady)	1.2×
0.558	cₛ² = 1 (steady)	1.26×
0.57	cₛ² = 1 (steady)	1.33×
0.582	cₛ² = 1 (steady)	1.39×
0.594	cₛ² = 1 (steady)	1.46×
0.606	cₛ² = 1 (steady)	1.54×
0.617	cₛ² = 1 (steady)	1.61×
0.629	cₛ² = 1 (steady)	1.7×
0.641	cₛ² = 1 (steady)	1.79×
0.653	cₛ² = 1 (steady)	1.88×
0.665	cₛ² = 1 (steady)	1.99×
0.677	cₛ² = 1 (steady)	2.09×
0.689	cₛ² = 1 (steady)	2.21×
0.701	cₛ² = 1 (steady)	2.34×
0.713	cₛ² = 1 (steady)	2.48×
0.724	cₛ² = 1 (steady)	2.63×
0.736	cₛ² = 1 (steady)	2.79×
0.748	cₛ² = 1 (steady)	2.97×
0.76	cₛ² = 1 (steady)	3.17×
0.772	cₛ² = 1 (steady)	3.38×
0.784	cₛ² = 1 (steady)	3.62×
0.796	cₛ² = 1 (steady)	3.89×
0.807	cₛ² = 1 (steady)	4.19×
0.819	cₛ² = 1 (steady)	4.54×
0.831	cₛ² = 1 (steady)	4.93×
0.843	cₛ² = 1 (steady)	5.37×
0.855	cₛ² = 1 (steady)	5.9×
0.867	cₛ² = 1 (steady)	6.51×
0.879	cₛ² = 1 (steady)	7.25×
0.891	cₛ² = 1 (steady)	8.14×
0.903	cₛ² = 1 (steady)	9.26×
0.914	cₛ² = 1 (steady)	10.7×
0.926	cₛ² = 1 (steady)	12.6×
0.938	cₛ² = 1 (steady)	15.2×
0.95	cₛ² = 1 (steady)	19×
0.012	cₛ² = 4	0.03×
0.024	cₛ² = 4	0.061×
0.036	cₛ² = 4	0.092×
0.048	cₛ² = 4	0.125×
0.059	cₛ² = 4	0.158×
0.071	cₛ² = 4	0.192×
0.083	cₛ² = 4	0.227×
0.095	cₛ² = 4	0.262×
0.107	cₛ² = 4	0.299×
0.119	cₛ² = 4	0.337×
0.131	cₛ² = 4	0.376×
0.142	cₛ² = 4	0.415×
0.154	cₛ² = 4	0.456×
0.166	cₛ² = 4	0.499×
0.178	cₛ² = 4	0.542×
0.19	cₛ² = 4	0.586×
0.202	cₛ² = 4	0.632×
0.214	cₛ² = 4	0.68×
0.226	cₛ² = 4	0.728×
0.237	cₛ² = 4	0.779×
0.249	cₛ² = 4	0.831×
0.261	cₛ² = 4	0.884×
0.273	cₛ² = 4	0.939×
0.285	cₛ² = 4	0.997×
0.297	cₛ² = 4	1.06×
0.309	cₛ² = 4	1.12×
0.321	cₛ² = 4	1.18×
0.332	cₛ² = 4	1.25×
0.344	cₛ² = 4	1.31×
0.356	cₛ² = 4	1.38×
0.368	cₛ² = 4	1.46×
0.38	cₛ² = 4	1.53×
0.392	cₛ² = 4	1.61×
0.404	cₛ² = 4	1.69×
0.416	cₛ² = 4	1.78×
0.427	cₛ² = 4	1.87×
0.439	cₛ² = 4	1.96×
0.451	cₛ² = 4	2.06×
0.463	cₛ² = 4	2.16×
0.475	cₛ² = 4	2.26×
0.487	cₛ² = 4	2.37×
0.499	cₛ² = 4	2.49×
0.511	cₛ² = 4	2.61×
0.522	cₛ² = 4	2.74×
0.534	cₛ² = 4	2.87×
0.546	cₛ² = 4	3.01×
0.558	cₛ² = 4	3.16×
0.57	cₛ² = 4	3.31×
0.582	cₛ² = 4	3.48×
0.594	cₛ² = 4	3.65×
0.606	cₛ² = 4	3.84×
0.617	cₛ² = 4	4.04×
0.629	cₛ² = 4	4.25×
0.641	cₛ² = 4	4.47×
0.653	cₛ² = 4	4.71×
0.665	cₛ² = 4	4.96×
0.677	cₛ² = 4	5.24×
0.689	cₛ² = 4	5.53×
0.701	cₛ² = 4	5.85×
0.713	cₛ² = 4	6.2×
0.724	cₛ² = 4	6.57×
0.736	cₛ² = 4	6.98×
0.748	cₛ² = 4	7.43×
0.76	cₛ² = 4	7.92×
0.772	cₛ² = 4	8.46×
