Model

What spilling costs

The slowdown is not proportional to how much spilled: each spilled byte costs r times more, twice.

time \propto \frac{( 1 - f ) + 2 f r}{B _{mem}}, r = \frac{B _{mem}}{B _{disk}}

$f$: Spill fraction; the share of the data that goes to disk
$r$: Bandwidth ratio; memory bandwidth ÷ disk bandwidth (about 50 per node)

Recurring principle: Fixed capacity meets growth

How it behaves

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

local NVMe (r = 50)network-attached disk (r = 500)

Data table

Slowdown by spill fraction
Spill fraction f	series	Slowdown (× in-memory)
0	local NVMe (r = 50)	1×
0.013	local NVMe (r = 50)	2.24×
0.025	local NVMe (r = 50)	3.48×
0.038	local NVMe (r = 50)	4.71×
0.05	local NVMe (r = 50)	5.95×
0.063	local NVMe (r = 50)	7.19×
0.075	local NVMe (r = 50)	8.43×
0.088	local NVMe (r = 50)	9.66×
0.1	local NVMe (r = 50)	10.9×
0.113	local NVMe (r = 50)	12.1×
0.125	local NVMe (r = 50)	13.4×
0.138	local NVMe (r = 50)	14.6×
0.15	local NVMe (r = 50)	15.8×
0.163	local NVMe (r = 50)	17.1×
0.175	local NVMe (r = 50)	18.3×
0.188	local NVMe (r = 50)	19.6×
0.2	local NVMe (r = 50)	20.8×
0.212	local NVMe (r = 50)	22×
0.225	local NVMe (r = 50)	23.3×
0.237	local NVMe (r = 50)	24.5×
0.25	local NVMe (r = 50)	25.8×
0.263	local NVMe (r = 50)	27×
0.275	local NVMe (r = 50)	28.2×
0.287	local NVMe (r = 50)	29.5×
0.3	local NVMe (r = 50)	30.7×
0.313	local NVMe (r = 50)	31.9×
0.325	local NVMe (r = 50)	33.2×
0.338	local NVMe (r = 50)	34.4×
0.35	local NVMe (r = 50)	35.6×
0.362	local NVMe (r = 50)	36.9×
0.375	local NVMe (r = 50)	38.1×
0.388	local NVMe (r = 50)	39.4×
0.4	local NVMe (r = 50)	40.6×
0.412	local NVMe (r = 50)	41.8×
0.425	local NVMe (r = 50)	43.1×
0.438	local NVMe (r = 50)	44.3×
0.45	local NVMe (r = 50)	45.5×
0.463	local NVMe (r = 50)	46.8×
0.475	local NVMe (r = 50)	48×
0.487	local NVMe (r = 50)	49.3×
0.5	local NVMe (r = 50)	50.5×
0.512	local NVMe (r = 50)	51.7×
0.525	local NVMe (r = 50)	53×
0.537	local NVMe (r = 50)	54.2×
0.55	local NVMe (r = 50)	55.5×
0.563	local NVMe (r = 50)	56.7×
0.575	local NVMe (r = 50)	57.9×
0.588	local NVMe (r = 50)	59.2×
0.6	local NVMe (r = 50)	60.4×
0.613	local NVMe (r = 50)	61.6×
0.625	local NVMe (r = 50)	62.9×
0.637	local NVMe (r = 50)	64.1×
0.65	local NVMe (r = 50)	65.3×
0.662	local NVMe (r = 50)	66.6×
0.675	local NVMe (r = 50)	67.8×
0.688	local NVMe (r = 50)	69.1×
0.7	local NVMe (r = 50)	70.3×
0.713	local NVMe (r = 50)	71.5×
0.725	local NVMe (r = 50)	72.8×
0.738	local NVMe (r = 50)	74×
0.75	local NVMe (r = 50)	75.3×
0.762	local NVMe (r = 50)	76.5×
0.775	local NVMe (r = 50)	77.7×
0.787	local NVMe (r = 50)	79×
0.8	local NVMe (r = 50)	80.2×
0.813	local NVMe (r = 50)	81.4×
0.825	local NVMe (r = 50)	82.7×
0.838	local NVMe (r = 50)	83.9×
0.85	local NVMe (r = 50)	85.2×
0.863	local NVMe (r = 50)	86.4×
0.875	local NVMe (r = 50)	87.6×
0.887	local NVMe (r = 50)	88.9×
0.9	local NVMe (r = 50)	90.1×
0.912	local NVMe (r = 50)	91.3×
0.925	local NVMe (r = 50)	92.6×
0.938	local NVMe (r = 50)	93.8×
0.95	local NVMe (r = 50)	95×
0.963	local NVMe (r = 50)	96.3×
0.975	local NVMe (r = 50)	97.5×
0.988	local NVMe (r = 50)	98.8×
1	local NVMe (r = 50)	100×
0	network-attached disk (r = 500)	1×
0.013	network-attached disk (r = 500)	13.5×
0.025	network-attached disk (r = 500)	26×
0.038	network-attached disk (r = 500)	38.5×
0.05	network-attached disk (r = 500)	51×
0.063	network-attached disk (r = 500)	63.4×
0.075	network-attached disk (r = 500)	75.9×
0.088	network-attached disk (r = 500)	88.4×
