A Holistic Approach for Bitcoin Confirmation Times & Optimal Fee Selection

TL;DR

The paper addresses the Bitcoin pay-for-speed problem by modelling transaction confirmation times through a Cramér-Lundberg (CL) framework that mirrors mempool dynamics under a fixed fee. It establishes that, under realistic scaling, the batch-service queue (BSQ) converges to the CL fluid limit and, in heavy traffic, to Brownian motion, enabling explicit density, tail, and mean formulas for confirmation times and facilitating fast optimal-fee decisions via BM approximations. The model is validated with real mempool data, showing good alignment with observed confirmation-time distributions and often outperforming data-driven fee estimators in predicting minimal-fee strategies for target delays. The work also connects BSQ, CL, and BM in a unified treatment, providing practical estimation methods and highlighting broad applicability to other pay-for-speed contexts.

Abstract

Bitcoin is currently subject to a significant pay-for-speed trade-off. This is caused by lengthy and highly variable transaction confirmation times, especially during times of congestion. Users can reduce their transaction confirmation times by increasing their transaction fee. In this paper, based on the inner workings of Bitcoin, we propose a model-based approach (based on the Cramér-Lundberg model) that can be used to determine the optimal fee, via, for example, the mean or quantiles, and models accurately the confirmation time distribution for a given fee. The proposed model is highly suitable as it arises as the limiting model for the mempool process (that tracks the unconfirmed transactions), which we rigorously show via a fluid limit and we extend this to the diffusion limit (an approximation of the Cramér-Lundberg model for fast computations in highly congested instances). We also propose methods (incorporating the real-time data) to estimate the model parameters, thereby combining model and data-driven approaches. The model-based approach is validated on real-world data and the resulting transaction fees outperform, in most instances, the data-driven ones.

Figures (13)

• Figure 1: Comparison of a real-world realisation of the mempool versus a CL model. On the left, a snapshot of the mempool taken on 11-4-2020 from 00:00 am to 8:00 am. The cumulative mempool weight (vMB) is outlined on the $y$-axis over time on the $x$-axis. Source: https://mempool.jhoenicke.de. On the right, a sample path of the CL model.
• Figure 2: Histograms of $n$ extracted confirmation times for $\phi\in\{3,6,12,50\}$ sat/byte (from left to right). We overlay (plotted in red) the density of the confirmation times to the theoretical distribution in Equation \ref{['eq:densityut']}.
• Figure 3: Example of the different stochastic processes introduced above.
• Figure 4: Sketch of the renewal process $N_k(t)$ that is used to determine the distribution of $U_k(x)$. For a block size $x$, only $B_k(x)$ transactions can fit, leaving an open space of $U_k(x)$.
• Figure 5: The density of the confirmation times, c.f. Equation \ref{['eq:densityut']}, for $c=0.4$ (a) and $c=0.7$ (b), and for $y=0$ and $y=5$.

Theorems & Definitions (36)

• remark 1: The definition of a confirmation time
• proposition 1: Density of confirmation time
• proposition 2: Confirmation time in blocks
• proposition 3: The expected confirmation time
• theorem 1: Convergence of the confirmation time via a fluid limit
• theorem 2: Convergence of the confirmation time via a diffusion limit
• remark 2: Correction for the undershoot
• proof : Proof of Proposition \ref{['col:Conftimedens']}.
• proof : Proof of Proposition \ref{['prop:DM1BP']}
• theorem 3: Expected time to confirmation