Consider a small time period T over which one wishes to restrict the data throughput:

VBV data throughput = (T + VBV buffer duration) * VBV maximum bit rate

VBV buffer duration = VBV buffer size / VBV maximum bit rate

VBV data throughput = (T + VBV buffer size / VBV maximum bit rate) * VBV maximum bit rate = T * VBV maximum bit rate + VBV buffer size


One may prefer to consider the average data rate over time T (i.e. a bit rate spike averaged over duration T):

'Spike' bit rate averaged over time T = VBV data throughput / T = VBV maximum bit rate + VBV buffer size / T


Beyond these mathematics, to find the range of VBV buffer sizes and VBV maximum bit rates that constrain the bit rate spikes per one's requirements:

VBV buffer duration = VBV buffer size / VBV maximum bit rate > frames / frame rate

VBV buffer size * frame rate / frames > VBV maximum bit rate > average bit rate

Mathieu Monnier advises that frames be at least 5 but x264 enforces a minimum of 3. The minimum VBV buffer duration is empirically derived from the relative size of the largest frames (I-frames) to the average frame size. Mathieu claims that I-frames are ~5x the average size of a P-frame.



$spike_rate = $maxrate + $bufsize / $spike_dur;   --- 1
$bufsize * $fps / $frames > $maxrate > $abr;      --- 2

To find the lower bounds:

$maxrate > $abr; from 2

$bufsize * $fps / $frames > $abr;
=> $bufsize > $abr * $frames / $fps; from 2

To find upper bounds:

$maxrate = $spike_rate - $bufsize / $spike_dur; from 1
but $maxrate > $abr
=> $spike_rate - $bufsize / $spike_dur > $abr;
=> $bufsize < ($spike_rate - $abr) * $spike_dur;

Similarly $bufsize * $fps / $frames > $maxrate; from 2
=> $maxrate < ($spike_rate - $abr) * $spike_dur * $fps / $frames;


$maxrate_lower = $abr;
$bufsize_lower = $abr * $frames / $fps;

$maxrate_upper = ($spike_rate - $abr) * $spike_dur * $fps / $frames;
$bufsize_upper = ($spike_rate - $abr) * $spike_dur;



Examples:

- Apple TV

ABR = 3 Mbps
T = 0.5 s
'Spike' rate = 7.5 Mbps

VBV maximum bit rate = VBV data throughput / T - VBV buffer size / T = 7.5 Mbps - VBV buffer size / 0.5 > 3 Mbps

=> (7.5 Mbps - 3 Mbps) * 0.5 = 2.25 Mb > VBV buffer size

2.25 Mb * frame rate / frames = 2.25 Mb * 24 / 5 = 10.8 Mbps > VBV buffer size * frame rate / frames > VBV maximum bit rate > 3 Mbps

As an example, try taking VBV maximum bit rate as 5 Mbps, then:

VBV buffer size = (7.5 Mbps - 5 Mbps) * 0.5 s = 1.25 Mb
10.8 Mbps > 1.25 Mb * 24 fps / 5 frames = 6 Mbps > 5 Mbps therefore a valid solution


- iPod 640x480

ABR = 1.5 Mbps
T = 0.5 s
'Spike' = 2.5 Mbps

VBV maximum bit rate = VBV data throughput / T - VBV buffer size / T = 2.5 Mbps - VBV buffer size / 0.5 > 1.5 Mbps

(2.5 Mbps - 1.5 Mbps) * 0.5 s = 0.5 Mb > VBV buffer size

0.5 Mb * frame rate / frames > VBV buffer size * frame rate / frames > VBV maximum bit rate > 1.5 Mb

0.5 Mb * frame rate / frames > VBV buffer size * frame rate / frames > 2.5 Mbps - VBV buffer size / 0.5 s > 1.5 Mbps

Try VBV buffer size = 0.5 Mb, then: