Abandonment Time Distributions Research Articles

An Equilibrium Analysis of a Multiclass Queue with Endogenous Abandonments in Heavy Traffic

This paper studies a multiclass queueing system with endogenous abandonments where congestion affects customers’ abandonment behavior, and vice versa. Our model captures this interaction by developing two closely related models: an abandonment model and a queueing model. In the abandonment model, customers take the virtual waiting time distribution as given. Class k customers receive a reward rk from service and incur a cost ck per period of waiting. Customers are forward looking and make wait or abandon decisions dynamically to maximize their expected discounted utilities. The queueing model takes the customers’ abandonment time distribution as an input and studies the resulting virtual waiting time distribution. Because the multiclass queueing system is not amenable to exact analysis, we resort to an approximate analysis in the conventional heavy traffic limit (under the hazard rate scaling). Leveraging the so-called state-space collapse property, we provide a characterization of the system performance. Combining the results for the two models, we show that there exists a unique equilibrium in which the customers’ abandonment time and the virtual waiting time for the various classes are consistent in the two models. Finally, we provide a computational scheme to calculate the equilibrium numerically and apply that using data from an Israeli bank call center. The online appendix is available at https://doi.org/10.1287/opre.2017.1638 .

An equilibrium analysis of a discrete-time Markovian queue with endogenous abandonments

This paper studies a Geo/Geo/1$$+GI$$+GI queue in which the abandonments are endogenous. One crucial feature of this model is that the abandonment behavior is affected by the system performance and vice versa. Our model captures this interaction by developing two closely related models: an abandonment model and a queueing model. In the abandonment model, customers take the virtual waiting time distribution as given. They receive a reward r from service and incur a cost c per period of waiting. Customers are forward-looking and maximize their expected discounted utilities by making wait or abandon decisions dynamically as they wait in the queue. The queueing model takes the customers' abandonment time distribution as an input and studies the resulting virtual waiting time distribution. In equilibrium, the customers' abandonment behavior and the system performance must be consistent across the two models. Therefore, combining the two models and imposing this consistency requirement, we show that there exists a unique equilibrium. Lastly, we provide a computational scheme to calculate the equilibrium numerically.

An Equilibrium Analysis of a Discrete-Time Markovian Queue with Endogenous Abandonments

This paper studies a Geo/Geo/1 GI queue in which the abandonments are endogenous. One crucial feature of this model is that the abandonment behavior is affected by the system performance and vice versa. Our model captures this interaction by developing two closely related models: an abandonment model and a queueing model. In the abandonment model, customers take the virtual waiting time distribution as given. They receive a reward r from service and incur a cost c per period of waiting. Customers are forward looking and maximize their expected discounted utilities by making wait or abandon decisions dynamically as they wait in the queue. The queueing model takes the customers' abandonment time distribution as an input and studies the resulting virtual waiting time distribution. In equilibrium, the customers' abandonment behavior and the system performance must be consistent across the two models. Therefore, combining the two models and imposing this consistency requirement, we show that there exists a unique equilibrium. Lastly, we provide a computational scheme to calculate the equilibrium numerically.

General hazard-type scaling of abandonment time distribution for a G/Ph/n+GI queue in the Halfin–Whitt heavy-traffic regime

We consider the diffusion approximation of a G/Ph/n queue with customer abandonment in the Halfin---Whitt heavy-traffic regime and extend the conventional locally Lipschitz hazard-type scaling of abandonment time distribution to a more general scaling in the study. Under that new scaling of abandonment, not only the non-locally Lipschitz hazard-type case such as the non-locally bounded hazard-rate scaling which has never been solved to date, but also a wider range of abandonment time distributions including the non-absolutely continuous ones becomes subject to the analysis of diffusion approximation. Due to the general character of our scaling scheme of abandonment time, the stochastic equation for the limit of C-tight, scaled, and centered customer-count processes contains a nonlinear drift term as the limit of abandonment-count process, which does not satisfy the local Lipschitz condition so that the continuous mapping method in the literature of this area is invalid in our case. Instead, applying the Girsanov transformation to the localized multidimensional equation satisfied by the limit of customer-count process in multiphase of service time, we show the uniqueness in law of the solution to establish the desired diffusion approximation.

Capacity Sizing in Queueing Models with a Random Number of Servers

One prevalent assumption in queueing theory is that the number of servers in a queueing model is deterministic. However, randomness in the number of servers often arises in practice, particularly when the servers themselves may be viewed as strategic decision makers, e.g., as in virtual call centers or ride-sharing services where agents are allowed to set their own schedules, often at very short time notice. In this paper, we study the problem of staffing many-server queues with general abandonment and a random number of servers. We model the number of agents available to work in a given period as a binomial random variable, and rely on a fluid model to determine cost-minimizing staffing levels. We provide theoretical support for the usage of the fluid model by demonstrating the asymptotic accuracy of fluid-based performance measures and staffing prescriptions. As an application, we characterize the optimal staffing policy with self-scheduling servers, and show that it is not straightforward: It may be optimal to either understaff or overstaff the system, depending on (i) self-scheduling agent behavior and (ii) the abandonment-time distribution.

G-RAND: A phase-type approximation for the nonstationary [formula omitted] queue

We present a Markov model to analyze the queueing behavior of the nonstationary G(t)/G(t)/s(t)+G(t) queue. We assume an exhaustive service discipline (where servers complete their current service before leaving) and use acyclic phase-type distributions to approximate the general interarrival, service, and abandonment time distributions. The time-varying performance measures of interest are: (1) the expected number of customers in queue, (2) the variance of the number of customers in queue, (3) the expected number of abandonments, and (4) the virtual waiting time distribution of a customer arriving at an arbitrary moment in time. We refer to our model as G-RAND since it analyzes a general queue using the randomization method. A computational experiment shows that our model allows the accurate analysis of small- to medium-sized problem instances.

