  The Lightning Network (LN) is a prominent payment channel network aimed at
addressing Bitcoin's scalability issues. Due to the privacy of channel
balances, senders cannot reliably choose sufficiently liquid payment paths and
resort to a trial-and-error approach, trying multiple paths until one succeeds.
This leaks private information and decreases payment reliability, which harms
the user experience. This work focuses on the reliability and privacy of LN
payments. We create a probabilistic model of the payment process in the LN,
accounting for the uncertainty of the channel balances. This enables us to
express payment success probabilities for a given payment amount and a path.
Applying negative Bernoulli trials for single- and multi-part payments allows
us to compute the expected number of payment attempts for a given amount,
sender, and receiver. As a consequence, we analytically derive the optimal
number of parts into which one should split a payment to minimize the expected
number of attempts. This methodology allows us to define service level
objectives and quantify how much private information leaks to the sender as a
side effect of payment attempts. We propose an optimized path selection
algorithm that does not require a protocol upgrade. Namely, we suggest that
nodes prioritize paths that are most likely to succeed while making payment
attempts. A simulation based on the real-world LN topology shows that this
method reduces the average number of payment attempts by 20% compared to a
baseline algorithm similar to the ones used in practice. This improvement will
increase to 48% if the LN protocol is upgraded to implement the channel
rebalancing proposal described in BOLT14.
