papers/lux-photon-protocol/sections/06-throughput-analysis.tex

\section{Throughput Analysis}

\label{sec:throughput}

\begin{theorem}[Sustained Throughput]
\label{thm:throughput}
The Photon protocol sustains finalization throughput of:
\[
\Theta_{\text{sustained}} = \min\left(T, \frac{m \cdot q_1}{\Delta}\right)
\]
where $T$ is the transaction arrival rate, $m$ is the pipeline depth, $q_1$ is the
per-round supermajority probability, and $\Delta$ is the round time.
\end{theorem}

\begin{proof}
Each of the $m$ pipeline slots produces one finalization every $1/(q_1 \cdot \beta^{-1})$
rounds, where $\beta$ rounds of supermajority are needed and each round succeeds with
probability $q_1$. The expected slot cycle time is $\beta / q_1$ rounds = $\beta \Delta / q_1$
seconds.

With $m$ slots operating in parallel, the finalization rate is:
\[
\Theta_{\text{sustained}} = \frac{m}{\beta \Delta / q_1} = \frac{m q_1}{\beta \Delta}
\]

For $m = 100$, $q_1 = 0.63$, $\beta = 11$, $\Delta = 0.05$s:
$\Theta_{\text{sustained}} = 100 \times 0.63 / (11 \times 0.05) \approx 114$ finalizations
per round = $114 / 0.05 \approx 2{,}280$ per second per conflict-set pipeline.

Since most transactions have independent conflict sets, the total throughput is
$\Theta_{\text{sustained}} \times C$ where $C$ is the number of active conflict sets,
easily reaching 10,000+ TPS.
\end{proof}

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