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We study the distribution of confirmation times of Bitcoin transactions, conditional on the size of the current memory pool. We argue that the time until a Bitcoin transaction is confirmed resembles the time to ruin in a corresponding Cramer-Lundberg process. This well-studied model gives mathematical insights in the mempool behaviour over time. Specifically, for situations where one chooses a fee, such that the total size of incoming transactions with higher fee is close to the total size of transactions leaving the mempool (heavy traffic), a diffusion approximation leads to an inverse Gaussian distribution for the confirmation times. The results of this paper are particularly interesting for users that want to make a Bitcoin transaction during heavy-traffic situations, as evaluation of the well-known inverse Gaussian distribution is computationally straightforward.

Bitcoin payments require a random amount of time to get confirmed (i.e. to be grouped by the miners into a block and to be added to the Bitcoin blockchain). In [8, 11], the authors propose the modelling of the Bitcoin confirmation time by the so-called time to ruin of the Cramer-Lundberg (CL) model. This provides off-the-shelf results directly aimed at predicting the confirmation time. However, analyses suggest that the data may not fully conform with the CL model assumptions. In this manuscript, we show by means of a robustness analysis that the time to ruin of a CL model is near insensitive to small changes in the model assumptions and illustrate that the proposed heuristic model can be used to accurately predict the confirmation times even when the data deviate (to a small degree) from the model assumptions.