0.1	network-attached disk (r = 500)	101×
0.113	network-attached disk (r = 500)	113×
0.125	network-attached disk (r = 500)	126×
0.138	network-attached disk (r = 500)	138×
0.15	network-attached disk (r = 500)	151×
0.163	network-attached disk (r = 500)	163×
0.175	network-attached disk (r = 500)	176×
0.188	network-attached disk (r = 500)	188×
0.2	network-attached disk (r = 500)	201×
0.212	network-attached disk (r = 500)	213×
0.225	network-attached disk (r = 500)	226×
0.237	network-attached disk (r = 500)	238×
0.25	network-attached disk (r = 500)	251×
0.263	network-attached disk (r = 500)	263×
0.275	network-attached disk (r = 500)	276×
0.287	network-attached disk (r = 500)	288×
0.3	network-attached disk (r = 500)	301×
0.313	network-attached disk (r = 500)	313×
0.325	network-attached disk (r = 500)	326×
0.338	network-attached disk (r = 500)	338×
0.35	network-attached disk (r = 500)	351×
0.362	network-attached disk (r = 500)	363×
0.375	network-attached disk (r = 500)	376×
0.388	network-attached disk (r = 500)	388×
0.4	network-attached disk (r = 500)	401×
0.412	network-attached disk (r = 500)	413×
0.425	network-attached disk (r = 500)	426×
0.438	network-attached disk (r = 500)	438×
0.45	network-attached disk (r = 500)	451×
0.463	network-attached disk (r = 500)	463×
0.475	network-attached disk (r = 500)	476×
0.487	network-attached disk (r = 500)	488×
0.5	network-attached disk (r = 500)	501×
0.512	network-attached disk (r = 500)	513×
0.525	network-attached disk (r = 500)	525×
0.537	network-attached disk (r = 500)	538×
0.55	network-attached disk (r = 500)	550×
0.563	network-attached disk (r = 500)	563×
0.575	network-attached disk (r = 500)	575×
0.588	network-attached disk (r = 500)	588×
0.6	network-attached disk (r = 500)	600×
0.613	network-attached disk (r = 500)	613×
0.625	network-attached disk (r = 500)	625×
0.637	network-attached disk (r = 500)	638×
0.65	network-attached disk (r = 500)	650×
0.662	network-attached disk (r = 500)	663×
0.675	network-attached disk (r = 500)	675×
0.688	network-attached disk (r = 500)	688×
0.7	network-attached disk (r = 500)	700×
0.713	network-attached disk (r = 500)	713×
0.725	network-attached disk (r = 500)	725×
0.738	network-attached disk (r = 500)	738×
0.75	network-attached disk (r = 500)	750×
0.762	network-attached disk (r = 500)	763×
0.775	network-attached disk (r = 500)	775×
0.787	network-attached disk (r = 500)	788×
0.8	network-attached disk (r = 500)	800×
0.813	network-attached disk (r = 500)	813×
0.825	network-attached disk (r = 500)	825×
0.838	network-attached disk (r = 500)	838×
0.85	network-attached disk (r = 500)	850×
0.863	network-attached disk (r = 500)	863×
0.875	network-attached disk (r = 500)	875×
0.887	network-attached disk (r = 500)	888×
0.9	network-attached disk (r = 500)	900×
0.912	network-attached disk (r = 500)	913×
0.925	network-attached disk (r = 500)	925×
0.938	network-attached disk (r = 500)	938×
0.95	network-attached disk (r = 500)	950×
0.963	network-attached disk (r = 500)	963×
0.975	network-attached disk (r = 500)	975×
0.988	network-attached disk (r = 500)	988×
1	network-attached disk (r = 500)	1,000×

Worked examples, recomputed

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

Where it appears

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

Memory & Resource Management
Who is allowed to hold how much memory, for how long, and what happens to the work when the answer is less than it asked for?
$time \propto [(1 - f) + 2 \cdot f \cdot r] / B_{mem}$
Read it in place →