Using the Erlang formula

In the previous section we saw that the Erlang formula can be used to compute the average
waiting time for a given number of agents, service times and traffic intensity. One would like to use the formula also for other types of questions, such as: for given _ and s, and a maximal acceptable ASA or given SL, what is the maximal call volume per time unit _ that the call center can handle? Because of the complexity of C(s, a) we cannot ”reverse” the formula, but by trial-and-error we can answer these types of questions. The question that is of course posed mostoften is to calculate the minimum number of agents needed for a given load and service level. This also can be done using trial-and-error, and software tools such as our Erlang calculator (to be found at www.math.vu.nl/~koole/ccmath/ErlangC) often do this automatically.
In our Erlang C calculator, fill in 1 and 5 at “Arrivals” and “Service time”, fill in “80” and “20” at “Service level” and select “Service level”. After pushing the “compute” button the computation shows that 8 agents are needed to reach this SL. Most software tools will give you an integer number of agents as answer. This makes sense, as we cannot employ say half an agent. However, we can employ an agent half of the time. Thus when a software tool requires you to schedule 17.4 agents during a half an hour, then you should schedule 17 agents during 18 minutes, and 18 agents during 12 Chapter 4 — The Erlang C formula 19
minutes. With 17 agents you are below the SL, with 18 you are above. Thus the ”bad” SL during 18 minutes is compensated by the better than required SL due to using 18 agents. In our Erlang C calculator we decided not to implement this, because we assume that the time interval is so short that a constant number of agents is required. Let us continue the example. Selecting “Number of agents” instead of “Service level”, and pushing the “compute” button again shows that the actual service level is 86% instead of only 80% that was required.
”Garbage in = garbage out”. This well-known phrase holds also for the Erlang formula: the input parameters should be determined with care. Especially with the value of the expected call durations _ one can easily make mistakes. The reason for this is that the entire time the agent is not available for taking a new call should be counted. For the Erlang formula the service starts the moment the ACD assigns a call to an agent, and ends when the agents becomes available, i.e., if the telephone switch has again the possibility to assign a call to that agent. Thus _ consists not only of the actual call duration, but also of the reaction time (that can be as long as 10 seconds!), plus the wrap-up time (that can be as long as the call itself). Note that the reaction time is seen by the caller as waiting time. This should be taken into account when calculating the service levels, by decreasing the acceptable waiting time with the average reaction time.
In a call center the reaction time is 3 seconds on average, the average call duration is 25 seconds and there is no finish time. On peak hours on average 200 calls per 15 minutes arrive. An average waiting time of 10 seconds is seen as an acceptable service level. We calculate first the load without reaction time. The number of calls per second is 200/(15 × 60) _ 0.2222 (_ means “approximately”), and the load is 0.2222×25 _ 5.555. The Erlang formula shows that we need 7 agents, giving an expected waiting time of 8.2 seconds. This seems alright, but in reality there is an expected waiting time of no less than 27.9 seconds! This follows from the Erlang formula, with a service time of 25+3 = 28 seconds (and thus a load of 0.2222 × 28 _ 6.222), and 7 agents. The waiting time is then 24.9 seconds, to which the 3 seconds reaction time should be added. To calculate the right number of agents we start with a service time of 28 seconds, and we look for the number of agents needed to get a maximal waiting time of 10 − 3 = 7 seconds. This is the case for 8 agents, with an average waiting time of 6.5 seconds. This way the average waiting time remains limited to 9.5 seconds.
A possible conclusion of the last example could be that agents should be stimulated to react faster in order to avoid that an extra agent should be scheduled. However, these types of measures, aimed at improving the quantitative aspects of the call center, can lead to a decrease of the quality of the call center work, due to the increased work pressure. We will not deal with the human aspects of call center work; let it just be noted that 100% productivity is in no situation possible, and the overcapacity calculated by the Erlang formula is one of the means forthe agents to get the necessary short breaks between calls.