How much offered traffic is carried in practice will depend on what happens to unanswered calls when all servers are busy.
[3][4] In Erlang's analysis of efficient telephone line usage he derived the formulae for two important cases, Erlang-B and Erlang-C, which became foundational results in teletraffic engineering and queueing theory.
A distinguishing assumption behind the Erlang B formula is that there is no queue, so that if all service elements are already in use then a newly arriving call will be blocked and subsequently lost.
Erlang's formulae apply quite widely, but they may fail when congestion is especially high causing unsuccessful traffic to repeatedly retry.
One erlang of carried traffic refers to a single resource being in continuous use, or two channels each being in use fifty percent of the time, and so on.
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.
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.
For example, one could estimate 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).
(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.)
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 queueing space 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 vanishes forever.
where: Note: The erlang is a dimensionless load unit calculated as the mean arrival rate, λ, multiplied by the mean call holding time, h. See Little's law to prove that the erlang unit has to be dimensionless for Little's Law to be dimensionally sane.
This may be expressed recursively[6] as follows, in a form that is used to simplify the calculation of tables of the Erlang B formula: Typically, instead of B(E, m) the inverse 1/B(E, m) is calculated in numerical computation in order to ensure numerical stability: or a Python version The Erlang B formula is decreasing and convex in m.[7] It requires that call arrivals can be modeled by a Poisson process, which is not always a good match, but is valid for any statistical distribution of call holding times with a finite mean.
Erlang B was developed as a trunk sizing tool for telephone networks with holding times in the minutes range, but being a mathematical equation it applies on any time-scale.
Extended Erlang B differs from the classic Erlang-B assumptions by allowing for a proportion of blocked callers to try again, causing an increase in offered traffic from the initial baseline level.
and the recall factor can be used to calculate the probability that all of a caller's attempts are lost, not just their first call but also any subsequent retries.
The Erlang C formula expresses the probability that an arriving customer will need to queue (as opposed to immediately being served).
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.
Once this action has been taken, congestion will return to reasonable levels and Erlang's equations can then be used to determine how exactly many circuits are really required.
[11] An example of an instance which would cause such a High Loss System to develop would be if a TV-based advertisement were to announce a particular telephone number to call at a specific time.
If the service provider had not catered for this sudden peak demand, extreme traffic congestion will develop and Erlang's equations cannot be used.