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Erlang Distribution Calculator

Enter the shape k, rate λ and value x to evaluate the Erlang distribution.

How it is calculated

Inputs are the shape k (a whole number of events), the rate λ (events per unit of time) and the time point x (in the same time unit). The tool rounds k to the nearest integer and needs 1 ≤ k ≤ 100,000, λ > 0 and x ≥ 0; otherwise there is no result. Density: f(x) = λk xk−1 e−λx / (k−1)!. Cumulative probability: F(x) = 1 − e−λx × Σ (λx)n/n! for n from 0 to k−1. Mean: k / λ. PDF and CDF are shown with six decimals, the mean with four. The PDF is a density per unit of x, not a probability, so it can exceed 1 when x is measured in small units.

Worked example

Calls reach a desk at a rate of λ = 0.5 per minute. How long until the third call (k = 3), and what is the chance it has arrived by x = 4 minutes? Here λx = 2. Density: 0.5³ × 4² × e−2 / 2! = 0.125 × 16 × 0.135335 / 2 = 0.135335. Cumulative: 1 − e−2 × (1 + 2 + 2) = 1 − 0.676676 = 0.323324, a 32.3% chance. Mean waiting time: 3 / 0.5 = 6.0000 minutes.

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Common mistakes

  • Using the mean time between events instead of the rate. If events come every 2 minutes on average, λ is 0.5 per minute, not 2.
  • Reading the PDF as a probability. It is a density; for a probability up to x, use the CDF value.
  • Entering a non-integer shape and expecting a gamma distribution. The tool rounds k to a whole number, so 2.6 behaves exactly like 3 and values below 0.5 return no result.
  • Mixing time units between λ and x. If λ is per minute, x must be in minutes, otherwise the CDF and the mean are meaningless.

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.

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  • Editorial team

    The density is not a probability, so for the chance that the k-th event lands in a time window, subtract two cumulative values. With 3 events and a rate of 0.5, the cumulative probability is 0.323324 at x of 4 and 0.576810 at x of 6, so the chance the third arrival falls between them is about 0.2535. What process are you modelling with it?

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