Definitions

# Erlang unit

The erlang (symbol E) as a dimensionless unit is used in telephony as a statistical measure of the volume of telecommunications traffic. It is named after the Danish telephone engineer A. K. Erlang, the originator of traffic engineering and queueing theory. Traffic of one erlang refers to a single resource being in continuous use, or two channels being at fifty percent use, and so on, pro rata. For example, if an office had two telephone operators who are both busy all the time, that would represent two erlangs (2 E) of traffic, or a radio channel that is occupied for thirty minutes during an hour is said to carry 0.5 E of traffic.

Alternatively, an erlang may be regarded as a "use multiplier" per unit time, so 100% use is 1 E, 200% use is 2 E, and so on. For example, if total cell phone use in a given area per hour is 180 minutes, this represents 180/60 = 3 E. In general, if the mean arrival rate of new calls is λ per unit time and the mean call holding time is h, then the traffic in erlangs A is:

$A = lambda h$

This may be used to determine if a system is over-provisioned or under-provisioned (has too many or too few resources allocated). For example, the traffic measured over many busy hours might be used for a T1 or E1 circuit group to determine how many voice lines are likely to be used during the busiest hours. If no more than 12 out of 24 channels are likely to be used at any given time, the other 12 might be made available as data channels.

Traffic measured in erlangs is used to calculate grade of service (GOS) or quality of service (QoS). There are a range of different Erlang formulae to calculate these, including Erlang B, Erlang C and the related Engset formula. These are discussed below, and may each be derived by means of a special case of continuous-time Markov processes known as a birth-death process.

## Erlang B formula

The Erlang B formula assumes an infinite population of sources (such as telephone subscribers), which jointly offer traffic to N servers (such as links in a trunk group). The rate of arrival of new calls (birth rate) is equal to λ and is constant, not depending on the number of active sources, because the total number of sources is assumed to be infinite. The rate of call departure (death rate) is equal to the number of calls in progress divided by h, the mean call holding time. The formula calculates blocking probability in a loss system, where if a request is not served immediately when it tries to use a resource, it is aborted. Requests are therefore not queued. Blocking occurs when there is a new request from a source, but all the servers are already busy. The formula assumes that blocked traffic is immediately cleared.

$B\left(N, A\right) = frac\left\{frac\left\{A^N\right\}\left\{N!\right\}\right\}\left\{sum_\left\{i=0\right\}^\left\{N\right\}\left\{frac\left\{A^i\right\}\left\{i!\right\}\right\}\right\}$

This may be expressed recursively as follows, in a form that is used to calculate tables of the Erlang B formula:

$B\left(0, A\right) = 1 ,$

$B\left(N,A\right) = \left\{ \left\{A B\left(N-1,A\right)\right\} over \left\{N+A B\left(N-1,A\right)\right\} \right\} ,$

where:

• B is the probability of blocking
• N is the number of resources such as servers or circuits in a group
• A = λh is the total amount of traffic offered in erlangs

The Erlang B formula applies to loss systems, such as telephone systems on both fixed and mobile networks, which do not provide traffic buffering, and are not intended to do so. It assumes that the call arrivals may be modeled by a Poisson process, but is valid for any statistical distribution of call holding times. Erlang B is a trunk sizing tool for voice switch to voice switch traffic.

## Erlang C formula

The Erlang C formula also assumes an infinite population of sources, which jointly offer traffic of A erlangs to N servers. However, if all the servers are busy when a request arrives from a source, the request is queued. An unlimited number of requests may be held in the queue in this way simultaneously. This formula calculates the probability of queuing offered traffic, assuming that blocked calls stay in the system until they can be handled. This formula is used to determine the number of agents or customer service representatives needed to staff a call centre, for a specified desired probability of queuing.

$P_W = \left\{\left\{frac\left\{A^N\right\}\left\{N!\right\} frac\left\{N\right\}\left\{N - A\right\}\right\} over sum_\left\{i=0\right\}^\left\{N-1\right\} frac\left\{A^i\right\}\left\{i!\right\} + frac\left\{A^N\right\}\left\{N!\right\} frac\left\{N\right\}\left\{N - A\right\}\right\} ,$

where:

• A is the total traffic offered in units of erlangs
• N is the number of servers
• PW is the probability that a customer has to wait for service

It is assumed that the call arrivals can be modeled by a Poisson process and that call holding times are described by a negative exponential distribution.

## Engset formula

The Engset formula, named after T. O. Engset, is related but deals with a finite population of S sources rather than the infinite population of sources that Erlang assumes:

$E\left(N, A, S\right) = frac\left\{A^N$
{left(begin{array}{c} S N end{array} right)}} {sum_{i=0}^NA^i {left(begin{array}{c} S i end{array} right)}}

This may be expressed recursively as follows, in a form that is used to calculate tables of the Engset formula:

$E\left(0, A, S\right) = 1 ,$

$E\left(N, A, S\right) = \left\{ \left\{A\left(S-N+1\right)E\left(N-1,A,S\right)\right\} over \left\{N+A\left(S-N+1\right)E\left(N-1,A,S\right)\right\} \right\} ,$

where:

• E is the probability of blocking
• A is the traffic in erlangs generated by each source when idle
• S is the number of sources
• N is the number of servers

Again, it is assumed that the call arrivals can be modeled by a Poisson process and that call holding times are described by a negative exponential distribution. However, because there are a finite number of sources, the arrival rate of new calls decreases as more sources (such as telephone subscribers) become busy and hence cannot originate new calls. When N = S, the formula reduces to a binomial distribution.