Model

Slow-start ramp

Doubles it every RTT.

t \approx RTT \cdot lo g_{2} \frac{W}{W _{0}}

$W_{0}$: Initial window; a small congestion window of about 10 segments ≈ 14.6 KB
$W$: Window needed; bytes in flight needed to fill the link (the bandwidth-delay product)
$RTT$: Round-trip time; time for a message to reach the other side and the answer to come back

How it behaves

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

RTT = 10 msRTT = 100 ms

Data table

Time to reach a window, from 14.6 KB
Window W	series	Time to reach it
19.5 KB	RTT = 10 ms	4.54 ms
22.4 KB	RTT = 10 ms	6.49 ms
25.6 KB	RTT = 10 ms	8.44 ms
29.3 KB	RTT = 10 ms	10.4 ms
33.5 KB	RTT = 10 ms	12.3 ms
38.4 KB	RTT = 10 ms	14.3 ms
44 KB	RTT = 10 ms	16.2 ms
50.3 KB	RTT = 10 ms	18.2 ms
57.6 KB	RTT = 10 ms	20.1 ms
66 KB	RTT = 10 ms	22.1 ms
75.5 KB	RTT = 10 ms	24.1 ms
86.5 KB	RTT = 10 ms	26 ms
99 KB	RTT = 10 ms	28 ms
113 KB	RTT = 10 ms	29.9 ms
130 KB	RTT = 10 ms	31.9 ms
149 KB	RTT = 10 ms	33.8 ms
170 KB	RTT = 10 ms	35.8 ms
195 KB	RTT = 10 ms	37.7 ms
223 KB	RTT = 10 ms	39.7 ms
255 KB	RTT = 10 ms	41.6 ms
292 KB	RTT = 10 ms	43.6 ms
334 KB	RTT = 10 ms	45.5 ms
383 KB	RTT = 10 ms	47.5 ms
438 KB	RTT = 10 ms	49.4 ms
502 KB	RTT = 10 ms	51.4 ms
574 KB	RTT = 10 ms	53.3 ms
657 KB	RTT = 10 ms	55.3 ms
753 KB	RTT = 10 ms	57.2 ms
862 KB	RTT = 10 ms	59.2 ms
987 KB	RTT = 10 ms	61.1 ms
1.1 MB	RTT = 10 ms	63.1 ms
1.26 MB	RTT = 10 ms	65 ms
1.45 MB	RTT = 10 ms	67 ms
1.65 MB	RTT = 10 ms	68.9 ms
1.89 MB	RTT = 10 ms	70.9 ms
2.17 MB	RTT = 10 ms	72.8 ms
2.48 MB	RTT = 10 ms	74.8 ms
2.84 MB	RTT = 10 ms	76.7 ms
3.25 MB	RTT = 10 ms	78.7 ms
3.73 MB	RTT = 10 ms	80.6 ms
4.26 MB	RTT = 10 ms	82.6 ms
4.88 MB	RTT = 10 ms	84.5 ms
5.59 MB	RTT = 10 ms	86.5 ms
6.4 MB	RTT = 10 ms	88.4 ms
7.33 MB	RTT = 10 ms	90.4 ms
8.39 MB	RTT = 10 ms	92.3 ms
9.6 MB	RTT = 10 ms	94.3 ms
11 MB	RTT = 10 ms	96.2 ms
12.6 MB	RTT = 10 ms	98.2 ms
14.4 MB	RTT = 10 ms	100 ms
16.5 MB	RTT = 10 ms	102 ms
18.9 MB	RTT = 10 ms	104 ms
21.6 MB	RTT = 10 ms	106 ms
24.7 MB	RTT = 10 ms	108 ms
28.3 MB	RTT = 10 ms	110 ms
32.4 MB	RTT = 10 ms	112 ms
37.1 MB	RTT = 10 ms	114 ms
42.5 MB	RTT = 10 ms	116 ms
48.7 MB	RTT = 10 ms	118 ms
55.7 MB	RTT = 10 ms	120 ms
63.8 MB	RTT = 10 ms	122 ms
73 MB	RTT = 10 ms	124 ms
83.6 MB	RTT = 10 ms	126 ms
95.7 MB	RTT = 10 ms	127 ms
110 MB	RTT = 10 ms	129 ms
125 MB	RTT = 10 ms	131 ms
144 MB	RTT = 10 ms	133 ms
164 MB	RTT = 10 ms	135 ms
188 MB	RTT = 10 ms	137 ms
215 MB	RTT = 10 ms	139 ms
247 MB	RTT = 10 ms	141 ms
282 MB	RTT = 10 ms	143 ms
323 MB	RTT = 10 ms	145 ms
370 MB	RTT = 10 ms	147 ms
424 MB	RTT = 10 ms	149 ms
485 MB	RTT = 10 ms	151 ms
