Erlang: Information from Answers.com

The erlang (symbol E^[1]) as a dimensionless unit is used in telephony as a statistical measure of offered load. 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 each, 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 E is:

E = λ 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.

1 Erlang B formula
- 1.1 Extended Erlang B
2 Erlang C formula
3 Engset formula
- 3.1 Example application
- 3.2 Technical details
4 See also
5 References
6 Tools
7 External links

Erlang B formula

Erlang-B (sometimes also written without the hyphen Erlang B), also known as the Erlang loss formula, is a formula for the blocking probability derived from the Erlang distribution to describe the probability of call loss on a group of circuits (in a circuit switched network, or equivalent). It is, for example, used in planning telephone networks. The formula was derived by Agner Krarup Erlang and is not limited to telephone networks, since it describes a probability in a queuing system (albeit a special case with a number of servers but no buffer spaces for incoming calls to wait for a free server).

The formula applies under the condition that an unsuccessful call, because the line is busy, is not queued or retried, but instead really lost forever. It is assumed that call attempts arrive following a poisson process. Further it is assumed that call arrivals are independent, and message length (holding times) are exponentially distributed (Markovian system) although the formula turns out to apply under general holding time distributions.

Erlangs are a dimensionless quantity calculated as the average arrival rate, λ, multiplied by the average call length, h. (see Little's Law) 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.

The formula provides the GoS (grade of service) which is the probability P_b that a new call arriving at the circuit group is rejected because all servers (circuits) are busy: B(E, m) when E Erlang of traffic are offered to m trunks (communication channels).

$P_b = B(E,m) = \frac{\frac{E^m}{m!}} { \sum_{i=0}^m \frac{E^i}{i!}}$

where:

$P b$ is the probability of blocking
m is the number of resources such as servers or circuits in a group
E = λh is the total amount of traffic offered in erlangs

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

$B(E,0) = 1 \,$ .

$B(E,j) = \frac{E B(E,j - 1)}{E B(E,j - 1) + j} \forall{j} = 1,2,...,m$

Typically, instead of B(E, m) the inverse 1/B(E, m) is calculated in numerical computation in order to ensure numerical stability:

$\frac{1}{B(E,0)} = 1$

$\frac{1}{B(E,j)} = 1 + \frac{j}{E} \frac{1}{B(E,j - 1)} \forall{j} = 1,2,...,m$

Function ErlangB (E as Double, m As Integer) As Double
Dim InvB As Double
Dim j As Integer

  InvB = 1.0
  For j = 1 To m
    InvB = 1.0 + j / E * InvB
  Next j
  ErlangB = 1.0 / InvB
End Function

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.

Extended Erlang B

Extended Erlang B is an iterative calculation, rather than a formula, that adds an extra parameter, the Recall Factor, which defines the recall attemps^[2].

The steps in the process are as follows^[3]:

1. Calculate

$P_b = B(E,m)\,$

as above for Erlang B.

2. Calculate the probable number of blocked calls

$B_e = EP_b\,$

3. Calculate the number of recalls, $R$ assuming a Recall Factor, $R f$ :

$R = B_e R_f\,$

4. Calculate the new offered traffic

$E_{i+1}=E_{i}+R\,$

where $E i$ is the traffic at the current step.

5. Return to step 1 and iterate until a stable value of $E$ is obtained.

Erlang C formula

The Erlang C formula expresses the waiting probability in a queuing system. Just as the Erlang B formula, Erlang C 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.

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

where:

A is the total traffic offered in units of erlangs
N is the number of servers
P_W 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. A common use for Erlang C is modeling and dimensioning call center agents in a call center environment.

Engset formula

The Engset calculation is a related formula, named after its developer, T. O. Engset, used to determine the probability of congestion occurring within a telephony circuit group. It deals with a finite population of S sources rather than the infinite population of sources that Erlang assumes. The formula requires that the user knows the expected peak traffic, the number of sources (callers) and the number of circuits in the network.

Example application

A business installing a PABX needs to know the minimum number of voice circuits it needs to have to and from the telephone network. An approximate approach is to use the Erlang-B formula. However, if the business has a small number of extensions, then it should instead use the more exact Engset calculation, which reflects the fact that extensions already in use will not make additional simultaneous calls. (For a large user population, the Engset and the Erlang-B calculations give the same result.)

Technical details

Engset's equation is similar to the Erlang-B formula; however it contains one major difference: Erlang's equation assumes an infinite source of calls, yielding a Poisson arrival process, while Engset specifies a finite number of callers^[4] ^[5]. Thus Engset's equation should be used when the source population is small (say less than 200 users, extensions or customers).

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

where

A = offered traffic intensity in erlangs, from all sources

S = number of sources of traffic

N = number of circuits in group

P(b) = probability of blocking or congestion

In practice, like Erlang's equations, Engset's formula requires recursion to solve for the blocking or congestion probability. There are several recursions that could be used^[5]. One way to determine this probability, one first determines an initial estimate. This initial estimate is substituted into the equation and the equation then is solved. The answer to this initial calculation is then substituted back into the equation, resulting in a new answer which is again substituted. This iterative process continues until the equation converges to a stable result.^[4]^[6].

Engset's equation follows^[4]:

$P(b)=\frac{\left[\frac{\left(S-1\right)!}{N!\cdot\left(S-1-N\right)!}\right]\cdot M^N}{\sum_{X=1}^N\left[\frac{\left(S-1\right)!}{X!\cdot\left(S-1-X\right)!}\right]\cdot M^X}$