Patience Time Research Articles

Routing and Staffing in Customer Service Chat Systems with Generally Distributed Service and Patience Times

Routing and Staffing in Customer Service Chat Systems with Generally Distributed Service and Patience Times

Perfect sampling of stochastic matching models with reneging

Abstract In this paper, we introduce a slight variation of the dominated-coupling-from-the-past (DCFTP) algorithm of Kendall, for bounded Markov chains. It is based on the control of a (typically non-monotonic) stochastic recursion by another (typically monotonic) one. We show that this algorithm is particularly suitable for stochastic matching models with bounded patience, a class of models for which the steady-state distribution of the system is in general unknown in closed form. We first show that the Markov chain of this model can easily be controlled by an infinite-server queue. We then investigate the particular case where patience times are deterministic, and this control argument may fail. In that case we resort to an ad-hoc technique that can also be seen as a control (this time, by the arrival sequence). We then compare this algorithm to the primitive coupling-from-the-past (CFTP) algorithm and to control by an infinite-server queue, and show how our perfect simulation results can be used to estimate and compare, for instance, the loss probabilities of various systems in equilibrium.

Analysis of a Multi-Server Queue with Group Service and Service Time Dependent on the Size of a Group as a Model of a Delivery System

In this paper, we consider a multi-server queue with a finite buffer. Request arrivals are defined by the Markov arrival process. Service is provided to groups of requests. The minimal and maximal group sizes are fixed. The service time of a group has a phase-type distribution with an irreducible representation depending on the size of the group. The requests are impatient. The patience time for an arbitrary request has an exponential distribution. After this time expires, the request is lost if all servers are busy or, if some server is idle, with a certain probability, all requests staying in the buffer start their service even if their number is below the required minimum. The behavior of the system is described by a multi-dimensional continuous-time Markov chain that does not belong to the class of level-independent quasi-birth-and-death processes. The algorithm for the computation of the stationary distribution of this chain is presented, and expressions for the computation of the queuing system’s performance characteristics are derived. The description of a delivery system operation in terms of the analyzed queuing model is given, and the problem of the optimization of its operation is numerically solved. Multi-server queues with a phase-type distribution for the group service time that are dependent on the size of the group, the account of request impatience, and the correlated arrival process have not previously been analyzed in the existing literature. However, they represent a precise model of many real-world objects, including delivery systems.

When drone delivery meets human courier: A co-sourcing perspective

With the constantly rising labor cost and deterioration in traffic congestion, the drone-based service emerges as a game changer for the urban last-mile delivery. Meanwhile, the human courier, as a traditional player in the last-mile delivery, appears as a typical form of gig economy, which is always available but with less revenue to harvest. To help the retailer with the fleet sizing decision-making, a co-sourcing mechanism is introduced, which is by nature a time-varying queue with a finite waiting space. And the orders are subject to cancellation when the waiting exceeds the customers’ patience time. Furthermore, three stochastic optimization models are formulated, namely, the pure stability, the pure profit, and the hybrid models, to jointly determine the optimal waiting space threshold and the drone fleet size, which are then approximated with fluid models. Based on the optimal control theory, the optimal decisions are obtained according to the outsourcing probability and time delay. As validated in the simulation, the proposed pure stability model can effectively stabilize the system performance at the control target, while the hybrid model takes into account both the stability and the profit. Finally, the tradeoff between profitability and volatility is analyzed, and the impacts of the parameters on the overall system performance are revealed.

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 .

