Patience Time Distributions Research Articles

Optimally Scheduling Heterogeneous Impatient Customers

Problem definition: We study scheduling multi-class impatient customers in parallel server queueing systems. At the time of arrival, customers are identified as being one of many classes, and the class represents the service and patience time distributions as well as cost characteristics. From the system’s perspective, customers of the same class at time of arrival get differentiated on their residual patience time as they wait in queue. We leverage this property and propose two novel and easy-to-implement multi-class scheduling policies. Academic/practical relevance: Scheduling multi-class impatient customers is an important and challenging topic, especially when customers’ patience times are nonexponential. In these contexts, even for customers of the same class, processing them under the first-come, first-served (FCFS) policy is suboptimal. This is because, at time of arrival, the system only knows the overall patience distribution from which a customer’s patience value is drawn, and as time elapses, the estimate of the customer’s residual patience time can be further updated. For nonexponential patience distributions, such an update indeed reveals additional information, and using this information to implement within-class prioritization can lead to additional benefits relative to the FCFS policy. Methodology: We use fluid approximations to analyze the multi-class scheduling problem with ideas borrowed from convex optimization. These approximations are known to perform well for large systems, and we use simulations to validate our proposed policies for small systems. Results: We propose a multi-class time-in-queue policy that prioritizes both across customer classes and within each class using a simple rule and further show that most of the gains of such a policy can be achieved by deviating from within-class FCFS for at most one customer class. In addition, for systems with exponential patience times, our policy reduces to a simple priority-based policy, which we prove is asymptotically optimal for Markovian systems with an optimality gap that does not grow with system scale. Managerial implications: Our work provides managers ways of improving quality of service to manage parallel server queueing systems. We propose easy-to-implement policies that perform well relative to reasonable benchmarks. Our work also adds to the academic literature on multi-class queueing systems by demonstrating the joint benefits of cross- and within-class prioritization. Funding: A. Bassamboo received financial support from the National Science Foundation [Grant CMMI 2006350]. C. (A.) Wu received financial support from the Hong Kong General Research Fund [Early Career Scheme, Project 26206419]. Supplemental Material: The online appendix is available at https://doi.org/10.1287/msom.2023.1190 .

Steady-state and first passage time distributions for waiting times in the $ MAP/M/s+G $ queueing model with generally distributed patience times

<p style='text-indent:20px;'>We study the <inline-formula><tex-math id="M2">\begin{document}$ MAP/M/s+G $\end{document}</tex-math></inline-formula> queueing model that arises in various multi-server engineering problems including telephone call centers, under the assumption of MAP (Markovian Arrival Process) arrivals, exponentially distributed service times, infinite waiting room, and generally distributed patience times. Using sample-path arguments, we propose to obtain the steady-state distribution of the virtual waiting time and subsequently the other relevant performance metrics of interest via the steady-state solution of a certain Continuous Feedback Fluid Queue (CFFQ). The proposed method is exact when the patience time is a discrete random variable and is asymptotically exact when it is continuous/hybrid, for which case discretization of the patience time distribution is required giving rise to a computational complexity depending linearly on the number of discretization levels. Additionally, a novel method is proposed to accurately obtain the first passage time distributions for the virtual and actual waiting times again using CFFQs while approximating the deterministic time horizons by Erlang distributions or more efficient Concentrated Matrix Exponential (CME) distributions. Numerical results are presented to validate the effectiveness of the proposed numerical method.</p>

Stationary distribution convergence of the offered waiting processes in heavy traffic under general patience time scaling

We study a sequence of single server queues with customer abandonment (\(GI/GI/1+GI\)) under heavy traffic. The patience time distributions vary with the sequence, which allows for a wider scope of applications. It is known Lee and Weerasinghe (Stochastic Process Appl 121(11):2507–2552, 2011) and Reed and Ward (Math Oper Res 33(3):606–644, 2008) that the sequence of scaled offered waiting time processes converges weakly to a reflecting diffusion process with nonlinear drift, as the traffic intensity approaches one. In this paper, we further show that the sequence of stationary distributions and moments of the offered waiting times, with diffusion scaling, converge to those of the limit diffusion process. This justifies the stationary performance of the diffusion limit as a valid approximation for the stationary performance of the \(GI/GI/1+GI\) queue. Consequently, we also derive the approximation for the abandonment probability for the \(GI/GI/1+GI\) queue in the stationary state.

