Model

Fixed cost per operation

total cost = fixed cost + (cost per byte × size)

T (x) = F + \frac{x}{B}, x_{1/2} = F \cdot B

$F$: Fixed cost; the setup cost paid for each operation, whatever its size
$B$: Transfer rate; bytes per second once data streams
$x$: Size; bytes moved per operation
$x_{1/2}$: Natural transfer size; the size at which half of your time is setup

Recurring principle: Fixed cost per operation

How it behaves

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

object storage (F ≈ 50 ms, B ≈ 80 MB/s)hard disk (F ≈ 13 ms, B ≈ 100 MB/s)

Data table

Share of time spent moving data, by request size
Request size x	series	Time spent transferring
1,000 B	object storage (F ≈ 50 ms, B ≈ 80 MB/s)	0.025%
1.16 KB	object storage (F ≈ 50 ms, B ≈ 80 MB/s)	0.03%
1.38 KB	object storage (F ≈ 50 ms, B ≈ 80 MB/s)	0.035%
1.64 KB	object storage (F ≈ 50 ms, B ≈ 80 MB/s)	0.042%
1.95 KB	object storage (F ≈ 50 ms, B ≈ 80 MB/s)	0.05%
2.32 KB	object storage (F ≈ 50 ms, B ≈ 80 MB/s)	0.059%
2.75 KB	object storage (F ≈ 50 ms, B ≈ 80 MB/s)	0.07%
3.27 KB	object storage (F ≈ 50 ms, B ≈ 80 MB/s)	0.084%
3.89 KB	object storage (F ≈ 50 ms, B ≈ 80 MB/s)	0.099%
4.62 KB	object storage (F ≈ 50 ms, B ≈ 80 MB/s)	0.118%
5.49 KB	object storage (F ≈ 50 ms, B ≈ 80 MB/s)	0.14%
6.53 KB	object storage (F ≈ 50 ms, B ≈ 80 MB/s)	0.167%
7.76 KB	object storage (F ≈ 50 ms, B ≈ 80 MB/s)	0.198%
9.22 KB	object storage (F ≈ 50 ms, B ≈ 80 MB/s)	0.235%
11 KB	object storage (F ≈ 50 ms, B ≈ 80 MB/s)	0.28%
13 KB	object storage (F ≈ 50 ms, B ≈ 80 MB/s)	0.332%
15.5 KB	object storage (F ≈ 50 ms, B ≈ 80 MB/s)	0.395%
18.4 KB	object storage (F ≈ 50 ms, B ≈ 80 MB/s)	0.469%
21.9 KB	object storage (F ≈ 50 ms, B ≈ 80 MB/s)	0.557%
26 KB	object storage (F ≈ 50 ms, B ≈ 80 MB/s)	0.661%
30.9 KB	object storage (F ≈ 50 ms, B ≈ 80 MB/s)	0.784%
36.7 KB	object storage (F ≈ 50 ms, B ≈ 80 MB/s)	0.931%
43.6 KB	object storage (F ≈ 50 ms, B ≈ 80 MB/s)	1.1%
51.8 KB	object storage (F ≈ 50 ms, B ≈ 80 MB/s)	1.31%
61.6 KB	object storage (F ≈ 50 ms, B ≈ 80 MB/s)	1.55%
73.2 KB	object storage (F ≈ 50 ms, B ≈ 80 MB/s)	1.84%
87 KB	object storage (F ≈ 50 ms, B ≈ 80 MB/s)	2.18%
103 KB	object storage (F ≈ 50 ms, B ≈ 80 MB/s)	2.58%
123 KB	object storage (F ≈ 50 ms, B ≈ 80 MB/s)	3.05%
