Bitcoin Math Primer: Competing Poisson Processes \ stacker news

In a previous post, I explained how Bitcoin mining can be modeled as a Poisson process.

Recap: A Poisson process is a probabilistic model of discrete events. It's defined by a parameter, $λ$ , known as the hazard rate. $λ$ measures the probability of an event occurring per time unit, i.e. $λ$ = 1 block per 10 minutes. In it, we showed that $λ = h / d$ where $h$ is the hash rate (e.g. measured in EH/s) and $d$ is an "effective difficulty", which roughly corresponds to the average number of exahashes (EH) required to find one block.

Competing Poisson ProcessesCompeting Poisson Processes

But Bitcoin mining is not really a single Poisson processes. It's actually a number of Poisson processes competing against each other. Every miner/mining pool is its own Poisson process, and the event a block is found by Antpool isn't the same as the event a block is found by Foundry.

Thus, Bitcoin mining can be better thought of as a "competing Poisson process". That is, we have multiple Poisson processes competing against each other, and we want to answer questions like:

What is the expected time until the next block found by Antpool?
What is the expected time until the next block found by Foundry?
What is the expected time until the next block found by anyone?
What is the probability that Antpool (or anyone else) finds the next block?

The super cool thing about competing Poisson processes is that they basically add up to one big Poisson process.

Antpool v. Foundry exampleAntpool v. Foundry example

Suppose there are just two miners, Antpool and Foundry. Antpool has hash rate $h_{A}$ and Foundry has hash rate $h_{F}$ . Antpool blocks are modeled by a Poisson process with rate $λ_{A} = h_{A} / d$ and Foundry blocks are modeled by a Poisson process with rate $λ_{F} = h_{F} / d$ .

The combined process (e.g. blocks found by anyone) is then simply modeled by a single Poisson process with rate

λ_{co mbin e d} = \frac{h _{A} + h _{F}}{d}

In other words: the combined process is just a Poisson process based on the combined hash rate.

Moreover, the expected time until Antpool finds another block is always $10 min / λ_{A}$ and the expected time until Foundry finds another block is always $10 min / λ_{F}$ . The expected time unitl any block is found is always $10 min / λ_{co mbin e d}$ .

The probability that Antpool finds the next block is $h_{A} / (h_{A} + h_{F})$ and the probability that Foundry finds the next block is $h_{F} / (h_{A} + h_{F})$ .

General case of N minersGeneral case of N miners

In the more general case, suppose there are $N$ miners with hash rates $h_{i}$ for $i = 1, \dots, N$ . Then, each miner's blocks are modeled by a Poisson process with rate $λ_{i} = h_{i} / d$ , and the combined process is modeled by a Poisson process with rate

λ_{co mbin e d} = \frac{1}{d} i = 1 \sum N h_{i} = \frac{Total hash rate}{d}

The expected time until miner $i$ finds another block is $10 min / λ_{i}$ , and the expected time until any block is found is $10 min / λ_{co mbin e d}$ .

The probability that miner $i$ finds the next block is $h_{i} / \sum_{j = 1}^{N} h_{j}$ , e.g. it's their share of the total hash rate.

Competing Poisson processes will be important for analyzing selfish mining, and for constructing other game theoretic models of miner behavior, because in strategic analysis it matters who finds the next block and when. (e.g. It matters whether the selfish miner finds the next block, or the honest miner.)