# Telephony Traffic Models

## Erlang: unit of traffic

Averaged over time, one erlang of telephone traffic occupies exactly one channel. However, the arrival and closing of telephone calls are a random processes. As time elapses, one erlang of traffic may occupy zero, one or multiple channels. The definition of the unit erlang does not say anything about how the traffic behaves statistically about this average. Thus one erlang of traffic can be generated for instance by
• One call of infinite duration, or
• A random process of many calls arriving and closing, such that the average number of active calls is one.
The unit of telephone traffic is called after Erlang, a Danish mathematician, who published in 1914 and 1917 the first basic results on the number of subscribers that can be served with a given number of channels at a required Quality of Service (blocking probability).
 Number of Users The random process running here illustrates the number of simultaneous telephone calls. New calls arrive according to a Poisson process. Call durations have an exponential distribution. The mean duration is taken quite short: around 10 seconds, which could occur in trunked networks for professional closed-user groups (e.g. taxi cabs, police, military). Traffic: 3.13 8 16 erlang. Call duration: 10 sec 3 min on average.

## Call handling

In a telephone system, a finite number of N channels are available. New calls are assigned a channel until all channels are full. Whenever all channels are occupied, a new call either is

## Model

Often it is assumed that new calls arrive according to Poisson process with rate l calls per unit of time. Note that this assumption may be less realistic is blocked calls result in new attempts by impatient subscribers. Calls have a (memoryless) exponential duration with mean 1/m. Under such assumptions, the number of active calls is a Markov process. If calls can always served, i.e., if the system has infinite resources, the successful traffic is l/m erlang.

The added feature of the Engset model is that it does not assume an infinite user population, like the Erlang model does. One can specify (or compute) the number of users. Traffic can now be expressed in erlang or in erlang per user. The traffic value in the dimensionless quantity 'erlang' signifies the number of lines that would be occupied on average if there were no blocking.