# The Power of the Erlang formula

In any system where a large population makes use of a finite number of resources, the blocking probability is given by the Erlang-B formula. There is no need to assume a special probability distribution for the holding time of a resource. The only assumption is that the arrival process is a Poisson process, which will be automatically satisfied when the user-population is large enough and when users place their calls independently. When the user-population is small (e.g. a small number of user within reach of one base station) then the more refined Engset formula is more accurate.

Another special case is the use of Voice over IP. Ariel Topasso created an Applet that computes the bandwidth / quality tradeoff for Voice over IP and combined it with the Erlang calculator you find below. His Applet can be downloaded from http://arielpablo.net/projects.

The general validity of the Erlang formula makes it so powerful. The formula can be found in my book (in Dutch). The applet below uses an efficient recursive method to compute the blocking probability (also in the book). It goes further than Westbay's calculator: you can set the number of lines to any positive integer below 32768 (that is more than 180). The applet also gives you the option to compute the number of lines needed or the amount of traffic that is permitted.

Fill in any two (out of three) values in the Erlang calculator. Click the button next to the value that you want to be computed.

### Block Probability

All resources may be occupied when a user asks for one (i.e. places a call). In that case the call is lost. The probability that a call is lost is called the blocking probability.

### trAffic (in erlang)

The total demand for the resources is often designated with a variable named "A" and is measured in the dimensionless unit 'erlang'. Traffic in erlang is a number that signifies the expected number of resources that would be occupied if there were no blocking. It is named in honor of the Danish Mathematician A.K. Erlang. 'Traffic' is found by multiplying the total expected number of calls per time unit with the average time that a resource is kept. For example, if the arrival rate is 3 calls per second and the average holding time of a line (the resource) is 20 seconds, then the traffic is 60 erlang.

### Number of lines

The Erlang formula is mostly applied in telephony, where the finite resources are a limited set of lines. Here you specify how many lines are available, or you can calculate how many lines should be available for a given amount of traffic and a given blocking probability. When the number of lines is calculated its value may be non-integer.

Acknowledgements to Frank Derks, who could write applets before I could and will always write them faster, and to Kaufman, who knew teletraffic theory before I did and knew it better.

Frits Schoute.