arrival times and the Erlang (gamma) distribution

If interarrival times are the gaps, arrival times are the cumulative clock readings: the time S1 of the first event, S2 of the second, and so on. They answer 'when does the n-th event happen?' rather than 'how long since the last one?'. Since each arrival sits at the end of a run of gaps, the n-th arrival time is simply the sum of the first n interarrival times: Sn = T1 + T2 + ... + Tn.

Adding n independent exponential waits, each with rate lambda, gives a named distribution: the Erlang distribution with shape n and rate lambda (the integer-shape special case of the gamma distribution). It has mean n/lambda and variance n/lambda^2 — n waits of average length 1/lambda, as you would hope. Its density is f(t) = lambda^n * t^(n-1) * e^(-lambda*t) / (n-1)! for t > 0. There is also a beautiful bridge to counting: 'the n-th arrival is after time t' is the very same event as 'fewer than n events have happened by time t', i.e. P(Sn > t) = P(N(t) < n), which links the Erlang directly to the Poisson count of N(t).

This is the distribution to reach for whenever you wait for a tally to reach a threshold: the time until the 5th customer, the 10th radioactive decay, the 3rd machine failure. It also explains why a sum of exponentials is more 'bell-like' and less spread out (relatively) than a single exponential — by the central-limit idea, many small random gaps average out, so for large n the arrival time Sn becomes approximately normal around n/lambda. The single exponential (n = 1) and the Erlang are the same family seen at different scales.