146 KB	object storage (F ≈ 50 ms, B ≈ 80 MB/s)	3.61%
174 KB	object storage (F ≈ 50 ms, B ≈ 80 MB/s)	4.26%
206 KB	object storage (F ≈ 50 ms, B ≈ 80 MB/s)	5.02%
245 KB	object storage (F ≈ 50 ms, B ≈ 80 MB/s)	5.91%
292 KB	object storage (F ≈ 50 ms, B ≈ 80 MB/s)	6.95%
346 KB	object storage (F ≈ 50 ms, B ≈ 80 MB/s)	8.15%
412 KB	object storage (F ≈ 50 ms, B ≈ 80 MB/s)	9.54%
489 KB	object storage (F ≈ 50 ms, B ≈ 80 MB/s)	11.1%
582 KB	object storage (F ≈ 50 ms, B ≈ 80 MB/s)	13%
691 KB	object storage (F ≈ 50 ms, B ≈ 80 MB/s)	15%
822 KB	object storage (F ≈ 50 ms, B ≈ 80 MB/s)	17.4%
977 KB	object storage (F ≈ 50 ms, B ≈ 80 MB/s)	20%
1.13 MB	object storage (F ≈ 50 ms, B ≈ 80 MB/s)	22.9%
1.35 MB	object storage (F ≈ 50 ms, B ≈ 80 MB/s)	26.1%
1.6 MB	object storage (F ≈ 50 ms, B ≈ 80 MB/s)	29.6%
1.9 MB	object storage (F ≈ 50 ms, B ≈ 80 MB/s)	33.3%
2.26 MB	object storage (F ≈ 50 ms, B ≈ 80 MB/s)	37.2%
2.69 MB	object storage (F ≈ 50 ms, B ≈ 80 MB/s)	41.3%
3.19 MB	object storage (F ≈ 50 ms, B ≈ 80 MB/s)	45.6%
3.8 MB	object storage (F ≈ 50 ms, B ≈ 80 MB/s)	49.9%
4.51 MB	object storage (F ≈ 50 ms, B ≈ 80 MB/s)	54.2%
5.36 MB	object storage (F ≈ 50 ms, B ≈ 80 MB/s)	58.4%
6.37 MB	object storage (F ≈ 50 ms, B ≈ 80 MB/s)	62.6%
7.58 MB	object storage (F ≈ 50 ms, B ≈ 80 MB/s)	66.5%
9 MB	object storage (F ≈ 50 ms, B ≈ 80 MB/s)	70.2%
10.7 MB	object storage (F ≈ 50 ms, B ≈ 80 MB/s)	73.7%
12.7 MB	object storage (F ≈ 50 ms, B ≈ 80 MB/s)	76.9%
15.1 MB	object storage (F ≈ 50 ms, B ≈ 80 MB/s)	79.8%
18 MB	object storage (F ≈ 50 ms, B ≈ 80 MB/s)	82.5%
21.4 MB	object storage (F ≈ 50 ms, B ≈ 80 MB/s)	84.8%
25.4 MB	object storage (F ≈ 50 ms, B ≈ 80 MB/s)	86.9%
30.2 MB	object storage (F ≈ 50 ms, B ≈ 80 MB/s)	88.8%
35.8 MB	object storage (F ≈ 50 ms, B ≈ 80 MB/s)	90.4%
42.6 MB	object storage (F ≈ 50 ms, B ≈ 80 MB/s)	91.8%
50.6 MB	object storage (F ≈ 50 ms, B ≈ 80 MB/s)	93%
60.2 MB	object storage (F ≈ 50 ms, B ≈ 80 MB/s)	94%
71.5 MB	object storage (F ≈ 50 ms, B ≈ 80 MB/s)	94.9%
85 MB	object storage (F ≈ 50 ms, B ≈ 80 MB/s)	95.7%
101 MB	object storage (F ≈ 50 ms, B ≈ 80 MB/s)	96.4%
120 MB	object storage (F ≈ 50 ms, B ≈ 80 MB/s)	96.9%
143 MB	object storage (F ≈ 50 ms, B ≈ 80 MB/s)	97.4%
170 MB	object storage (F ≈ 50 ms, B ≈ 80 MB/s)	97.8%
202 MB	object storage (F ≈ 50 ms, B ≈ 80 MB/s)	98.1%
