Erlang (unit)

The erlang (symbol E^[1]) is a dimensionless unit that is used in telephony as a statistical measure of offered load or carried load on service-providing elements such as telephone circuits or telephone switching equipment. It is named after the Danish telephone engineer A. K. Erlang, the originator of traffic engineering and queueing theory.

1 Traffic measurements of a telephone circuit
2 Erlang B formula
- 2.1 Extended Erlang B
3 Erlang C formula
4 Engset formula
- 4.1 Example application
- 4.2 Technical details
5 See also
6 References
7 Tools
8 External links

Traffic measurements of a telephone circuit

When used to represent carried traffic, a value (which can be a non-integer such as 43.5) followed by “erlangs” represents the average number of concurrent calls carried by the circuits (or other service-providing elements), where that average is calculated over some reasonable period of time. The period over which the average is calculated is often one hour, but shorter periods (e.g., 15 minutes) may be used where it is known that there are short spurts of demand and a traffic measurement is desired that does not mask these spurts. One erlang of carried traffic refers to a single resource being in continuous use, or two channels being in use fifty percent of the time, and so on. For example, if an office has 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 one hour continuously is said to have a load of 1 Erlang.

When used to describe offered traffic, a value followed by “erlangs” represents the average number of concurrent calls that would have been carried if there were an unlimited number of circuits (that is, if the call-attempts that were made when all circuits were in use had not been rejected). The relationship between offered traffic and carried traffic depends on the design of the system and user behavior. Three common models are (a) callers whose call-attempts are rejected go away and never come back, (b) callers whose call-attempts are rejected try again within a fairly short space of time, and (c) the system allows users to wait in queue until a circuit becomes available.

A third measurement of traffic is instantaneous traffic, expressed as a certain number of erlangs, meaning the exact number of calls taking place at a point in time. In this case the number is an integer. Traffic-level-recording devices, such as moving-pen recorders, plot instantaneous traffic.

The concepts and mathematics introduced by Agner Krarup Erlang have broad applicability beyond telephony. They apply wherever users arrive more or less at random to receive exclusive service from any one of a group of service-providing elements without prior reservation, for example, where the service-providing elements are ticket-sales windows, toilets on an airplane, or motel rooms. (Erlang’s models do not apply where the server-providing elements are shared between several concurrent users or different amounts of service are consumed by different users, for instance, on circuits carrying data traffic.)

Offered traffic (in erlangs) is related to the call arrival rate, λ, and the average call-holding time, h, by:

$E = \lambda h$

provided that h and λ are expressed using the same units of time (seconds and calls per second, or minutes and calls per minute).

The practical measurement of traffic is typically based on continuous observations over several days or weeks, during which the instantaneous traffic is recorded at regular, short intervals (such as every few seconds). These measurements are then used to calculate a single result, most commonly the busy hour traffic (in erlangs). This is the average number of concurrent calls during a given one-hour period of the day, where that period is selected to give the highest result. (This result is called the time-consistent busy hour traffic). An alternative is to calculate a busy hour traffic value separately for each day (which may correspond to slightly different times each day) and take the average of these values. This generally gives a slightly higher value than the time-consistent busy hour value.

The goal of Erlang’s traffic theory is to determine exactly how many service-providing elements should be provided in order to satisfy users, without wasteful over-provisioning. To do this, a target is set for the grade of service (GoS) or quality of service (QoS). For example, in a system where there is no queuing, the GoS may be that no more than 1 call in 100 is blocked (i.e., rejected) due to all circuits being in use (a GoS of 0.01), which becomes the target probability of call blocking, P_b, when using the Erlang B formula.

There are several Erlang formulae, including Erlang B, Erlang C and the related Engset formula, based on different models of user behavior and system operation. 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.

