For general concepts of queueing theory that apply to this queueing model, see About Queueing Models.

In the Erlang C queueing model, customers arrive at a queueing
system having * n* servers and

Since there are infinitely many waiting positions, the waiting
positions can never be exhausted. Therefore, every arriving customer will
eventually be served, even if after a long wait. Note that in contrast
to the Erlang B queueing model, no customer ever experiences blockage!
However, if the traffic load * x* in erlangs is greater
than or equal to the number

Part of the definition of the Erlang C queue specifies how waiting
customers are selected for service: we assume that whenever there exist waiting
customers, the longest-waiting customer will always be the next customer to
begin service. In other words, we assume a **FIFO**
queueing discipline.

The **Erlang C Function** * C(n,x)*
is defined by

** C**(

where *B(n, x) = (x^{n}/n!) / (1
+ x + x^{2}/2! + x^{3}/3! + ... +
x^{n}/n!)* is the Erlang B function.
(See Erlang B Queueing Model.)

The Erlang C function computes the probability that an arriving
customer in the Erlang C queueing model will find that all servers are busy.
This is the same as the fraction of arriving customers that are delayed (*i.e*.,
must wait) before beginning service. This function is used in computing
many other functions in the Erlang Library for Excel. The Erlang C function
occurs in the Erlang Library for Excel under the name ErlcFractionDelayed.

## Erlang C Definitions |
## Erlang C Formulas |

Average
Handle Time (average duration of service)
Average
Wait for All customers (customers
with zero wait time are included in the average); see ErlcWait,
ErlcWait4
Average
Wait for Delayed customers (customers
with zero wait time are not included in the average)
Number
of Waiting customers (average number in queue); see
ErlcNwaiting, ErlcNwaiting4
Number
of Busy servers
Number
of customers in System
)
= Probability that the wait time of a customer will be less than or equal
to t, where t any
non-negative number. This is the same as the fraction of callers
whose wait time is less than or equal to t. See ErlcFractionOk.t |
The Erlang C function ,n)
is defined asx
,n)
= x*n(B,n)
/ (x − n*(1−x(B,n)))xwhere ,n)
is the Erlang B function. Thenx
< μ andn
/λμ
,n)x
μ
/ (C*(μ−n))x= *C/ (AHT −n)x
/ (AHT−n)x
+ AWAAHT
* C/ (x −n)x
x
+ NBNW= + (x*C/ (x −n))x
/xn
*Ce^{−(n−x)*t/AHT} |

In these formulas, the quantities **C**, **AHT**,
etc., are theoretical averages that are approached as a limit under the assumption
that the queue operates for a very long period of time without any change to
the queue parameters (number of servers, arrival rate, service rate).

Call center analysts often use the Erlang C queueing model to help understand the functioning of a group of agents taking incoming calls in a call center. As in any application of queueing theory, there are three parts that fit together: (1) a queueing model, (2) a real-world system, and a (3) mapping of the queueing model to the real-world system (see About Queueing Models). In this example, the queueing model is Erlang C, and the real-world system is a group of agents handling calls in a call center. The mapping of the queueing model to the real-world system looks like this:

Erlang C Entity |
Call Center Entity |

Server | Agent |

Position in the system | Trunk |

Customer | Caller |

Arrival rate | Calls arriving per second |

Average Handle Time | Average Talk Time + Average Wrap Time |

NW |
Average Queue Length |

etc. |

Clearly some of the assumptions of the Erlang C model are not true for a call center. For example, there are only a finite number of trunks, hence only a finite number of places where calls can be parked while waiting for service. But, if the number of trunks is quite large, then as a practical matter, there may be no situation when all trunks are busy, hence the assumption of infinitely many waiting positions might "almost" be true. Similarly, the assumptions of Poisson arrivals and exponential service times will not hold exactly. Nevertheless, experience over many decades has shown that using the Erlang C model can give helpful insights into the operation of agent groups in call centers, provided that

(1) there are a large number of trunks (hence many waiting slots),

(2) the assumptions of Poisson arrivals and exponential service times are approximately correct over the period being studied,

(3) wating calls are handled first-come-first-served.

There are many other applications of the Erlang C model—in telecom and in other fields as well.

About Queueing Models, Erlang B Queueing Model

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