Optimally Scheduling Heterogeneous Impatient Customers

We study the question of scheduling impatient customers in parallel server queuing systems. At the time of arrival, customers can be identified as one of many classes, where the class represents the service time and patience time distributions, and cost characteristics. From the system's perspective, customers that are of the same type at time of arrival get further differentiated on their residual patience time as they wait in the system. This is due to the fact that at time of arrival, the system only knows the overall patience distribution from which the customer's patience value is drawn, and as time elapses, this estimate can be further updated for customers who are still in the system using the information that the customer has not yet abandoned. For non-exponential patience distributions, such an update indeed reveals additional information. In this paper, we use a fluid approach to characterize the cost-minimizing policy that schedules customers on two dimensions of heterogeneity: class and time-in-queue information. We propose a multi-class time-in-queue policy that prioritizes both across customer classes, and within each customer class using a simple rule, and further we show that most of the gains of such a policy can be achieved by deviating from within-class First Come First Serve (FCFS) for at most one customer class. Finally, for systems with exponential abandonment times, our policy reduces to a simple priority-based policy, which we prove to be asymptotically optimal with an optimality gap that does not grow with system scale.

Customer Service Chat Systems with General Service and Patience Times

We study customer service chat (CSC) systems with generally distributed service and patience times by developing measure-valued processes to model and analyze the system dynamics. We first prove that these processes are tight in the many-server asymptotic regime and then show that their limits satisfy a set of fluid model equations. We then establish the invariant states of these limits and use these invariant states to obtain (non-asymptotic) approximations for various performance metrics of CSC systems in the steady state. We also demonstrate the accuracy of these approximations using extensive numerical experiments. These approximations allow us to establish the impact of service and patience time distributions on the system performance and to devise effective dynamic routing policies.

Existence and uniqueness of a fluid model for many-server queues with abandonment

In this paper, we establish a new (direct) proof for existence and uniqueness of the fluid model for Gt/GI/N+GI queues proposed recently by Kang and Ramanan under mild conditions on the arrival, service and patience time distributions. In particular, the existence of the fluid model is established directly from the fluid model itself. Then main technique here is the application of two non-linear functional integral equations, one of which is of Volterra type.

Convergence to equilibrium states for fluid models of many-server queues with abandonment

Fluid models, in particular their equilibrium states, have become an important tool for the study of many-server queues with general service and patience time distributions. However, it remains an open question whether the solution to a fluid model converges to the equilibrium state and under what condition. We show in this paper that the convergence holds under some conditions. Our method builds on the framework of measure-valued processes, which keeps track of the remaining patience and service times.

Validity of heavy-traffic steady-state approximations in many-server queues with abandonment

We consider $$GI/Ph/n+M$$ G I / P h / n + M parallel-server systems with a renewal arrival process, a phase-type service time distribution, $$n$$ n homogenous servers, and an exponential patience time distribution with positive rate. We show that in the Halfin---Whitt regime, the sequence of stationary distributions corresponding to the normalized state processes is tight. As a consequence, we establish an interchange of heavy-traffic and steady-state limits for $$GI/Ph/n+M$$ G I / P h / n + M queues.

Fluid limits of many-server queues with abandonments, general service and continuous patience time distributions

This paper extends the works of Kang and Ramanan (2010) and Kaspi and Ramanan (2011), removing the hypothesis of absolute continuity of the service requirement and patience time distributions. We consider a many-server queueing system in which customers enter service in the order of arrival in a non-idling manner and where reneging is considerate. Similarly to Kang and Ramanan (2010), the dynamics of the system are represented in terms of a process that describes the total number of customers in the system as well as two measure-valued processes that record the age in service of each of the customers being served and the “potential” waiting times. When the number of servers goes to infinity, fluid limit is established for this triple of processes. The convergence is in the sense of probability and the limit is characterized by an integral equation.

On queues with impatience: stability, and the optimality of Earliest Deadline First

In this paper, we consider a queue with impatient customers, under general assumptions. We introduce a convenient representation of the system by a stochastic recursive sequence keeping track of the remaining service and patience times of the customers. This description allows us (i) to provide a comprehensive stability condition in the general case, (ii) to give a rigorous proof of the optimality of the Earliest Deadline First (EDF) service discipline in several cases, and (iii) to show that the abandonment probability of the system follows inversely the stochastic ordering of the generic patience time distribution.

