How good is the Erlang formula?

In this section we consider the weak points of the Erlang formula and its underlying assumptions. It will motivate some of the more sophisticated models. It might come as a surprise that the ASA is bigger than 0 although there is overcapacity. The reason for this is the variability in arrival times and service durations. If all arrival times were equally spaced and if all call holding times were constant, then no waiting would occur. However, in the random environment of the call center under capacity occurs during short periods of time. This is the reason why queueing occurs. The queue will always empty again if on average there is overcapacity. The Erlang formula quantifies the amount of waiting (in terms of ASA or TSF) for a particular type of random arrival and service times. The mathematical random processes that model the arrivals and departures are therefore nothing more than approximations. The quality of the approximation and the sensitivity of the formula to changes with respect to the different aspects of the model decide whether the Erlang formula gives acceptable results. We deal with the underlying assumptions one by one and discuss the consequences for the approximation.
Abandonments In a well-dimensioned call center there are few abandonments. Not modeling these abandonments is therefore not a gross simplification. However, there are call centers that show a completely different behavior than predicted by the Erlang formula because abandonments are not explicitly modeled. In general we can state that abandonments reduce the waiting time of other customers, thus it is good for the SL that abandonments occur! In call centers with a close to or even exceeding s it is crucial to model abandonments as well. Luckily this is possible Retrials Abandonments are relatively well understood and the Erlang C formula can be extended to account for abandonments without too much difficulty. This is no longer true
if the customers who abandoned start to call again and thus generate retrials. Little is known about the behavior of customers concerning retrials and about good mathematical models. Unfortunately retrials are a common phenomenon in most call centers. Peaks in offered load Formally speaking, the Erlang formula allows no fluctuations in offered load. However, in every call center there are daily changes in load. As long as these changes remain limited, and, more importantly, if there are no periods with undercapacity, then the Erlang formula performs well for periods where there are little fluctuations in load and number of agents. By using the Erlang formula for different time intervals we can get the whole picture by averaging (as explained in Chapter 3).

However, as soon as undercapacity occurs then the backlog of calls from one period is shifted to the next. This backlog should be explicitly modeled, which is not possible within the framework of the Erlang formula. Therefore the Erlang formula cannot be used in the case of undercapacity. For a short peak in offered traffic (e.g., reactions to a tv commercial) straightforward capacity calculations ignoring the random behavior can give quite good results. See also Type of call durations The Erlang formula is based on the assumption that the service times come from a so-called exponential distribution. Without going in the mathematical details, we just note that all positive values are possible as call durations, thus also very long or short ones, but that most of the durations are below the average. Certain measurements on standard telephone traffic show that call durations are approximately exponential, although the results in the literature do not completely agree on this subject. A typical case where call durations are not exponential is when there are multiple types of calls with different call length averages, or if a call always takes a certain minimum amount of time. In these cases one should wonder what the influence is of the different service time distributions on the Erlang formula. We can state that this influence decreases as the call center increases in size. With some care it can be concluded that only the average call duration is of major importance to the performance of the call center.
Human behavior Up to now we ignored the behavior of the agents, apart from the time it takes to take up to phone. However, agent behavior is not as simple as that. Employees take small breaks to get coffee, to discuss things, etc. Modeling explicitly the human behavior is a difficult task; describing and quantifying the behavior is even more difficult! In most situations these small breaks are taken when there are no calls in the queue. It can therefore be expected that they are of minor importance to the SL. In other situations it has a bigger impact, and it can ceriously limit the possibilities of quantitative modeling.