$$ \newcommand{\R}{\mathbb{R}} \newcommand{\N}{\mathbb{N}} \newcommand{\E}{\mathbb{E}} \newcommand{\Csmooth}{\bar{C}} \newcommand{\Craw}{C} \newcommand{\tasktype}{\tau} \newcommand{\bps}{\mathrm{BPS}} $$

Formal Model

Network Graph

A payment flow network is a directed graph \(G = (V, E)\) where \(V = S \cup K\) with \(S\) the set of sources (payment senders) and \(K\) the set of sinks (service providers). Each edge \(e = (s, k) \in E\) represents a potential payment flow from source \(s \in S\) to sink \(k \in K\).

We consider a slotted time model with discrete epochs \(t \in \{0, 1, 2, \ldots\}\). The system tracks multiple task types \(\tasktype \in \mathcal{T}\), which serve as commodities in the multi-commodity flow formulation.

Each sink \(k \in K\) declares a capacity signal \(\Craw(k, t, \tasktype) \in \R_{\geq 0}\) for each task type \(\tasktype\) it serves, representing its maximum throughput (payment absorbable per epoch). The EWMA-smoothed capacity is: $$ \begin{equation} \label{eq:ewma} \Csmooth(k, t, \tasktype) = \alpha \cdot \Craw(k, t, \tasktype) + (1 - \alpha) \cdot \Csmooth(k, t-1, \tasktype) \end{equation} $$ where \(\alpha \in (0, 1]\) is the smoothing parameter.

For each sink \(k\) and task type \(\tasktype\), the virtual queue backlog \(Q(k, t, \tasktype)\) tracks unprocessed payment accumulation: $$ \begin{equation} \label{eq:queue} Q(k, t+1, \tasktype) = \max\bigl{Q(k, t, \tasktype) + F(k, t, \tasktype) - \Csmooth(k, t, \tasktype),\; 0\bigr} \end{equation} $$ where \(F(k, t, \tasktype)\) is the payment flow routed to sink \(k\) at time \(t\).

Backpressure Payment Routing

In the standard Tassiulas–Ephremides formulation, the max-weight scheduler selects links to maximize \(\sum_{(i,j)} W_{ij}(t) \cdot \mu_{ij}(t)\) where \(W_{ij}(t)\) is the differential backlog and \(\mu_{ij}(t)\) the allocated rate. For BPE’s single-hop architecture (sources connect directly to sinks), we adapt this as follows.

The differential backlog for sink \(k\) from source \(s\) under task type \(\tasktype\) is: $$ \begin{equation} W(k, t, \tasktype) = Q(s, t, \tasktype) - Q(k, t, \tasktype) \end{equation} $$ where \(Q(s, t, \tasktype)\) represents the source’s pending payment queue.

Definition (Max-Weight Routing)

At each epoch \(t\), for each task type \(\tasktype\), allocate the total source flow \(\Lambda(\tasktype, t) = \sum_{s \in S_\tasktype} \lambda_s(t)\) to sinks proportionally to their capacity-weighted backpressure: $$ \begin{equation} \label{eq:maxweight} F(k, t, \tasktype) = \frac{W(k, t, \tasktype)^+ \cdot \Csmooth(k, t, \tasktype)} {\sum_{k' \in K_\tasktype} W(k', t, \tasktype)^+ \cdot \Csmooth(k', t, \tasktype)} \cdot \Lambda(\tasktype, t) \end{equation} $$ where \(x^+ = \max(x, 0)\).

In the steady state where all sinks have similar backlog levels, this reduces to proportional-to-capacity allocation—which is the mechanism implemented in our Superfluid GDA pool (member units proportional to smoothed capacity).

Multi-Commodity Extension

Task types serve as commodities. Each source \(s\) emits flow for a specific task type \(\tasktype_s\), and sinks may register for multiple task types. The virtual queues are maintained per \((k, \tasktype)\) pair, achieving commodity-specific backpressure identical to the multi-commodity extension of Tassiulas–Ephremides (Tassiulas & Ephremides, 1992).

EWMA Smoothing

The smoothing parameter \(\alpha\) trades off responsiveness against oscillation. Following Jacobson’s EWMA for TCP RTT estimation (Jacobson, 1988), we set \(\alpha = 0.3\) (validated in Evaluation).

For a sink with constant true capacity \(c\), the EWMA-smoothed signal converges geometrically: \(|\Csmooth(k, t) - c| \leq (1-\alpha)^t \cdot |\Csmooth(k, 0) - c|\).

Proof

Proof. By direct expansion of Eq. (EWMA) with \(\Craw(k, t) = c\) for all \(t\). ◻

Monetary No-Drop Constraint

Unlike data packets, money cannot be dropped. When total source flow exceeds total sink capacity, BPE deploys an overflow buffer \(B(\tasktype, t)\): $$ \begin{equation} B(\tasktype, t+1) = B(\tasktype, t) + \Lambda(\tasktype, t) - \sum_{k \in K_\tasktype} \min\bigl{F(k,t,\tasktype),\; \Csmooth(k,t,\tasktype)\bigr} \end{equation} $$

The buffer absorbs excess and drains proportionally to capacity as capacity becomes available. We prove in Throughput Optimality that \(B(t)\) is bounded under sufficient capacity.