240 MB	object storage (F ≈ 50 ms, B ≈ 80 MB/s)	98.4%
285 MB	object storage (F ≈ 50 ms, B ≈ 80 MB/s)	98.7%
338 MB	object storage (F ≈ 50 ms, B ≈ 80 MB/s)	98.9%
402 MB	object storage (F ≈ 50 ms, B ≈ 80 MB/s)	99.1%
478 MB	object storage (F ≈ 50 ms, B ≈ 80 MB/s)	99.2%
568 MB	object storage (F ≈ 50 ms, B ≈ 80 MB/s)	99.3%
675 MB	object storage (F ≈ 50 ms, B ≈ 80 MB/s)	99.4%
802 MB	object storage (F ≈ 50 ms, B ≈ 80 MB/s)	99.5%
954 MB	object storage (F ≈ 50 ms, B ≈ 80 MB/s)	99.6%
1,000 B	hard disk (F ≈ 13 ms, B ≈ 100 MB/s)	0.077%
1.16 KB	hard disk (F ≈ 13 ms, B ≈ 100 MB/s)	0.091%
1.38 KB	hard disk (F ≈ 13 ms, B ≈ 100 MB/s)	0.109%
1.64 KB	hard disk (F ≈ 13 ms, B ≈ 100 MB/s)	0.129%
1.95 KB	hard disk (F ≈ 13 ms, B ≈ 100 MB/s)	0.153%
2.32 KB	hard disk (F ≈ 13 ms, B ≈ 100 MB/s)	0.182%
2.75 KB	hard disk (F ≈ 13 ms, B ≈ 100 MB/s)	0.216%
3.27 KB	hard disk (F ≈ 13 ms, B ≈ 100 MB/s)	0.257%
3.89 KB	hard disk (F ≈ 13 ms, B ≈ 100 MB/s)	0.305%
4.62 KB	hard disk (F ≈ 13 ms, B ≈ 100 MB/s)	0.363%
5.49 KB	hard disk (F ≈ 13 ms, B ≈ 100 MB/s)	0.431%
6.53 KB	hard disk (F ≈ 13 ms, B ≈ 100 MB/s)	0.511%
7.76 KB	hard disk (F ≈ 13 ms, B ≈ 100 MB/s)	0.607%
9.22 KB	hard disk (F ≈ 13 ms, B ≈ 100 MB/s)	0.721%
11 KB	hard disk (F ≈ 13 ms, B ≈ 100 MB/s)	0.856%
13 KB	hard disk (F ≈ 13 ms, B ≈ 100 MB/s)	1.02%
15.5 KB	hard disk (F ≈ 13 ms, B ≈ 100 MB/s)	1.2%
18.4 KB	hard disk (F ≈ 13 ms, B ≈ 100 MB/s)	1.43%
21.9 KB	hard disk (F ≈ 13 ms, B ≈ 100 MB/s)	1.69%
26 KB	hard disk (F ≈ 13 ms, B ≈ 100 MB/s)	2.01%
30.9 KB	hard disk (F ≈ 13 ms, B ≈ 100 MB/s)	2.37%
36.7 KB	hard disk (F ≈ 13 ms, B ≈ 100 MB/s)	2.81%
43.6 KB	hard disk (F ≈ 13 ms, B ≈ 100 MB/s)	3.32%
51.8 KB	hard disk (F ≈ 13 ms, B ≈ 100 MB/s)	3.92%
61.6 KB	hard disk (F ≈ 13 ms, B ≈ 100 MB/s)	4.63%
73.2 KB	hard disk (F ≈ 13 ms, B ≈ 100 MB/s)	5.45%
87 KB	hard disk (F ≈ 13 ms, B ≈ 100 MB/s)	6.42%
103 KB	hard disk (F ≈ 13 ms, B ≈ 100 MB/s)	7.53%
123 KB	hard disk (F ≈ 13 ms, B ≈ 100 MB/s)	8.83%
146 KB	hard disk (F ≈ 13 ms, B ≈ 100 MB/s)	10.3%
174 KB	hard disk (F ≈ 13 ms, B ≈ 100 MB/s)	12%
206 KB	hard disk (F ≈ 13 ms, B ≈ 100 MB/s)	14%
245 KB	hard disk (F ≈ 13 ms, B ≈ 100 MB/s)	16.2%
292 KB	hard disk (F ≈ 13 ms, B ≈ 100 MB/s)	18.7%
346 KB	hard disk (F ≈ 13 ms, B ≈ 100 MB/s)	21.4%
