Erlang Distribution Calculator

About this calculator

The Erlang distribution describes the total waiting time until the k-th event occurs in a Poisson process — equivalently, the sum of k independent exponential random variables each with rate λ. Its probability density function is f(x) = (λ^k · x^(k−1) · e^(−λx)) ÷ (k−1)!, and its mean is simply k/λ; the cumulative distribution gives the probability that all k events have occurred by time x.

This calculator takes a shape parameter k, a rate λ, and a value x, and returns the Erlang PDF, CDF, and mean. The distribution takes its name from Agner Krarup Erlang, who developed it for telephone traffic engineering, and telecom and call-center capacity planners still use it (via the related Erlang B and Erlang C formulas) to size trunk lines and staffing levels, while reliability and queueing-theory engineers use it more broadly to model waiting times for a fixed number of sequential arrivals.