Equivalently, it is the distribution of the time until the kth event of a Poisson process with a rate of

The Erlang and Poisson distributions are complementary, in that while the Poisson distribution counts the events that occur in a fixed amount of time, the Erlang distribution counts the amount of time until the occurrence of a fixed number of events.

The Erlang distribution was developed by A. K. Erlang to examine the number of telephone calls that might be made at the same time to the operators of the switching stations.

This work on telephone traffic engineering has been expanded to consider waiting times in queueing systems in general.

An alternative, but equivalent, parametrization uses the scale parameter

It can therefore be regarded as a generalized chi-squared distribution for even numbers of degrees of freedom.

The CDF may also be expressed as The Erlang-k distribution (where k is a positive integer)

is defined by setting k in the PDF of the Erlang distribution.

An asymptotic expansion is known for the median of an Erlang distribution,[2] for which coefficients can be computed and bounds are known.

) using the following formula:[6] Events that occur independently with some average rate are modeled with a Poisson process.

The waiting times between k occurrences of the event are Erlang distributed.

(The related question of the number of events in a given amount of time is described by the Poisson distribution.)

The Erlang distribution, which measures the time between incoming calls, can be used in conjunction with the expected duration of incoming calls to produce information about the traffic load measured in erlangs.

This can be used to determine the probability of packet loss or delay, according to various assumptions made about whether blocked calls are aborted (Erlang B formula) or queued until served (Erlang C formula).

The Erlang-B and C formulae are still in everyday use for traffic modeling for applications such as the design of call centers.

The age distribution of cancer incidence often follows the Erlang distribution, whereas the shape and scale parameters predict, respectively, the number of driver events and the time interval between them.

[7][8] More generally, the Erlang distribution has been suggested as good approximation of cell cycle time distribution, as result of multi-stage models.

[9][10] The kinesin is a molecular machine with two "feet" that "walks" along a filament.

The waiting time between each step is exponentially distributed.