412 KB	hard disk (F ≈ 13 ms, B ≈ 100 MB/s)	24.5%
489 KB	hard disk (F ≈ 13 ms, B ≈ 100 MB/s)	27.8%
582 KB	hard disk (F ≈ 13 ms, B ≈ 100 MB/s)	31.4%
691 KB	hard disk (F ≈ 13 ms, B ≈ 100 MB/s)	35.3%
822 KB	hard disk (F ≈ 13 ms, B ≈ 100 MB/s)	39.3%
977 KB	hard disk (F ≈ 13 ms, B ≈ 100 MB/s)	43.5%
1.13 MB	hard disk (F ≈ 13 ms, B ≈ 100 MB/s)	47.8%
1.35 MB	hard disk (F ≈ 13 ms, B ≈ 100 MB/s)	52.1%
1.6 MB	hard disk (F ≈ 13 ms, B ≈ 100 MB/s)	56.4%
1.9 MB	hard disk (F ≈ 13 ms, B ≈ 100 MB/s)	60.5%
2.26 MB	hard disk (F ≈ 13 ms, B ≈ 100 MB/s)	64.6%
2.69 MB	hard disk (F ≈ 13 ms, B ≈ 100 MB/s)	68.4%
3.19 MB	hard disk (F ≈ 13 ms, B ≈ 100 MB/s)	72%
3.8 MB	hard disk (F ≈ 13 ms, B ≈ 100 MB/s)	75.4%
4.51 MB	hard disk (F ≈ 13 ms, B ≈ 100 MB/s)	78.4%
5.36 MB	hard disk (F ≈ 13 ms, B ≈ 100 MB/s)	81.2%
6.37 MB	hard disk (F ≈ 13 ms, B ≈ 100 MB/s)	83.7%
7.58 MB	hard disk (F ≈ 13 ms, B ≈ 100 MB/s)	85.9%
9 MB	hard disk (F ≈ 13 ms, B ≈ 100 MB/s)	87.9%
10.7 MB	hard disk (F ≈ 13 ms, B ≈ 100 MB/s)	89.6%
12.7 MB	hard disk (F ≈ 13 ms, B ≈ 100 MB/s)	91.1%
15.1 MB	hard disk (F ≈ 13 ms, B ≈ 100 MB/s)	92.4%
18 MB	hard disk (F ≈ 13 ms, B ≈ 100 MB/s)	93.5%
21.4 MB	hard disk (F ≈ 13 ms, B ≈ 100 MB/s)	94.5%
25.4 MB	hard disk (F ≈ 13 ms, B ≈ 100 MB/s)	95.3%
30.2 MB	hard disk (F ≈ 13 ms, B ≈ 100 MB/s)	96.1%
35.8 MB	hard disk (F ≈ 13 ms, B ≈ 100 MB/s)	96.7%
42.6 MB	hard disk (F ≈ 13 ms, B ≈ 100 MB/s)	97.2%
50.6 MB	hard disk (F ≈ 13 ms, B ≈ 100 MB/s)	97.6%
60.2 MB	hard disk (F ≈ 13 ms, B ≈ 100 MB/s)	98%
71.5 MB	hard disk (F ≈ 13 ms, B ≈ 100 MB/s)	98.3%
85 MB	hard disk (F ≈ 13 ms, B ≈ 100 MB/s)	98.6%
101 MB	hard disk (F ≈ 13 ms, B ≈ 100 MB/s)	98.8%
120 MB	hard disk (F ≈ 13 ms, B ≈ 100 MB/s)	99%
143 MB	hard disk (F ≈ 13 ms, B ≈ 100 MB/s)	99.1%
170 MB	hard disk (F ≈ 13 ms, B ≈ 100 MB/s)	99.3%
202 MB	hard disk (F ≈ 13 ms, B ≈ 100 MB/s)	99.4%
240 MB	hard disk (F ≈ 13 ms, B ≈ 100 MB/s)	99.5%
285 MB	hard disk (F ≈ 13 ms, B ≈ 100 MB/s)	99.6%
338 MB	hard disk (F ≈ 13 ms, B ≈ 100 MB/s)	99.6%
402 MB	hard disk (F ≈ 13 ms, B ≈ 100 MB/s)	99.7%
478 MB	hard disk (F ≈ 13 ms, B ≈ 100 MB/s)	99.7%
568 MB	hard disk (F ≈ 13 ms, B ≈ 100 MB/s)	99.8%
675 MB	hard disk (F ≈ 13 ms, B ≈ 100 MB/s)	99.8%
802 MB	hard disk (F ≈ 13 ms, B ≈ 100 MB/s)	99.8%
954 MB	hard disk (F ≈ 13 ms, B ≈ 100 MB/s)	99.9%