A Markov Model for Measuring Service Levels in Nonstationary G(t)/G(t)/s(t)+G(t) Queues

We present a Markov model to approximate the queueing behavior at the G(t)=G(t)=s(t) + G(t) queue with exhaustive discipline and abandonments. The performance measures of interest are: (1) the average number of customers in queue, (2) the variance of the number of customers in queue, (3) the average number of abandonments and (4) the virtual waiting time distribution of a customer when arriving at an arbitrary moment in time. We use acyclic phase-type distributions to approximate the general interarrival, service and abandonment time distributions. An ecient, iterative algorithm allows the accurate analysis of small- to medium-sized problem instances. The validity and accuracy of the model are assessed using a simulation study.

Dynamic scheduling of a GI/GI/1+GI queue with multiple customer classes

We consider a dynamic control problem for a GI/GI/1+GI queue with multiclass customers. The customer classes are distinguished by their interarrival time, service time, and abandonment time distributions. There is a cost c k >0 for every class k?{1,2,?,N} customer that abandons the queue before receiving service. The objective is to minimize average cost by dynamically choosing which customer class the server should next serve each time the server becomes available (and there are waiting customers from at least two classes). It is not possible to solve this control problem exactly, and so we formulate an approximating Brownian control problem. The Brownian control problem incorporates the entire abandonment distribution of each customer class. We solve the Brownian control problem under the assumption that the abandonment distribution for each customer class has an increasing failure rate. We then interpret the solution to the Brownian control problem as a control for the original dynamic scheduling problem. Finally, we perform a simulation study to demonstrate the effectiveness of our proposed control.

A fluid approximation for large-scale service systems

We introduce and analyze a deterministic fluid model that serves as an approximation for the Gt/GI/st + GI manyserver queueing model, which has a general time-varying arrival process (the Gt), a general service-time distribution (the first GI), a time-dependent number of servers (the st) and allows abandonment from queue according to a general abandonment-time distribution (the +GI). This fluid model approximates the associated queueing system when the arrival rate and number of servers are both large. We also show that the system dynamics greatly simplifies in two special cases: (i) when the service time distribution is exponential (M) and (ii) when the service time distribution is deterministic (D) and the model is stationary. We develop an efficient algorithm to compute all standard performance functions in both cases. In case (i), we establish an asymptotic loss of memory (ALOM) property, i.e., asymptotic independence from the initial conditions as time evolves. We show that the difference in the performance functions with different initial conditions dissipates over time exponentially fast, under regularity conditions. In contrast, in case (ii) we show that ALOM fails dramatically. Instead, although all model parameters are constants, we show that the performance rapidly approaches a periodic steady state (PSS) with a period equal to the service time, whenever the system does not start with the unique stationary distribution. Moreover, the form of the PSS depends on the initial condition. Simulation and a heavy-traffic limit confirm that this anomalous behavior also occurs in the large-scale queueing model.

Real-Time Delay Estimation in Overloaded Multiserver Queues with Abandonments

We use heavy-traffic limits and computer simulation to study the performance of alternative real-time delay estimators in the overloaded GI/GI/s+GI multiserver queueing model, allowing customer abandonment. These delay estimates may be used to make delay announcements in call centers and related service systems. We characterize performance by the expected mean squared error in steady state. We exploit established approximations for performance measures with a nonexponential abandonment-time distribution to obtain new delay estimators that effectively cope with nonexponential abandonment-time distributions.

Approximations for the M/GI/N+GI type call center

In this paper, we propose approximations to compute the steady-state performance measures of the M/GI/N+GI queue receiving Poisson arrivals with N identical servers, and general service and abandonment-time distributions. The approximations are based on scaling a single server M/GI/1+GI queue. For problems involving deterministic and exponential abandon times distributions, we suggest a practical way to compute the waiting time distributions and their moments using the Laplace transform of the workload density function. Our first contribution is numerically computing the workload density function in the M/GI/1+GI queue when the abandon times follow general distributions different from the deterministic and exponential distributions. Then we compute the waiting time distributions and their moments. Next, we scale-up the M/GI/1+GI queue giving rise to our approximations to capture the behavior of the multi-server system. We conduct extensive numerical experiments to test the speed and performance of the approximations, which prove the accuracy of their predictions.

Adaptive Behavior of Impatient Customers in Tele-Queues: Theory and Empirical Support

We address the modeling and analysis of abandonments from a queue that is invisible to its occupants. Such queues arise in remote service systems, notably the Internet and telephone call centers; hence, we refer to them as tele-queues. A basic premise of this paper is that customers adapt their patience (modeled by an abandonment-time distribution) to their service expectations, in particular to their anticipated waiting time. We present empirical support for that hypothesis, and propose an M/M/m-based model that incorporates adaptive customer behavior. In our model, customer patience depends on the mean waiting time in the queue. We characterize the resulting system equilibrium (namely, the operating point in steady state), and establish its existence and uniqueness when changes in customer patience are bounded by the corresponding changes in their anticipated waiting time. The feasibility of multiple system equilibria is illustrated when this condition is violated. Finally, a dynamic learning model is proposed where customer expectations regarding their waiting time are formed through accumulated experience. We demonstrate, via simulation, convergence to the theoretically anticipated equilibrium, while addressing certain issues related to censored-sampling that arise because of abandonments.