Adaptive behavior of impatient customers in tele-queues: theory and empirical support.(Abstract)

1. Introduction

Customer characteristics in service systems are largely dependent upon the system performance characteristics as perceived by its users. For example, the arrival rate is likely to increase as the typical waiting time decreases. This dependence interacts with the queueing process to determine the system operating point, and may have a considerable effect on performance.

Our focus in this paper is on the modeling of customer abandonments and their interplay with the system performance. We consider a queueing system with impatient customers, who may abandon the queue if not admitted to service soon enough. We assume that the queue is invisible, in the sense that waiting customers do not obtain any information regarding the queue size or their remaining waiting time before admitted to service. Queues of this type are especially relevant to remote service systems, such as telephone call centers or Internet-based services; hence, we refer to them as tele-queue. For a discussion of the central role that customer patience plays in tele-queues see Garnett et al. (1999).

The foundation for our model is the hypothesis that customers' patience significantly depends on their expectations regarding the waiting time in the system. These expectations, in turn, are formed through accumulated experience and affected by subjective factors--time perception, the importance of the service being sought, and so on. As an example, customers who expect to wait a few seconds will behave differently, in terms of their abandonment time, in case they expect to wait several minutes or even hours. These expectations, in turn, conceivably differ if past experience consists of short waits, or long waits, or short and long waits intertwined. Patience is obviously influenced by numerous factors related to customer profiles and environment characteristics (see, for example, Maister 1985, Zakay and Hornik 1996, Levine 1997). However, for the purpose of performance analysis, most of these factors can be taken as a priori given and fixed. The waiting time distribution is singled out in this respect since it is the outcome of the queueing process (hence, in fact, itself is influenced by the patience profile).

Empirical Support--A Preview. Inconsistent with the above adaptivity hypothesis, the prevalent assumption in traditional queueing theory is that patience (the time-to-abandon or its probability distribution) is "assigned" to individual customers independently of any system performance characteristic (see Garnett et al. 1999 for a recent literature review). In particular, patience is unaltered by possible changes in congestion. Such models, however, cannot accommodate the scatterplot in Figure 1 that exhibits remarkable patience-adaptivity.

[FIGURE 1 OMITTED]

The data is from a bank call center as reported in Mandelbaum et al. (2000); see also [section] 4. We are scatterplotting abandonment fraction against average delay, for delayed customers (positive waiting time) who seek technical Internet support. It is seen that average delay during 8:30-8:45 A.M., 17:45-18:00 P.M., 18:30-18:45 P.M., and 23:30-23:45 P.M. is about 100, 140, 180, and 240 seconds, respectively. Nonetheless, the fraction of abandoning customers (among those delayed) is remarkably stable at 38%, for all periods. This stands in striking contrast to traditional queueing models, where patience is assumed unrelated to system performance: Such models would predict a strict increase of the abandonment fraction with the waiting time, as in Figure 3. The behavior indicated in Figure 1 clearly suggests that customers do adapt their patience to system performance.

[FIGURE 3 OMITTED]

A Descriptive Approach. Several recent papers have proposed an optimization-based model for customer patience, where abandonment decisions are based on a personal cost function that balances service utility against the cost associated with the expected remaining time to service. In particular, Hassin and Haviv (1995) and Haviv and Ritov (2001) analyze systems with a single customer type, and Mandelbaum and Shimkin (2000) consider a heterogeneous customer population, in terms of utility functions and the resulting abandonment profiles. In these models, the optimal abandonment decision depends on the entire waiting-time distribution offered by the system.

Unlike this prescriptive approach, we consider here a descriptive model, where the dependence of patience on system performance is explicitly specified within the model primitives, in much the same way that a demand function is assumed to be given in economic models. Such an explicit model can be more directly related to experimental data, and is not restricted by the assumption and consequences of strictly rational behavior of the customers.

Our model is highly simplified by assuming that customers' patience depends on the waiting time in the queue only through its average, namely the mean wait; thus, the patience depends on a single performance parameter rather than an entire distribution. The motivation for this simplified model is threefold. First, the mean arguably presents a natural parameter that summarizes customers' expectations regarding their waiting time; indeed, a typical customer can hardly be expected to form a clear estimate of the entire waiting time distribution based on limited experience. Second, the dependence on a single parameter makes it much easier to relate the model to empirical data; see [section] 4. And third, it offers a considerable simplification in performance analysis (compared, say, with Mandelbaum and Shimkin 2000).

Outline of the Paper. Section 2 presents the basic queueing model, which incorporates the dependence of the patience profile on the average waiting time, and defines the system equilibrium point. (1) We distinguish between the average waiting time assumed by the customers (denoted x), which determines the patience profile, and between the actual quantity, namely the offered expected wait that results from this patience profile. Simply put, equilibrium is achieved when the two coincide.

In [section] 3, we analyze the equilibrium and its properties, focusing first on existence and uniqueness. Assuming that customer patience decreases as the (assumed) average wait x increases, existence and uniqueness of equilibrium follow from basic monotonicity considerations, as shown in [section] 3.1. The more interesting case is when patience is allowed to increase with x ([section] 3.2). Here customers adjust their behavior to comply with their expectations. When patience can grow not more than proportionally with x, existence and uniqueness of the equilibrium can still be established and the equilibrium point may be calculated. When this growth condition is violated, multiple equilibria are feasible, as we explicitly demonstrate there.

In [section] 3.3, we apply the proposed model to address the following question: What is the required dependence of customer patience, so that the abandonment fraction is kept constant despite varying congestion conditions. This question is motivated by the relative insensitivity of the abandonment fraction that was revealed in Figure 1.

Section 4 presents additional empirical support for the dependence of customer patience on the anticipated waiting time. Section 5 provides a brief survey of the literature on patience modeling.

Our basic equilibrium model assumes that the system is in steady state, in the sense that the system characteristics are stationary and the customers are well acquainted with those characteristics that are relevant to their behavior. In [section] 6, we complement the static equilibrium viewpoint with a dynamic learning model, which incorporates the additional ingredient of learning by the customers, and traces the system evolution towards a possible equilibrium. Indeed, the average waiting time parameter x is not initially known, but may be estimated by the customers based on their accumulated experience. We briefly address the issue of censored sampling that arises here: In those customer's visits that end up with abandonment, the offered wait itself is not observed but rather a lower bound on it, namely the abandonment time. As consistent estimation of the mean is quite complicated in this case, we also consider a simpler nonconsistent estimator and its effect on the equilibrium point. The dynamics of the queueing system which incorporates the proposed learning process is examined via simulation, and its convergence to the anticipated equilibrium is demonstrated. We conclude in [section] 7 with a brief summary and comments concerning future work.

2. Model Formulation

Consider an M/M/m queue with Poisson arrivals at rate [lambda], and an exponential service time with mean [[mu].sup.-1] at each of the m servers. The service discipline is first-come-first-served. Waiting customers may abandon the queue at any time before admitted to service. Potential abandonment times of individual customers are assumed independent and identically distributed, according to a probability distribution G(*) over the nonnegative real line. We shall refer to G as the patience distribution function. Let [bar]G = 1 - G denote the survival function; thus [bar]G(t) is the probability that a waiting customer will not abandon…