Our work relates to two streams of literature. The first stream consists of studies of customer behavior in queueing settings. The second stream is research regarding callbacks. In this section we review both of these streams and articulate our contribution to each.
2.2.1 Customer Behavior in Queueing Settings
There exists a rich stream of theoretical models and empirical studies of customer behavior in queueing settings. The seminal theoretical model in this stream is by Naor (1969), who models an M/M/1 system in which utility-maximizing customers decide whether to join the queue or balk. Naor’s model began a long stream of queueing models that treat customers in queues as utility-maximizing agents. Because callers in our setting may abandon after joining the online queue, the models in this stream that are most related to ours are those that deal with caller abandonment. These include models where callers abandon due to a shrinking reward (Hassin and Haviv, 1995), due to a belief that they are in a “fault state” where they will never receive service (Mandelbaum and Shimkin, 2000), and due to nonlinear waiting costs (Shimkin and Mandelbaum, 2004). For an overview of other studies in this stream see Hassin and Haviv (2003).
Since callers in our setting may receive a delay estimate upon arriving to the system, we also review the theoretical models of customer behavior in systems that provide delay announce- ments. Guo and Zipkin (2007) capture the impact on system performance of providing arriving callers various levels of information regarding the state of the system. Allon et al. (2011a) consider a firm that strategically chooses its delay announcement. They show that providing delay announcements improves firm profits and expected customer utility. Furthermore, Allon and Bassamboo (2011) show that rather than immediately providing delay announcements as customers arrive, waiting a short period to provide the announcement can lead to even greater firm profits and customer utility. Finally, Yu et al. (2018) study how delay announcements can be used to determine the extent to which different types of customers value the service. We also mention that recently several other customer behaviors in queues have been modeled, including returning after balking (Cui et al., 2018b), selecting queues based on service quality (Yang et al., 2018), and trading spots (cutting) in line (Allon and Hanany 2012, Yang et al. 2016).
A number of recent empirical studies have explored how customers behave in a variety of queueing settings, including caller abandonment in call centers (Brown et al. 2005, Mandelbaum and Zeltyn 2013, Aksin et al. 2013, Ak¸sin et al. 2016, Yu et al. 2016, Emadi and Swaminathan 2018, Hathaway et al. 2019a), patient abandonment in health care settings (Batt and Terwiesch 2015, Osadchiy and Kc 2017), customer defection due to waiting times in a banking setting (Buell et al., 2016), and customer purchase decisions in food service settings (Allon et al. 2011b, Lu et al. 2013). The studies in this vein that are most related to ours are those that model caller abandonment decisions using a structural estimation approach (Aksin et al. 2013, Ak¸sin et al. 2016, Yu et al. 2016, Emadi and Swaminathan 2018). Note that structural estimation is gaining prevalence in Operations Management research as a way to impute the underlying preferences of customers and firms. Some of the settings have included firm decisions in health care (Olivares et al., 2008), airlines (Deshpande and Arıkan, 2012), supermarkets (Bray et al., 2018), and the automotive sector (Bray and Mendelson 2015, Colak and Bray 2018), as well as customer decisions in air travel (Li et al., 2014), online retailing (Fisher et al. 2017, Moon et al. 2017), bike-sharing systems (Kabra et al., 2018), and food service (Allon et al. 2011b, Kim et al. 2014).
The first structural estimation study of caller behavior is that of Aksin et al. (2013), who model callers waiting in queue as solving an optimal-stopping problem to determine when to abandon. Given their reward for service and their per unit waiting cost, callers decide when to stop waiting by abandoning the system. Aksin, et al. estimate the reward and cost parameters for four different service groups and find that the estimates of the caller parameters differ by group. Emadi and Swaminathan (2018) relax the rational expectations equilibirirum assumption and instead model callers in a Bayesian learning framework, where callers update their beliefs about the distribution of waiting times in the call center each time they call. They find that callers who have no experience with the call center are overly optimistic about their probability of receiving service in a given period. Ak¸sin et al. (2016) and Yu et al. (2016) extend the approach of Aksin et al. (2013) to call centers that provide delay announcements. In both studies callers use the information in their delay announcement as a signal of the current waiting times in the call center and change their abandonment behavior accordingly. Our contribution to this stream of research is the formulation of a structural model of caller behavior in the presence of a callback option.
