Bitcoin

The granddaddy of crypto.

The stochastic process of block discovery boils down to miners cranking blind searches in the space of hashes. Each hash attempt is a Bernoulli process, which in the limit of large numbers passes to a Poisson process. The parameter of the process is the average time between blocks.

Although total mining power has gone through exponential growth, different timeframes have kept a fairly stable average: In particular the block finality rate and the 4-year gap between reward halving events.

This is due to periodic difficulty adjustments, by which the protocol estimates total hashrate with the Poisson parameter, then retargets the probability on the inner Bernoulli process.