Erlang (unit) - Erlang B Formula

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).

where:

• 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:

.

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

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.