555 MB	RTT = 10 ms	153 ms
636 MB	RTT = 10 ms	155 ms
728 MB	RTT = 10 ms	157 ms
833 MB	RTT = 10 ms	159 ms
954 MB	RTT = 10 ms	161 ms
19.5 KB	RTT = 100 ms	45.4 ms
22.4 KB	RTT = 100 ms	64.9 ms
25.6 KB	RTT = 100 ms	84.4 ms
29.3 KB	RTT = 100 ms	104 ms
33.5 KB	RTT = 100 ms	123 ms
38.4 KB	RTT = 100 ms	143 ms
44 KB	RTT = 100 ms	162 ms
50.3 KB	RTT = 100 ms	182 ms
57.6 KB	RTT = 100 ms	201 ms
66 KB	RTT = 100 ms	221 ms
75.5 KB	RTT = 100 ms	241 ms
86.5 KB	RTT = 100 ms	260 ms
99 KB	RTT = 100 ms	280 ms
113 KB	RTT = 100 ms	299 ms
130 KB	RTT = 100 ms	319 ms
149 KB	RTT = 100 ms	338 ms
170 KB	RTT = 100 ms	358 ms
195 KB	RTT = 100 ms	377 ms
223 KB	RTT = 100 ms	397 ms
255 KB	RTT = 100 ms	416 ms
292 KB	RTT = 100 ms	436 ms
334 KB	RTT = 100 ms	455 ms
383 KB	RTT = 100 ms	475 ms
438 KB	RTT = 100 ms	494 ms
502 KB	RTT = 100 ms	514 ms
574 KB	RTT = 100 ms	533 ms
657 KB	RTT = 100 ms	553 ms
753 KB	RTT = 100 ms	572 ms
862 KB	RTT = 100 ms	592 ms
987 KB	RTT = 100 ms	611 ms
1.1 MB	RTT = 100 ms	631 ms
1.26 MB	RTT = 100 ms	650 ms
1.45 MB	RTT = 100 ms	670 ms
1.65 MB	RTT = 100 ms	689 ms
1.89 MB	RTT = 100 ms	709 ms
2.17 MB	RTT = 100 ms	728 ms
2.48 MB	RTT = 100 ms	748 ms
2.84 MB	RTT = 100 ms	767 ms
3.25 MB	RTT = 100 ms	787 ms
3.73 MB	RTT = 100 ms	806 ms
4.26 MB	RTT = 100 ms	826 ms
4.88 MB	RTT = 100 ms	845 ms
5.59 MB	RTT = 100 ms	865 ms
6.4 MB	RTT = 100 ms	884 ms
7.33 MB	RTT = 100 ms	904 ms
8.39 MB	RTT = 100 ms	923 ms
9.6 MB	RTT = 100 ms	943 ms
11 MB	RTT = 100 ms	962 ms
12.6 MB	RTT = 100 ms	982 ms
14.4 MB	RTT = 100 ms	1 s
16.5 MB	RTT = 100 ms	1.02 s
18.9 MB	RTT = 100 ms	1.04 s
21.6 MB	RTT = 100 ms	1.06 s
24.7 MB	RTT = 100 ms	1.08 s
28.3 MB	RTT = 100 ms	1.1 s
32.4 MB	RTT = 100 ms	1.12 s
37.1 MB	RTT = 100 ms	1.14 s
42.5 MB	RTT = 100 ms	1.16 s
48.7 MB	RTT = 100 ms	1.18 s
55.7 MB	RTT = 100 ms	1.2 s
63.8 MB	RTT = 100 ms	1.22 s
73 MB	RTT = 100 ms	1.24 s
83.6 MB	RTT = 100 ms	1.26 s
95.7 MB	RTT = 100 ms	1.27 s
110 MB	RTT = 100 ms	1.29 s
125 MB	RTT = 100 ms	1.31 s
144 MB	RTT = 100 ms	1.33 s
164 MB	RTT = 100 ms	1.35 s
188 MB	RTT = 100 ms	1.37 s
215 MB	RTT = 100 ms	1.39 s
247 MB	RTT = 100 ms	1.41 s
282 MB	RTT = 100 ms	1.43 s
323 MB	RTT = 100 ms	1.45 s
370 MB	RTT = 100 ms	1.47 s
424 MB	RTT = 100 ms	1.49 s
485 MB	RTT = 100 ms	1.51 s
555 MB	RTT = 100 ms	1.53 s
636 MB	RTT = 100 ms	1.55 s
728 MB	RTT = 100 ms	1.57 s
833 MB	RTT = 100 ms	1.59 s
954 MB	RTT = 100 ms	1.61 s

Worked examples, recomputed

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

14.6 KB → 125 MB at 100 ms ≈ 1.4 s

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?
$lo g_{2} (125, 000, 000/14, 600) \approx 13 - 14 doublings$
Read it in place →