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.

Physical Substrate & Performance Laws
Why do trade-offs exist at all?
$total cost = fixed cost + (cost per byte \times size)$
Read it in place →
Physical Substrate & Performance Laws
Why do trade-offs exist at all?
$positioning cost \times transfer rate \approx 1.3 MB$
Read it in place →
Data Movement
How do records cross between two systems that have independent rates, independent availability and independent lifetimes, and what are you willing to lose when the assumption underneath the crossing breaks?
$\frac{F}{B} per record; T (B) = F + B \cdot c$
F = $F$ · B (batch size) = $x$ · c (marginal cost per row) = $1/ B$
Read it in place →
Data Movement
How do records cross between two systems that have independent rates, independent availability and independent lifetimes, and what are you willing to lose when the assumption underneath the crossing breaks?
$per-record cost = F \div B$
Read it in place →
Storage: Physical Organization
What should sit next to what, given that I must decide at write time and find out at read time?
$Characteristic size = latency \times bandwidth$
latency = $F$ · bandwidth = $B$ · characteristic size = $x_{1/2}$
Read it in place →
Persistence & Durability
When a system says "yes, your data is safe", what exactly has it promised, and what did it have to do to keep that promise?
$B \approx \frac{T _{flush} \times bandwidth}{record_size}$
T_flush = $F$ · bandwidth = $B$ · B (records per flush) = $x_{1/2} \div record size$
Read it in place →
Execution Engine Runtime
Once a query plan is chosen, why does the machine still sit idle, and what does each mechanism of a query runtime pay to buy that idle time back?
$per-tuple overhead = F / N$
F (boundary cost) = $F$ · N (tuples per call) = $x$
Read it in place →
Execution Engine Runtime
Once a query plan is chosen, why does the machine still sit idle, and what does each mechanism of a query runtime pay to buy that idle time back?
$calls \times 5 μ s + rows \times 50 ns$
Read it in place →
Cost Mechanics
What is metered, and what is the smallest increment you are charged for?
$cost per byte = \frac{c _{req}}{S} + c_{byte}, S^{*} = \frac{c _{req}}{c _{byte}}$
c_req = $F$ · 1 / c_byte = $B$ · S = $x$
Read it in place →
Table & Storage Maintenance
Who pays for the gap between the order data arrives in and the order it is read in: the writer, every reader, or a background job, and what is the break-even?
$overhead fraction = \frac{F}{F + s / b}$
F (per-file cost) = $F$ · b (stream rate) = $B$ · s (file size) = $x$
Read it in place →
Orchestration & Workflow
How do you make unreliable, unobservable, irreversible side effects behave like a pure function of a time interval?
$useful share = \frac{work}{work + launch cost}$
launch cost = $F$ · work = $x / B$
Read it in place →