# Erlang C

Erlang C is a synonymous expression for a queue model that was developed by the Danish mathematician Agner Krarup Erlang at the beginning of the 20th century to denote the probability and the mean duration of waiting times at the telephone exchange.

In Kendall notation , it's an M / M / c model. The term Erlang C is also used synonymously for the Erlang-C formula , which reflects the distribution of the waiting time in this model. The Erlang-C model and the formula are used, for example, for call center plans in order to determine a service level or (indirectly via a service level specification) a staffing requirement from the specified call volume , number of operator stations (agents) and average service time . The Erlang-C model is criticized as part of the service level calculation for call centers because the model does not take into account several real conditions such as a limited number of lines or waiting places, the impatience of callers or heterogeneous groups of agents and callers. Erlang B exists to determine the line capacity .

The original formula Erlang came up with for this problem is

${\ displaystyle {\ text {volume of work}} a = {\ frac {{\ text {number of calls}} \ cdot {\ text {processing time}}} {3600 {\ text {seconds}}}}}$ .

Erlang assumes hourly intervals here. The formula can be generalized as follows ( t is a freely selectable time interval):

${\ displaystyle a = {\ frac {{\ text {number of calls}} \ cdot {\ text {average processing time}}} {t}}}$ Several parameters can be derived from the Erlang-C model:

If there are c operator stations (agents) available, the probability that a caller will have to wait at all and not be served immediately is:

${\ displaystyle P_ {1} = P [W] = {\ frac {{\ frac {a ^ {c}} {c!}} \ cdot {\ frac {c} {ca}}} {\ left (\ sum _ {n = 0} ^ {c-1} {\ frac {a ^ {n}} {n!}} \ right) + {\ frac {a ^ {c}} {c!}} \ cdot { \ frac {c} {ca}}}}}$ If the service rate or, equivalently, the mean time of service (of a conversation) is used, then the following applies . The probability that a typical customer will have to wait less than seconds is then added to ${\ displaystyle \ mu}$ ${\ displaystyle E [X]}$ ${\ displaystyle \ mu = E [X] ^ {- 1}}$ ${\ displaystyle t}$ ${\ displaystyle P [W \ leq t] = 1-P_ {1} \ cdot e ^ {- \ mu (ca) \ cdot t}}$ For an application example, see service level .

## Criticism of Erlang C

Although the use of the Erlang C model is widespread, there are numerous points of criticism. The reality differs from the model in many ways . The caller will not wait in the queue indefinitely , but will hang up after a certain period of time. In addition, the waiting room is limited by the number of lines available in the call center. If these are busy, the caller hears a busy tone . Above a certain service level threshold (80% - 90%), the use of additional employees brings only marginal improvements in accessibility. This is known as the Income Law . Employee breaks are not included in the formula, but must be assessed separately. The Erlang-C formula sometimes gives different results with small variations of the parameters λ, μ and c. This is especially the case when a is close to c. The assumption of arrival distribution is also often incorrect. Shortly after a commercial , there is a massive accumulation of calls. Another problem is heterogeneity . As a rule, the agents are not all on the same level of knowledge, but have certain “specialty areas”. In addition , the callers often do not form a homogeneous group, but rather several heterogeneous groups - for example with " premium customers". Overall, these inaccuracies lead to coverage (more agents are employed than Erlang C requires).

## Alternatives to the Erlang-C model

Algorithms based on Erlang C and adapted to the requirements are used in workforce management systems that deliver better results. However, these algorithms are not published for commercial reasons. Better queuing models exist , but they are not widely used. Instead, simulation programs are increasingly being used as a basis for planning.

## literature

• Florian Schümann, Horst Tisson: Call Center Controlling . Gabler, 2006, ISBN 3-409-12680-5 .