In 1917 a Danish mathematician named Agner Krarup Erlang first published his Erlang C formula giving mathematical insight into the fields of traffic engineering and queueing theory. Today the Erlang model is still communally used in major call centre forecasting and workforce management solutions, to help answer questions such as the number of agents required to answer forecasted call volumes, forecasting a service level and the probability of calls being answered or being put into wait.

I have over the years read many articles and papers seeking knowledge on the workings of Erlang, but unless you own a high aptitude for math’s it can often be difficult to understand. This is what led me to Ged Koole, who not only is an expert in this field but who has an uncanny ability to translate and bridge the gap between highly complex mathematical theory to everyday use. Ger has kindly allowed me to re-post his latest paper which not only highlights some of associated problems with the use of Erlang C theory but also offers an alternative solution.

**Ger Koole - www.gerkoole.com - ger@gerkoole.com**Recently a call center consultant contacted me concerning the Erlang C formula. He had filled in some numbers in the calculator on my website and found a service level of 93% within 3 minutes. Then he increased the average handling time by 10 seconds and his service level went down to 0%! He wrote me that "this is not correct in reality" and asked if the computation was right.

The computation was right, but the consultant
was also right in saying that this does not represent reality. In reality the
service level changes much more slowly when one of the input parameters
changes. The reason is

*customer abandonment*.**Stability**

In the Erlang C system, the only way
customers leave is through service. Thus it is essential that the average
amount of customers that enter the queue is lower than the maximum that can be
served. When this is not the case then the queue slowly builds up. Because
Erlang C is a

*long-run calculation*it predicts an infinite queue and thus a 0% service level. We call this an unstable queue. By the way, in such a situation the ASA prediction is infinity. You can get unpleasant surprises when using this. Suppose that your staffing levels are ok except for one slightly understaffed interval. Then your overall ASA prediction for the whole day is infinity!**Abandonments**

In reality unstable systems do not exist.
In physical queues less people enter when queues are long; in call centers,
where often waiting time information is not given, some customers abandon. The
longer the queue, the more customers will abandon. This will make any call
center stable and it will assure that the service level will never be 0%. Thus
in our service level calculation we should take customer abandonment into
acoount.

**Erlang X**

Erlang himself did not take abandonments
into account in his queueing systems. The first to do this was the Swedish
mathematician Conny Palm in the sixties. This evolved into what we call the
Erlang X formula. The Erlang X takes the effect of abandonments on the queue
length and on the waiting times of future arrivals into account. As an extra
variable the average patience of callers should be entered. When the Erlang X
is applied to the same example, with an average patience of 5 minutes, then 5%
of the calls abandon and the service level is 98%. In this situation the 5%
abandonments make room for almost all other customers to have a short waiting
time.

**Patience**

To use the Erlang X one should know the
average patience. Determining the average patience is a complicated statistical
question, because we do not know the patience of all customers, but only of
those who abandoned, and they were mostly impatient. Therefore we have a
tendency to underestimate the patience of our customers.

Further possibilities of the Erlang X
include limiting the number of lines and the inclusion of redials.

Further information on Erlang X can be found in Ger Koole's recently published book "Call center optimization", which describes all aspects of the WFM cycle as well as many other related subjects. This can be bought at Lulu.com.In addition all of the Erlang calculators referred in this article can be found at www.gerkoole.com/CCO. A license and other WFM software can be obtained from www.ccmath.com.

Ger Koole is a mathematician with a background in operations research and applied probability theory, especially queueing models. He applies these techniques to the service sector, mainly in call centers and health care.

He is full professor at VU University Amsterdam is the co-founder of CCmath, a software firm specialized in call center workforce management, and regularly gives training sessions to professionals in the call center and health care industry.