Reliability Modeling of a Food Industrial System with two Types of Repair Persons wherein Demand is Season Dependent

The present study pertains to developing a reliability model concerning a food industrial system that runs throughout the month except for Sundays. There are two types of repair persons engaged with the system, one known as operator while another is fitter. The operator is responsible for minor repairs, while the fitter is responsible for major repairs. It is also noticed that the operator is the first person who attends the failed unit in case of a major failure. If the operator is found incapable of repairing the unit within some patience time, then the fitter is called. While gathering the actual month-wise data of the plant, two types of seasons, namely normal and festival seasons, have been recorded. The festival season is from July to November, while the normal season is from December to June, based on consumer demand. During the months of the normal season, the system is shut down for a few days as the demand gets accomplished before the month ends. However, during the festival season, the system has to be kept operational throughout the month to fulfill the high demand for the product. The semi-Markov process and regenerative point technique are used to assess the reliability measures, namely mean time to system failure (MTSF), system availability in both seasons, expected busy period of the repair persons, and expected downtime of the system. The overall profitability of the system is also demonstrated by Graphical interpretation.

Many-Server Heavy-Traffic Limits for Queueing Systems with Perfectly Correlated Service and Patience Times

We characterize heavy-traffic process and steady-state limits for systems staffed according to the square-root safety rule, when the service requirements of the customers are perfectly correlated with their individual patience for waiting in queue. Under the usual many-server diffusion scaling, we show that the system is asymptotically equivalent to a system with no abandonment. In particular, the limit is the Halfin-Whitt diffusion for the [Formula: see text] queue when the traffic intensity approaches its critical value 1 from below, and is otherwise a transient diffusion, despite the fact that the prelimit is positive recurrent. To obtain a refined measure of the congestion due to the correlation, we characterize a lower-order fluid (LOF) limit for the case in which the diffusion limit is transient, demonstrating that the queue in this case scales like [Formula: see text]. Under both the diffusion and LOF scalings, we show that the stationary distributions converge weakly to the time-limiting behavior of the corresponding process limit. Funding: This work was supported by the National Natural Science Foundation of China [Grant 72188101] and the Division of Civil, Mechanical and Manufacturing Innovation [Grants 1763100 and 2006350].

Increases or discounts: Price strategies based on customers’ patience times

This study analyzes dynamic pricing in a non-preemptive M/M/1 service system in which customers exhibit patience in regard to waiting time. We use a stylized, single-server queue to incorporate customer patience into the queuing problem, with the assumption that customers will not enter the queue when the waiting time exceeds their inherent patience time, reflecting the decision whether to wait. We focus on queue design, structure, and revenue and consider two homogeneous and heterogeneous customer scenarios. Our findings show that, for homogeneous customers, price-raising and price-discounting strategies can mitigate revenue loss. We examined which policy is better for service providers in different scenarios. For heterogeneous customers, we draw two conclusions. First, a revenue-maximizing service provider should increase her priority fee if customers have patience in regard to waiting time, regardless of whether they are regular customers, priority customers, or a mix of both. Second, contrary to conventional wisdom, we show that the service provider should abandon a portion of his or her regular customers to attract more customers who are willing to pay a higher priority fee for better service when the value of regular customers is small. Under this situation, customers’ patience times can bring more revenue to the service provider. If the value of regular customers is high, however, then an increased priority fee cannot compensate for the revenue loss from abandoning these customers and losing their revenue. Overall, our findings confirm that customers’ patience times result in a high net monetary cost for the time-sensitive customer.

Self-reporting and screening: Data with right-censored, left-censored, and complete observations.

We consider survival data that combine three types of observations: uncensored, right‐censored, and left‐censored. Such data arises from screening a medical condition, in situations where self‐detection arises naturally. Our goal is to estimate the failure‐time distribution, based on these three observation types. We propose a novel methodology for distribution estimation using both semiparametric and nonparametric techniques. We then evaluate the performance of these estimators via simulated data. Finally, as a case study, we estimate the patience of patients who arrive at an emergency department and wait for treatment. Three categories of patients are observed: those who leave the system and announce it, and thus their patience time is observed; those who get service and thus their patience time is right‐censored by the waiting time; and those who leave the system without announcing it. For this third category, the patients' absence is revealed only when they are called to service, which is after they have already left; formally, their patience time is left‐censored. Other applications of our proposed methodology are discussed.

