Modeling the Energy Consumption of a Public Blockchain as a Stochastic Process

In this paper, we propose a multilevel stochastic model for the energy consumption of public proof-of-work blockchains. The main novelty is the proposal of a closed form for the expected energy consumption in one proof of work mining round. In the case of homogeneous per-hash efficiency, this proposition shows that the expected spending is $e_0/p$ depending only on the protocol difficulty and not on the distribution of the hash power among the miners. The proposal connects three levels of analysis: a local model of mining at the node level, a semi-global model of competitive block discovery and propagation, and a global stochastic model of workload, computational capacity, network connectivity and power consumption. This leads to the above closed form energy result. The mining process is approximated locally by exponential waiting times of Bernoulli hash trials. This extends to the semi-global model where the competition among miners and the delay in the propagation lead to the wasted computation. The global layer is modeled as a set of stochastic differential equations which models the interaction between workload dynamics, capacity constraints and communication overheads. The core analysis does not need Bayesian or Markov decision components but these are recommended for modeling estimation and adaptive control. We start with preliminary simulations on the VIBES platform and find qualitative properties of the full model: the total energy cost scales roughly linearly with the size of the network, the average energy per node decreases with increasing network size, the propagation latency is the primary source of wasted computation due to stale blocks and nodes tend to operate in a capacity-depleted regime with the workload-induced degradation being substantially higher than the recovery rate. The results give a structural analysis of how the design of the protocol and the network conditions affect the energy consumption and emphasize the importance of quantitatively calibrating with empirical data from Bitcoin.