A Quantization Problem Posed by Adaptive Streaming
Abstract
In adaptive bitrate (ABR) streaming, the delivery technology behind most Internet video, each title is encoded at several bitrates, forming an \emph{encoding ladder} of \emph{renditions}. Each client plays the highest rendition that its network bandwidth can sustain. We show that choosing the ladder is a problem of \emph{scalar quantization} of the bandwidth distribution. However, this quantization problem is of an unusual kind: the client's logic pins each quantization cell's reproduction value to the cell's \emph{left edge}, and the distortion measure is a one-sided quality loss rather than a squared error. The constraint reshapes the classical theory. The \emph{quality gap} of a ladder is the distance between its average delivered quality and the \emph{quality limit} that infinitely many renditions would attain. For optimal n-rendition ladders the gap 𝒢n* decays as Θ(1/n), not the textbook Θ(1/n2). Moreover, with large n, n 𝒢n*→ Cw=1/2(∫ √(p Q') d B)2, with the limit attained when the ladder's rates follow the density √(p Q'), which replaces Panter--Dite's p1/3; here p is the bandwidth density and Q the quality--rate curve of the content. The law yields closed-form design rules: quantile rung placement, the rendition count n(ε)≈ Cw/ε needed to reach tolerance ε, and the economic ladder size n*=√(λCw/κ) when a rendition costs κ to operate and the quality gap is priced at λ. The same analysis extends to the design of ladders employing different video resolutions, codecs, perceptual quality metrics, and the two-dimensional adaptation logic of modern web players.
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BibTeX
@misc{reznik2026quantization,
title = {{A Quantization Problem Posed by Adaptive Streaming}},
author = {Yuriy A. Reznik},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.03745},
eprint = {2609.03745},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.03745},
}
RIS
TY - UNPB
TI - A Quantization Problem Posed by Adaptive Streaming
AU - Yuriy A. Reznik
PY - 2026
DA - 2026/09/03
PB - arXiv
DO - 10.48550/arxiv.2609.03745
UR - https://arxiv.org/abs/2609.03745
AB - In adaptive bitrate (ABR) streaming, the delivery technology behind most Internet video, each title is encoded at several bitrates, forming an \emph{encoding ladder} of \emph{renditions}. Each client plays the highest rendition that its network bandwidth can sustain. We show that choosing the ladder is a problem of \emph{scalar quantization} of the bandwidth distribution. However, this quantization problem is of an unusual kind: the client's logic pins each quantization cell's reproduction value to the cell's \emph{left edge}, and the distortion measure is a one-sided quality loss rather than a squared error. The constraint reshapes the classical theory. The \emph{quality gap} of a ladder is the distance between its average delivered quality and the \emph{quality limit} that infinitely many renditions would attain. For optimal n-rendition ladders the gap 𝒢_n^* decays as Θ(1/n), not the textbook Θ(1/n^2). Moreover, with large n, n 𝒢_n^*→ C_w=1/2(∫ √(p Q') d B)^2, with the limit attained when the ladder's rates follow the density √(p Q'), which replaces Panter--Dite's p^1/3; here p is the bandwidth density and Q the quality--rate curve of the content. The law yields closed-form design rules: quantile rung placement, the rendition count n(ε)≈ C_w/ε needed to reach tolerance ε, and the economic ladder size n^*=√(λC_w/κ) when a rendition costs κ to operate and the quality gap is priced at λ. The same analysis extends to the design of ladders employing different video resolutions, codecs, perceptual quality metrics, and the two-dimensional adaptation logic of modern web players.
ER -