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Restructured from flat .tex/.pdf files into per-paper subdirectories. Added Antje Worring as co-author on all papers. Synced PDFs to pdfs/ for download site.
34 lines
1.4 KiB
TeX
34 lines
1.4 KiB
TeX
\section{Throughput Scaling}
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\label{sec:throughput}
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\begin{theorem}[Linear Scaling]
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\label{thm:scaling}
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The aggregate throughput of Prism with $B$ spectrum bands is:
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\[
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\Theta_{\text{Prism}} = B \cdot \Theta_{\text{Photon}} \cdot \left(1 - O\left(\frac{\sqrt{\log B}}{B}\right)\right)
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\]
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which is $\Theta(B \cdot \Theta_{\text{Photon}})$ for moderate $B$.
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\end{theorem}
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\begin{proof}
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Each band runs an independent Photon pipeline with throughput $\Theta_{\text{Photon}}$.
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The aggregate throughput is the sum across bands, minus the overhead of the spectral
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combiner and the load imbalance.
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The combiner overhead is $O(1)$ per transaction (one comparison and one append), negligible.
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The load imbalance causes the busiest band to handle $T/B + O(\sqrt{T\log B / B})$
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transactions (Theorem~\ref{thm:balance}). The slowest band determines the overall
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throughput, giving:
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\[
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\Theta_{\text{Prism}} = B \cdot \Theta_{\text{Photon}} \cdot \frac{T/B}{T/B + O(\sqrt{T\log B/B})}
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= B \cdot \Theta_{\text{Photon}} \cdot \left(1 - O\left(\frac{\sqrt{\log B}}{B}\right)\right)
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\]
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For $B = 16$: the correction factor is $1 - O(\sqrt{\ln 16}/16) \approx 1 - 0.12 = 0.88$,
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so scaling is 88\% efficient with 16 bands.
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\end{proof}
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%% -----------------------------------------------------------------------
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%% 7. EVALUATION
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%% -----------------------------------------------------------------------
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