Where the existing busy-hour carried traffic, E_c, is measured on an already-overloaded system, with a significant level of blocking, it is necessary to take account of the blocked calls in estimating the busy-hour offered traffic E_o (which is the traffic value to be used in the Erlang formula). The offered traffic can be estimated by E_o = E_c/(1 - P_b). For this purpose, where the system includes a means of counting blocked calls and successful calls, P_b can be estimated directly from the proportion of calls that are blocked. Failing that, P_b can be estimated by using E_c in place of E_o in the Erlang formula and the resulting estimate of P_b can then be used in E_o = E_c/(1 - P_b) to estimate E_o. Another method of estimating E_o in an overloaded system is to measure the busy-hour call arrival rate, λ (counting successful calls and blocked calls), and the average call-holding time (for successful calls), h, and then estimate E_o using the formula E = λh.

For a situation where the traffic to be handled is completely new traffic, the only choice is to try to model expected user behavior, estimating active user population, N, expected level of use, U (number of calls/transactions per user per day), busy-hour concentration factor, C (proportion of daily activity that will fall in the busy hour), and average holding time/service time, h (expressed in minutes). A projection of busy-hour offered traffic would then be E_o = (NUC/60)h erlangs. (The division by 60 translates the busy-hour call/transaction arrival rate into a per-minute value, to match the units in which h is expressed.)

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). Hence, the formula is also used in certain inventory systems with lost sales.

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, so call arrivals are independent. Further it is assumed that 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^[2] as follows, in a form that is used to simplify the calculation of tables of the Erlang B formula:

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

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

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

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

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

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 with finite mean. Erlang B is a trunk sizing tool for voice switch to voice switch traffic. The Erlang B formula is decreasing and convex in m.^[3]

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 attempts.^[4]

The steps in the process are as follows^[5]:

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

$E_{i%2B1}=E_{0}%2BR\,$

where $E_{0}$ is the initial (baseline) level of traffic.

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.

$P_W = {{\frac{A^N}{N!} \frac{N}{N - A}} \over \sum_{i=0}^{N-1} \frac{A^i}{i!} %2B \frac{A^N}{N!} \frac{N}{N - A}} \,$

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^[6] .^[7] 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.^[7] 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.^[6]^[8]

Engset's equation follows^[6]:

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

$M=\frac{A}{S-A\cdot\left(1-P(b)\right)}$

References

^ How Many? A Dictionary of Units of Measurement
^ Guoping Zeng (June 2003), Two common properties of the erlang-B function, erlang-C function, and Engset blocking function, Elsevier Science, http://www.sciencedirect.com/science?_ob=ArticleURL&_udi=B6V0V-4B82PB9-5&_user=10&_coverDate=06/30/2003&_rdoc=1&_fmt=high&_orig=search&_origin=search&_sort=d&_docanchor=&view=c&_acct=C000050221&_version=1&_urlVersion=0&_userid=10&md5=ab0c0972cc704de106a1d6e1c9463d87&searchtype=a, retrieved 2011-02-03
^ Messerli, E.J., 1972. 'Proof of a convexity property of the Erlang B formula'. Bell System Technical Journal 51, 951-953.
^ 'Designing optimal voice networks for businesses, government, and telephone companies' by J. Jewett, J. Shrago, B. Yomtov, TelCo Research, Chicago, 1980.
^ Inayatullah, M., Ullah, F.K., Khan., A.N., 'An Automated Grade Of Service Measuring System', IEEE—ICET 2006, 2nd International Conference on Emerging Technologies, Peshawar, Pakistan 13–14 November 2006, pp.230-237
^ ^a ^b ^c Parkinson, R.. "Traffic Engineering Techniques in Telecommunications" (pdf). Infotel Systems Inc.. http://www.tarrani.net/mike/docs/TrafficEngineering.pdf. Retrieved 2005-10-17.
^ ^a ^b Zukerman, M.. "An Introduction to Queueing Theory and Stochastic Teletraffic Models" (pdf). http://www.ee.cityu.edu.hk/~zukerman/classnotes.pdf. Retrieved 2008-08-05.
^ ITU-T Study Group 2. "Teletraffic Engineering Handbook" (PDF). http://www.com.dtu.dk/teletraffic/handbook/telenook.pdf. Retrieved June 2006.

Erlang (unit)

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