Editing Erlang (unit) (section)

== Erlang C formula ==

The ''Erlang C formula'' expresses the probability that an arriving customer will need to queue (as opposed to immediately being served).<ref name="kleinrock">{{cite book | title = Queueing Systems Volume 1: Theory | first1=Leonard | last1=Kleinrock | author-link = Leonard Kleinrock | isbn = 978-0471491101 | year=1975 | page=103}}</ref> Just as the Erlang B formula, Erlang C assumes an infinite population of sources, which jointly offer traffic of <math>E</math> erlangs to <math>m</math> 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.  However, the Erlang C formula assumes that callers never hang up while in queue, which makes the formula predict that more agents should be used than are really needed to maintain a desired service level.
: <math>P_\text{w} = {{\frac{E^m}{m!} \frac{m}{m - E}} \over \left ( \sum\limits_{i=0}^{m-1} \frac{E^i}{i!} \right ) + \frac{E^m}{m!} \frac{m}{m - E}} </math>
where:
* <math>E</math> is the total traffic offered in units of erlangs
* <math>m</math> is the number of servers
* <math>P_\text{w}</math> 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 an [[exponential distribution]], therefore the Erlang C formula follows from the assumptions of the [[M/M/c queue]] model.