How to Calculate Call Center Stats with the Erlang B Formula

Erlang B Calculators

Telecommunications traffic is measured in dimensionless units called Erlangs. 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 know the traffic level in Erlangs and the number of trunks (e.g. phone lines or servers), then you can use the Erlang B formula to calculate the probability that a call is dropped or blocked. This probability is known as the Grade of Sevice (GoS) for the call center, and it is always a number between 0 and 1.

When the probability is low, i.e., less than .05, then the call center is efficient at handling traffic. When the GoS is high, i.e., greater than 0.1, then the call center has a high rate of annoyed and impatient callers!

The Erlang B formula can also be used to determine how many phone lines are needed to attain a desired Grade of Service, given a certain level of traffic. The Erlang B formula and some examples are described below; you can also use the Erlang B calculators at left.

The first calculator tells you how many phone lines are needed if you input the traffic and desired probability of blocked calls. The second calculator tells you the probability that a call is dropped if you input the traffic and number of lines.

The Erlang B Formula

If the Erlangs of traffic is E and the number of trunk lines is M, then the well-known Erlang B formula for the probability that a call is dropped, the GoS, is given by the equation

GoS = (E^M/M!)/(∑^M_n=0 Eⁿ/n!).

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. (Computing Erlangs requires that call frequency and call duration be in the same units of time. You will get the same number no matter which units you use.) Thus, the probability that a call is blocked is

GoS = (5¹⁰/10!)/(∑¹⁰_n=0 5ⁿ/n!) = 0.0183846.

This means about 1.84% of the calls get dropped. As M becomes large, calculating GoS by hand becomes unwieldy, thus Erlang calculators must be used.

Example 2: Recursive iterations can be used to figure M from a given value of E and a desired value of GoS. Suppose that a call center receives 300 calls per hour, the average call duration is 5 minutes, and the center would like a GoS value of 0.025.

First, we calculate the number of Erlangs as (300)(1/12) = 25, since 5 minutes is 1/12 of an hour. Then using an iterative procedure similar to guess-and-check, we find that the call center must have a minimum of 33 phone lines to achieve this Grade of Service.

If the system places calls in a queue rather than drops them, you must use the Erlang C Formula instead. If a significant portion of dropped calls retry, then use the Extended Erlang B Formula, which factors in a recall rate. And for systems with a finite source of callers, use the Engset Blocking Function

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