Reply by rucknium:

If I understand your question right, I think you’re looking for the inverse cumulative distribution function (a.k.a. quantile function) of the Erlang distribution.

The random length of time to mine the next block has an exponential distribution with rate parameter 1/t. The length of time to mine n blocks has an Erlang distribution with shape parameter n and rate parameter 1/t.

The Erlang distribution is a special case of the Gamma distribution. The Erlang distribution’s shape parameter must be an integer, but the Gamma distribution’s shape parameter can be any positive real number. We can use the Gamma distribution if Erlang isn’t given to us by our calculator.

You would compute T in the R language with:

qgamma(p = p, shape = n, rate = 1/t)/n

The results of this simulation match the closed-form computation:

t <- 120
n <- 15
p <- 0.4

set.seed(314)

mining.times <- matrix(rexp(n * 100000, rate = 1/t), ncol = n)

mining.times <- rowSums(mining.times)

quantile(mining.times/n, probs = p)

qgamma(p = p, shape = n, rate = 1/t)/n
# Divide by n to get the mean instead of the total