Erlang distribution

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Erlang distribution

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The Erlang distribution, due to the Danish telecommunications engineer, A K Erlang, is a form of Gamma distribution, with γ=0, and α restricted to the integers, and usually denoted by the letter k. The parameter β is usually replaced by the symbol λ, representing a measure of rate (e.g. call arrival rate, the average number of telephone calls expected in a given unit of time). Taking the general form of the Gamma distribution, we have:

hence for the Erlang distribution we have a two parameter distribution:

As noted above, the distribution arises from telecommunication traffic theory. Erlang analyzed problems involving the random arrival of telephone calls (hence a Poisson process). The intervals between call arrivals is then an Exponential distribution, and the sum of k such distributions is an Erlang distribution (i.e. the time before the kth call arrives), so the Poisson, Exponential, Erlang and Gamma distributions are very closely related to one another.

The cumulative Poisson distribution, with mean=λ, to the k-1th event is:

and the cumulative Erlang distribution to the k-1th event is simply:

The form of the Erlang is the same as the form of the two-parameter Gamma with α restricted to the integers.

Erlang was particularly concerned with calls arriving at random to a telephone exchange or manually operated switchboard with a capacity to handle N simultaneous calls. The call arrival rate and average duration was assumed constant over a finite period (e.g. over an hour) and the problem was to determine the probability that callers would find the facility busy. This probability is known as the Grade of Service (GoS) and Erlang showed that it can be expressed as:

where N is the number of operators or line or exchange facilities available, and A is the traffic offered in Erlangs, which is a dimensionless measure typically described as "call hours per hour", or "call minutes per minute". The traffic measure A can be estimated as the expected number of calls per unit of time multiplied by the average length of these calls (the distribution of call lengths actually does not matter, although Erlang assumed this was Exponential). The formula above is known as the Erlang-B formula, and applies to systems without queuing. A variant known as the Erlang-C formula allows for calls to be queued. Typically these models work well for low GoS (i.e. a high probability of calls being serviced immediately or very quickly), but are less effective as GoS levels deteriorate because the assumption of call arrival patterns being random independent events often breaks down (e.g. individuals keep re-dialing when they get calls blocked by an engaged signal, or data traffic is automatically re-transmitted if it encounters congestion).

Key measures for the Erlang distribution are provided in the table below:

Mean

k/λ

Variance

k/λ2

Skewness

2/k

Kurtosis

6/k

MGF

References

[JOH1] Johnson N L, Kotz S (1970) Continuous Univariate Distributions, I. Houghton Mifflin/J Wiley & Sons, New York

Wikipedia: Erlang Distribution: http://en.wikipedia.org/wiki/Erlang_distribution