Many-server limits for service systems with dependent service and patience times

Many-server limits for service systems with dependent service and patience times

A System with Two Spare Units, Two Repair Facilities, and Two Types of Repairers

Assuming exponential lifetime and repair time distributions, we study the limiting availability A∞ as well as the per unit time-limiting profit ω of a one-unit system having two identical, cold standby spare units using semi-Markov processes. The failed unit is repaired either by an in-house repairer within an exponential patience time T or by an external expert who works faster but charges more. When there are two repair facilities, we allow the regular repairer to begin repair or to continue repair beyond T if the expert is busy. Two models arise accordingly as the expert repairs one or all failed units during each visit. We show that (1) adding a second spare to a one-unit system already backed by a spare raises A∞ as well as ω; (2) thereafter, adding a second repair facility improves both criteria further. Finally, we determine whether the expert must repair one or all failed units to maximize these criteria and fulfill the maintenance management objectives better than previously studied models.

Dynamic Rebalancing Strategy in Free-Float Bicycle Sharing Systems: Orbit Queues and Two-Sided Matching

In populous metropolitan areas, the free-floating bicycle-sharing system (FFBSS) acts as an innovative urban mobility as a service, which provides an ease-of-use feature and extra flexibility in contrast to the traditional shared bicycles with docks. In consideration of customer behaviors, such as abandonment and retrial, which occur in FFBSS, a redistribution strategy for shared bicycles among different user-density locations is presented with an aim to diminish the total operational cost while enhancing the overall service level. To formulate the user and multitype shared bicycle–arrival patterns as nonhomogeneous queues, our results provide a tractable analytical paradigm for a time-varying balancing strategy for FFBSS. The bicycle variation at each virtual zone after each redistribution is determined via a nonstationary queueing model, in which the service time, patience time, and research delay are all subject to general distribution. Then, the bicycle-deployment strategy is evaluated with respect to average queueing length and abandonment rate during a normal workday based on a tailored nonhomogeneous probabilistic matching queue. To verify the efficacy and cost-effectiveness of the proposed bicycle-redistribution strategy, multiple simulation runs are conducted with respect to various times of the day. It shows that the resulting optimal rebalancing strategy is batch-based in synchrony with the time heterogeneity in the traffic demand. Furthermore, several managerial insights are provided to shed light on the rule of thumb in practical FFBSS redistribution coordination.

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>

When Service Times Depend on Customers’ Delays: A Relationship Between Two Models of Dependence

When Service Times Depend on Customers’ Delays: A Relationship Between Two Models of Dependence Service times of customers often depend on the delay they experience in queue, as was recently demonstrated empirically in restaurants, call centers, and intensive care units. Two forms of dependence mechanisms in service systems with customer abandonment are studied in this paper: First, the service requirement of a customer may evolve while waiting in queue. Second, customers may arrive to the system with an exogenous service and patience time that are stochastically dependent. Because either dependence mechanism can have significant impacts on a system's performance, it should be identified and taken into consideration for performance evaluation and decision-making purposes. However, identifying the source of dependence from observed data is hard because both the service times and patience times are censored due to customer abandonment. Further, even if the dependence is known to be the latter exogenous one, there remains the difficult task of fitting a joint service-patience times distribution to the censored data. In “When Service Times Depend on Customers’ Delays: A Relationship Between Two Models of Dependence”, Wu, Bassamboo, and Perry provide a solution to address these statistical challenges.

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.

A Repairable System Supported by Two Spare Units and Serviced by Two Types of Repairers

We study a one-unit repairable system, supported by two identical spare units on cold standby, and serviced by two types of repairers. The model applies, for instance, to ANSI (American National Standard Institute) centrifugal pumps in a chemical plant, and hydraulic systems in aviation industry. The failed unit undergoes repair either by an in-house repairer within a random or deterministic patience time, or else by a visiting expert repairer. The expert repairs one or all failed units before leaving, and does so faster but at a higher cost rate than the regular repairer. Four models arise depending on the number of repairs done by the expert and the nature of the patience time. We compare these models based on the limiting availability A∞, and the limiting profit per unit time ω, using semi-Markov processes, when all distributions are exponential. As anticipated, to maximize A∞, the expert should repair all failed units. To maximize ω, a suitably chosen deterministic patience time is better than a random patience time. Furthermore, given all cost parameters, we determine the optimum number of repairs the expert should complete, and the optimum patience time given to the regular repairer in order to maximize ω.

