Erlang-S: A Data-Based Model of Servers in Queueing Networks

Classical queueing theory has typically focused on customers, and server availability has been taken for granted. However, data accessibility and the emergence of complex service systems, for example, call centers, revealed the need to stochastically model the complex behavior of servers. In this work, we propose a new model that accommodates such behavior; we call it Erlang-S, where “S” stands for servers. Our model assumes a pool of present servers, some of whom are available to serve customers from the queue while others are not, and the process of becoming available or unavailable is modeled explicitly. Our focus here is on applying the model to real systems, specifically call centers. Estimating the parameters of the new model from call-center data is challenging because reality is observed discretely in time as opposed to its continuous evolution. We therefore use an expectation-maximization algorithm that computes the expected number of relevant quantities given the discrete-time data. Erlang-S differs from the Erlang-A model, which has been commonly used for modeling call centers: the latter model assumes that all agents who are present are, in fact, available for service. When comparing predictions of the two models against call-center data, we find that Erlang-A generally overestimates queue length and consequently also the fraction of abandonment, and Erlang-S predicts reality more closely and usefully. Our conclusion is that it is important to model explicitly server dynamics to obtain accurate and valuable models of complex service systems.