# How to Calculate Waiting Times with the Erlang C Formula

## Erlang C Calculators

Telecommunications traffic is measured in dimensionless units called Erlangs, named after the Danish telephone engineer, Agner Krarup Erlang. The number of Erlangs of traffic is equal to the average number of calls received per unit of time, multiplied by the average duration of a call.

If you have a system in which calls are queued rather than dropped, then you can use the Erlang C formula to calculate the probability that a call is queued and the average call waiting time, aka the Average Speed of Answer (ASA). To use the Erlang C formula, you need to know the traffic level and the number of trunks or lines that are available to take calls.

The Erlang C formula can also be used to determine how many phone lines are needed to attain a desired probability of waiting. The Erlang C formula and some examples of its application are described below. You can also use the Erlang C calculators at left.

The first calculator tells you how many phone lines are needed if you input the traffic and desired probability that a call is queued. The second calculator tells you the probability that a call is queued and the average wait time when you input the traffic level and number of trunks.

## The Erlang C Formula

If the Erlangs of traffic is E and the number of trunks is M, then the Erlang C formula for the probability that a call is queued is given by the equationP

_{queued}= (E

^{M}/M!)/[E

^{M}/M! + (1 - E/M)∑

^{M-1}

_{n=0}E

^{n}/n!].

The average call waiting or Average Speed of Answer is given by the equation

ASA = (T*P

_{queued})/(M-E),

where T is the average duration of the call. The units of T determine the units of ASA. The two formulas above are only valid if M is greater than E, that is, the number of trunks in a system is greater than the number of Erlangs.

**Example 1:**Suppose a call center has 10 phone lines, receives 480 calls per day, and the average duration of a call is 15 minutes. Since 15 minutes = 1/96 days, the number of Erlangs is (480)(1/96) = 5. (To compute Erlangs, one must express the call frequency and call duration using the same units of time.) Thus, the probability that a call is placed in the waiting line is

P

_{queued}= (5

^{10}/10!)/[5

^{10}/10! + (1 - 5/10)∑

^{10-1}

_{n=0}5

^{n}/n!]

= (5

^{10}/10!)/[5

^{10}/10! + 0.5∑

^{9}

_{n=0}5

^{n}/n!] = 0.0361054

and the average waiting time for a call in the queue is

(15*0.0361054)/(10-5) = .108316 minutes, or about 6.5 seconds.

One should be careful when interpreting the ASA since it is only an average. Many calls in the system may experience much longer waits.

**Example 2:**Recursive iterations can be used to figure M from a given value of E and a desired value of P

_{queued}. Suppose that a call center receives 300 calls per hour, the average call duration is 5 minutes, and the center would like no more than 3% of the calls to be queued.

First, we calculate the number of Erlangs as (300)(1/12) = 25, since 5 minutes is 1/12 of an hour. Next we set P

_{queued}= 0.03. Then using an iterative procedure similar to guess-and-check, we find that the call center must have a minimum of 36 phone lines to achieve this probability of queuing.

If the system drops or blocks calls rather than places them in a queue, you must use the Erlang B Formula instead.

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