4. Erlang

The origin of the Erlang distribution is attributed to Agner Erlang, a Danish engineer for the Copenhagen Telephone Company, who in 1908 was seeking how many circuits are needed to accommodate the voice traffic on their telephone system. The distribution has two parameters, k, θ, where k represents the number of exponential variables that are summed to form the Erlang variable. The exponential variables have the same parameter, θ, as the Erlang. The Erlang has shapes that range from exponential to normal. This distribution is heavily used in the study of queuing systems, representing the time between arrivals, and also the time to service a unit. The chapter shows how to compute the cumulative probability, F(x), where x is the random variable. When the parameter values are not known, and sample data is available, measures from the data is used to estimate the parameter values. When the parameter values are not known, and no sample data is available, approximates of some measures from the distribution, usually by an expert, are obtained and these allow estimates of the parameter values. The Erlang is a special case of the gamma distribution, as will be seen in Chap. 5, where the parameter k is a positive integer for the Erlang, and is any positive number for the gamma.