0.784	cₛ² = 4	9.06×
0.796	cₛ² = 4	9.73×
0.807	cₛ² = 4	10.5×
0.819	cₛ² = 4	11.3×
0.831	cₛ² = 4	12.3×
0.843	cₛ² = 4	13.4×
0.855	cₛ² = 4	14.7×
0.867	cₛ² = 4	16.3×
0.879	cₛ² = 4	18.1×
0.891	cₛ² = 4	20.4×
0.903	cₛ² = 4	23.1×
0.914	cₛ² = 4	26.7×
0.926	cₛ² = 4	31.4×
0.938	cₛ² = 4	37.9×
0.95	cₛ² = 4	47.5×
0.012	cₛ² = 19 (a few very slow requests)	0.12×
0.024	cₛ² = 19 (a few very slow requests)	0.243×
0.036	cₛ² = 19 (a few very slow requests)	0.369×
0.048	cₛ² = 19 (a few very slow requests)	0.499×
0.059	cₛ² = 19 (a few very slow requests)	0.631×
0.071	cₛ² = 19 (a few very slow requests)	0.767×
0.083	cₛ² = 19 (a few very slow requests)	0.907×
0.095	cₛ² = 19 (a few very slow requests)	1.05×
0.107	cₛ² = 19 (a few very slow requests)	1.2×
0.119	cₛ² = 19 (a few very slow requests)	1.35×
0.131	cₛ² = 19 (a few very slow requests)	1.5×
0.142	cₛ² = 19 (a few very slow requests)	1.66×
0.154	cₛ² = 19 (a few very slow requests)	1.83×
0.166	cₛ² = 19 (a few very slow requests)	1.99×
0.178	cₛ² = 19 (a few very slow requests)	2.17×
0.19	cₛ² = 19 (a few very slow requests)	2.35×
0.202	cₛ² = 19 (a few very slow requests)	2.53×
0.214	cₛ² = 19 (a few very slow requests)	2.72×
0.226	cₛ² = 19 (a few very slow requests)	2.91×
0.237	cₛ² = 19 (a few very slow requests)	3.11×
0.249	cₛ² = 19 (a few very slow requests)	3.32×
0.261	cₛ² = 19 (a few very slow requests)	3.54×
0.273	cₛ² = 19 (a few very slow requests)	3.76×
0.285	cₛ² = 19 (a few very slow requests)	3.99×
0.297	cₛ² = 19 (a few very slow requests)	4.22×
0.309	cₛ² = 19 (a few very slow requests)	4.47×
0.321	cₛ² = 19 (a few very slow requests)	4.72×
0.332	cₛ² = 19 (a few very slow requests)	4.98×
0.344	cₛ² = 19 (a few very slow requests)	5.25×
0.356	cₛ² = 19 (a few very slow requests)	5.53×
0.368	cₛ² = 19 (a few very slow requests)	5.83×
0.38	cₛ² = 19 (a few very slow requests)	6.13×
0.392	cₛ² = 19 (a few very slow requests)	6.44×
0.404	cₛ² = 19 (a few very slow requests)	6.77×
0.416	cₛ² = 19 (a few very slow requests)	7.11×
0.427	cₛ² = 19 (a few very slow requests)	7.47×
0.439	cₛ² = 19 (a few very slow requests)	7.84×
0.451	cₛ² = 19 (a few very slow requests)	8.22×
0.463	cₛ² = 19 (a few very slow requests)	8.63×
0.475	cₛ² = 19 (a few very slow requests)	9.05×
0.487	cₛ² = 19 (a few very slow requests)	9.49×
0.499	cₛ² = 19 (a few very slow requests)	9.95×