2.2.2 Callback Research
Only a handful of studies have been conducted regarding call centers that offer callbacks. The two studies that are most pertinent to our work are those of Armony and Maglaras (2004a,b). In Armony and Maglaras (2004a) the authors model a call center as an M/M/N
multiclass system with an online queue for real-time service and an offline queue for callers who postpone service by receiving a callback. Upon entering the system, callers are told the steady-state mean waiting time in the online queue and receive a callback offer with a fixed guarantee of the maximum delay they will experience before receiving a callback. Callers then choose either to immediately balk, join the online queue, or accept the callback offer. Relying on diffusion approximations, Armony and Maglaras find the unique equilibrium of the system and characterize the system performance. They show under a wide range of caller preferences that offering callbacks decreases the average waiting time of callers who join the online queue, while increasing the throughput of the system. Armony and Maglaras (2004b) extend their analysis to a system where callers receive an estimate of their waiting time in the online queue upon
arrival, which is based on the current state of the system. The authors show that providing real-time delay estimates magnifies the performance improvements from their first analysis.
Our structural model of caller behavior under a callback option differs from the models of Armony and Maglaras in two ways. First, callers in our model may abandon after joining the online queue, whereas callers in the models of Armony and Maglaras always remain in the online queue until receiving service. Second, not all callers in our model answer the callback when it arrives, whereas callers in the models of Armony and Maglaras always answer a callback when it arrives. Because we capture these additional ways in which callers may exit the system without receiving service, we find in our counterfactual analysis that under certain conditions offering callbacks has only minimal impact on system throughput. This differs from the results of Armony and Maglaras who find that system throughput increases under a wide variety of conditions.
A pair of more recent papers have concentrated on the decision of when to offer callbacks. Legros et al. (2016) explore this question using an MDP approach, where the manager’s ob- jective is to minimize the sum of the expected online waiting costs, offline waiting costs, and abandonment costs. They show in the case of two servers that callbacks should only be offered when the number of callers waiting in the offline queue is below some threshold. They also numerically characterize a policy for how many servers should be reserved to answer callers in the online queue. Ata and Peng (2018) study the question of when to offer callbacks in a system where the arrival rate is a time-varying stochastic process. They propose a threshold policy which depends on the length of the online queue, the current arrival rate, and the maximum service rate of the system. They also demonstrate that their policy performs well in simulations. Our work differs from the work of Legros et al. (2016) and Ata and Peng (2018) in two ways. First, none of the policies analyzed by Legros et al. (2016) and Ata and Peng (2018) are im- pacted by callers who do not to answer callbacks when they arrive, while we capture the impact of this behavior on system throughput. Second, whereas Legros et al. (2016) and Ata and Peng (2018) assume that the callers’ decisions of whether to accept a callback offer and whether to abandon after joining the online queue are exogeneously determined, in our model we character- ize the caller decision-making process by treating them as utility-maximizing agents who make their decisions based on a set of preferences that we impute from actual data. This allows us
to perform counterfactual analyses of callback policies that could potentially be implemented in the future, but for which no historical data is available.
To summarize, our contribution to the stream of callback research is two-fold. First, to the best of our knowledge we perform the only empirical study of caller behavior in a call center that offers callbacks. We use caller decisions to impute their callback preferences and use these preferences to quantify the magnitude of the impact of various callback policies. Second, we provide a framework for conducting counterfactual analyses of callback policies and explore how various callback policies would affect the service quality and system throughput of this call center.