We study the probabilistic distribution of the confirmation time of Bitcoin transactions, conditional on the current memory pool (i.e., the queue of transactions awaiting confirmation). The results of this paper are particularly interesting for users that want to make a Bitcoin transaction during `heavy-traffic situations', when the transaction demand exceeds the block capacity. In such situations, Bitcoin users tend to bid up the transaction fees, in order to gain priority over other users that pay a lower fee. We argue that the time until a Bitcoin transaction is confirmed can be modelled as a particular stochastic fluid queueing process (to be precise: a Cramér-Lundberg process). We approximate the queueing process in two different ways. The first approach leads to a lower bound on the confirmation probability, which becomes increasingly tight as traffic decreases. The second approach relies on a diffusion approximation with a continuity correction, which becomes increasingly accurate as traffic intensifies. The accuracy of the approximations under different traffic loads are evaluated in a simulation study.