0.511	cₛ² = 19 (a few very slow requests)	10.4×
0.522	cₛ² = 19 (a few very slow requests)	10.9×
0.534	cₛ² = 19 (a few very slow requests)	11.5×
0.546	cₛ² = 19 (a few very slow requests)	12×
0.558	cₛ² = 19 (a few very slow requests)	12.6×
0.57	cₛ² = 19 (a few very slow requests)	13.3×
0.582	cₛ² = 19 (a few very slow requests)	13.9×
0.594	cₛ² = 19 (a few very slow requests)	14.6×
0.606	cₛ² = 19 (a few very slow requests)	15.4×
0.617	cₛ² = 19 (a few very slow requests)	16.1×
0.629	cₛ² = 19 (a few very slow requests)	17×
0.641	cₛ² = 19 (a few very slow requests)	17.9×
0.653	cₛ² = 19 (a few very slow requests)	18.8×
0.665	cₛ² = 19 (a few very slow requests)	19.9×
0.677	cₛ² = 19 (a few very slow requests)	20.9×
0.689	cₛ² = 19 (a few very slow requests)	22.1×
0.701	cₛ² = 19 (a few very slow requests)	23.4×
0.713	cₛ² = 19 (a few very slow requests)	24.8×
0.724	cₛ² = 19 (a few very slow requests)	26.3×
0.736	cₛ² = 19 (a few very slow requests)	27.9×
0.748	cₛ² = 19 (a few very slow requests)	29.7×
0.76	cₛ² = 19 (a few very slow requests)	31.7×
0.772	cₛ² = 19 (a few very slow requests)	33.8×
0.784	cₛ² = 19 (a few very slow requests)	36.2×
0.796	cₛ² = 19 (a few very slow requests)	38.9×
0.807	cₛ² = 19 (a few very slow requests)	41.9×
0.819	cₛ² = 19 (a few very slow requests)	45.4×
0.831	cₛ² = 19 (a few very slow requests)	49.3×
0.843	cₛ² = 19 (a few very slow requests)	53.7×
0.855	cₛ² = 19 (a few very slow requests)	59×
0.867	cₛ² = 19 (a few very slow requests)	65.1×
0.879	cₛ² = 19 (a few very slow requests)	72.5×
0.891	cₛ² = 19 (a few very slow requests)	81.4×
0.903	cₛ² = 19 (a few very slow requests)	92.6×
0.914	cₛ² = 19 (a few very slow requests)	107×
0.926	cₛ² = 19 (a few very slow requests)	126×
0.938	cₛ² = 19 (a few very slow requests)	152×
0.95	cₛ² = 19 (a few very slow requests)	190×

Worked examples, recomputed

Every number the mechanisms quote for this relationship is recomputed from it, so none of them disagrees with it.

cₛ² = 19 against 1 at the same ρ → 10× the wait

Where it appears

Each mechanism states it in its own names. The general symbol each name stands for is shown with it.

Physical Substrate & Performance Laws
Why do trade-offs exist at all?
$W_{q} \approx (\frac{ρ}{1 - ρ}) \times (\frac{c _{a}^{2} + c _{s}^{2}}{2}) \times S$
Read it in place →