the Erlang distribution
/ ER-lang /
How long until exactly k telephone calls have arrived at a switchboard, when calls come at a steady rate? The Danish engineer A. K. Erlang asked precisely this for early phone networks. The answer is the Erlang distribution: the gamma distribution restricted to a whole-number shape, so it really is the sum of k identical exponential waits.
Write X ~ Erlang(k, lambda): wait for k independent Exponential(lambda) events and add them. It is exactly Gamma(k, theta = 1/lambda) with k a positive integer. The density is f(x) = lambda^k x^(k-1) e^(-lambda x) / (k - 1)! for x >= 0 — note (k - 1)! replaces the gamma function because k is an integer. The mean is E[X] = k/lambda and the variance is Var(X) = k/lambda^2: k times the exponential's mean and variance, as you would expect for a sum of k independent copies. For k = 1 it collapses to the plain exponential.
Erlang distributions still anchor queueing theory and telecommunications — sizing call centres, modelling service times made of several stages, and reliability of systems that fail after k shocks. Think of it as 'the gamma you reach by counting whole arrivals'. The only subtlety is the name overlap: Erlang IS a gamma, just with the shape forced to be an integer, so any general gamma result applies, but software labelled 'Erlang' will refuse non-integer k.
A machine needs 4 independent shocks before it fails, and shocks arrive at lambda = 2 per year. Total time to failure is Erlang(k = 4, lambda = 2), with mean 4/2 = 2 years.
The Erlang is the integer-shape gamma — a clean sum of k exponential waits.
Erlang and gamma are the same family; 'Erlang' simply signals that the shape k is a whole number, so the factorial (k - 1)! can stand in for the gamma function.