# The Erlang-C Formula

The widely-used Erlang-C formula is given here for people who want to see the actual mathematical definition.

The Erlang-C calculations are described step by step below, using and example of 360 calls per half hour, with an average call duration of 4 minutes, and 55 agents. The target answer time for service level is 15 seconds. |
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## (1) Specify Call Arrival RateThe first parameter needed is the average customer arrival rate. It doesn't matter what time unit is used to specify the arrival rate, as long as the same time unit is used for the average call duration. Also, the results we shall get for waiting times will be in these same time units. Generally, for call-center applications of Erlang-C, the best time unit to use is seconds. |
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## (2) Specify Call DurationThe second factor to be specified is the average call duration. This must be expressed in the same time unit used for the call arrival rate. |
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## (3) Specify Number of AgentsThe third factor is the number of agents available. |
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## (4) Calculate Traffic IntensityThe term "traffic intensity" comes from the original application of Erlang-C, which was for telephone networks, and the volume of calls was described as the "traffic". We need to calculate the traffic intensity as a preliminary step to the main Erlang-C formula, but traffic intensity can be interpreted in several useful ways. |
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## (5) Calculate Agent OccupancyThe agent occupancy, or utilisation, is now calculated by dividing the traffic intensity by the number of agents. The agent occupancy will be between 0 and 1. If it is not less than 1 then the agents are overloaded, and the Erlang-C calculations cannot be done. For reporting results we ususually quote occupancy as a percentage, so we multiply by 100% to get the agent occupancy as a percentage. |
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## (6) Calculate the Erlang-C FormulaNow we can calculate the main Erlang-C formula. This formula
looks complicated, but is straightforward to calculate with a few
lines of programming. The value of E |
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## (7) Calculate Probability A Call WaitsE |
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## (8) Calculate Average Speed Of AnswerHaving calculated E |
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## (9) Calculate Service LevelFrequently we want to calculate the probability that a call will be answered in less than a target waiting time. The formula for this is given here. Remember that, again, the probability will be on the scale 0 to 1 and should be multiplied by 100% to get a percentage. |
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## (10) Calculate Agents NeededIf the service level is specified and you want to calculate the number of agents needed, then you must do a bit of (intelligent) trial and error. You have to find the number of agents that will just achieve the service-level you want. A good approach is to start with u, the traffic intensity, as the number of agents. If u is fractional then take the next highest integer. You will need at least this number of agents to avoid overload. Calculate the service-level you will get with this number of agents, then increase by one agent at a time until you just get the service-level you want. |