This is a question regarding the probability of orphaned blocks depending on the block interval. I am using Bitcoin vs Litecoin as an example.
Assuming equal hash rate and number of transactions, what is the relative increase in the probability of orphaned blocks for Litecoin, given that its block interval is a quarter?
The answer might be as simple as 4, but I'm wondering what is the quantitative justification?
Equation 2 in the following paper derives the math for estimating orphan rate if you know the propagation delay of winning block headers travelling across the mining network. The propagation delay has a hashrate-weighted probability distribution that complicates the calculation. It also depends on the block time. https://sites.cs.ucsb.edu/~rich/class/cs293b-cloud/papers/bitcoin-delay
An approximation, you can just use average propagation delay and the exponential CDF equation which is the probability of finding a block in "x" seconds:
CDF = 1 - e^(-x/T)
x = average propagation delay
T = block time = 600
This is the probability of an orphan occurring per block found.
The reasoning is like this: at any moment a winning header is found and propagating, this equation is the probability that someone else will find a block before that block has been seen by slightly less than half of the hashrate-weighted mining network. It gives a slight over-estimate of the orphan rate compared to the above paper. Using the median instead is an under-estimate. For a given orphan rate, you can use this equation backwards to estimate x.
The average propagation delay in BTC these days appears to be about x = 250 ms, so that CDF = 0.000416. The inverse of this 2,400, i.e. 1 orphan per 2,400 blocks which is the data I saw a couple of years ago. Assuming the same propagation delay x = 0.25 and T=150 for Litecoin, this gives 1 orphan per 600 blocks. I don't know the actual rate.
Since e^(-y) = 1-y when y is small, the equation simplifies to:
orphan_probability = x/T.
If you know the propagation delay and the orphan rate is higher than expected, it's evidence a big mining pool is doing a mild selfish mining attack. E.g., holding their winning blocks for 1 or 2 seconds in hopes of finding another, giving it better long-term success.