Hazard Rate Scaling of the Abandonment Distribution for the GI/M/n + GI Queue in Heavy Traffic

We obtain a heavy traffic limit for the GI/M/n + GI queue, which includes the entire patience time distribution. Our main approach is to scale the hazard rate function of the patience time distribution in such a way that our resulting diffusion approximation contains the entire hazard rate function. We then show through numerical studies that for various performance measures, our approximations tend to outperform those commonly used in practice. The robustness of our results is also demonstrated by applying them to solving constraint satisfaction problems arising in the context of telephone call centers.

Fluid models of many-server queues with abandonment

We study many-server queues with abandonment in which customers have general service and patience time distributions. The dynamics of the system are modeled using measure- valued processes, to keep track of the residual service and patience times of each customer. Deterministic fluid models are established to provide first-order approximation for this model. The fluid model solution, which is proved to uniquely exists, serves as the fluid limit of the many-server queue, as the number of servers becomes large. Based on the fluid model solution, first-order approximations for various performance quantities are proposed.

Many-server queues with customer abandonment: A survey of diffusion and fluid approximations

The performance of a call center is sensitive to customer abandonment. In this survey paper, we focus on G / GI / n + GI parallel-server queues that serve as a building block to model call center operations. Such a queue has a general arrival process (the G), independent and identically distributed (iid) service times with a general distribution (the first GI), and iid patience times with a general distribution (the +GI). Following the square-root safety staffing rule, this queue can be operated in the quality- and efficiency-driven (QED) regime, which is characterized by large customer volume, the waiting times being a fraction of the service times, only a small fraction of customers abandoning the system, and high server utilization. Operational efficiency is the central target in a system whose staffing costs dominate other expenses. If a moderate fraction of customer abandonment is allowed, such a system should be operated in an overloaded regime known as the efficiency-driven (ED) regime. We survey recent results on the many-server queues that are operated in the QED and ED regimes. These results include the performance insensitivity to patience time distributions and diffusion and fluid approximate models as practical tools for performance analysis.

Many-server diffusion limits for G/Ph/n+GI queues

This paper studies many-server limits for multi-server queues that have a phase-type service time distribution and allow for customer abandonment. The first set of limit theorems is for critically loaded $G/Ph/n+GI$ queues, where the patience times are independent and identically distributed following a general distribution. The next limit theorem is for overloaded $G/ Ph/n+M$ queues, where the patience time distribution is restricted to be exponential. We prove that a pair of diffusion-scaled total-customer-count and server-allocation processes, properly centered, converges in distribution to a continuous Markov process as the number of servers $n$ goes to infinity. In the overloaded case, the limit is a multi-dimensional diffusion process, and in the critically loaded case, the limit is a simple transformation of a diffusion process. When the queues are critically loaded, our diffusion limit generalizes the result by Puhalskii and Reiman (2000) for $GI/Ph/n$ queues without customer abandonment. When the queues are overloaded, the diffusion limit provides a refinement to a fluid limit and it generalizes a result by Whitt (2004) for $M/M/n/+M$ queues with an exponential service time distribution. The proof techniques employed in this paper are innovative. First, a perturbed system is shown to be equivalent to the original system. Next, two maps are employed in both fluid and diffusion scalings. These maps allow one to prove the limit theorems by applying the standard continuous-mapping theorem and the standard random-time-change theorem.

The Busy Period of an M/G/1 Queue with Customer Impatience

We consider an M/G/1 queue in which an arriving customer does not enter the system whenever its virtual waiting time, i.e. the amount of work seen upon arrival, is larger than a certain random patience time. We determine the busy period distribution for various choices of the patience time distribution. The main cases under consideration are exponential patience and a discrete patience distribution.

The Busy Period of an M/G/1 Queue with Customer Impatience

We consider an M/G/1 queue in which an arriving customer does not enter the system whenever its virtual waiting time, i.e. the amount of work seen upon arrival, is larger than a certain random patience time. We determine the busy period distribution for various choices of the patience time distribution. The main cases under consideration are exponential patience and a discrete patience distribution.