Fluid and diffusion models for a system of taxis and customers with delayed matching

We study a system of taxis and customers with Poisson arrivals and exponential patience times. We model a delayed matching process between taxis and customers using a matching rate $$\theta $$ as follows: if there are i taxis and j customers in the system, the next pairing will occur after an exponential amount of time with rate $$\theta i^{\delta _1}j^{\delta _2}$$ ( $$\delta _1, \delta _2 \in (0,+\infty $$ )). We formulate the system as a CTMC and study the fluid and diffusion approximations for this system, which involve the solutions to a system of differential equations. We consider two approximation methods: Kurtz’s method (KA) derived from Kurtz’s results (Kurtz in J Appl Probab 7(1):49–58, 1970; Kurtz in J Appl Probab 8(2):344–356, 1971) and Gaussian approximation (GA) that works for the case $$\delta _1 = \delta _2 = 1$$ (we call this the bilinear case) based on the infinitesimal analysis of the CTMC. We compare their performance numerically with simulations and conclude that GA performs better than KA in the bilinear case. We next formulate an optimal control problem to maximize the total net revenue over a fixed time horizon T by controlling the arrival rate of taxis. We solve the optimal control problem numerically and compare its performance to the real system. We also use Markov decision processes to compute the optimal policy that maximizes the long-run revenue rate. We finally propose a heuristic control policy (HPKA) and show that its expected regret is a bounded function of T. We also propose a version of this policy (HPMDP) that can actually be implemented in the real queueing system and study its performance numerically.

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.

Hazard rate estimation for call center customer patience time

Estimating the hazard function of customer patience time has become a necessary component of effective operational planning such as workforce staffing and scheduling in call centers. When customers get served, their patience times are right-censored. In addition, the exact event times in call centers are sometimes unobserved and naturally binned into time intervals, due to the design of data collection systems. We develop a TunT (Transform-unTransform) estimator that turns the difficult problem of nonparametric hazard function estimation into a regression problem on binned and right-censored data. Our approach starts with binning event times and transforming event count data with a mean-matching transformation, which enables a simpler characterization of the heteroscedastic variance function. A nonparametric regression technique is then applied to the transformed data. Finally, the estimated regression function is back-transformed to yield an estimator for the original hazard function. The proposed estimation procedure is illustrated using call center data to reveal interesting customer patience behavior, and health insurance plan trial data to compare the effect between treatment and control groups. The numerical study shows that our approach yields more accurate estimates and better staffing decisions than existing methods.

Application of queueing models with abandonment for Call Center congestion analysis

Abstract This paper studies and applies queueing systems to Call Centers regarding the possibility of customer abandonment from the system before being served due to their impatience in waiting for a service. Call Centers are service organizations that predominantly serve customers via phone calls. One of the main concerns in managing them is to provide quality service at a minimum cost. Noticing the quality of services offered is expressed by customers, for example by abandonment from the queue. This paper shows that the M/M/c+G analytical queueing models with abandonment, with patience time represented by generic distributions (particularly mixed distributions), are more effective than the M/M/c+M analytical queueing models with abandonment, with Exponential patience, commonly used to evaluate congestion problems in Call Centers and support sizing and operational decisions in these systems. We conducted a study using data extracted from a Bank Call Center located in Israel and the parameters and some performance measures are determined based on this data. These sampling measures are compared with the same measures achieved by the M/M/c+M and M/M/c+G analytical queueing models considered in this research, which use parameters obtained empirically and the mixed and non-mixed distributions based on Exponential and Lognormal to represent user patience. An experimental discrete simulation model was also used to explore an alternative scenario, showing the potential of using the approaches based on analytical models with abandonment for Call Center analysis.