Impatient Customers Research Articles

Analysis of queues in a random environment with impatient customers

We study M/M/c queues (c = 1, 1 < c < ∞ and c = ∞) in a Markovian environment with impatient customers. The arrivals and service rates are modulated by the underlying continuous-time Markov chain. When the external environment operates in phase 2, customers become impatient. We focus our attention on the explicit expressions of the performance measures. For each case of c, the corresponding probability generating function and mean queue size are obtained. Several special cases are studied and numerical experiments are presented.

Equilibrium behaviour and social optimization in Markovian queues with impatient customers and variant of working vacations

We study customers’ equilibrium behaviour and social optimization in a single-server Markovian queue with impatient customers and variant of multiple working vacations, where the impatience is due to slow service rate. Under the variant of multiple working vacations, the server takes a working vacation as soon as the system gets empty. When an arriving customer joins the vacation system, it activates an impatience timer. If its patience timer expires before it gets service, the customer abandons the system, and never returns. The server is allowed to take at most J successive working vacations, if at the end of a working vacation the system remains empty. An arriving customer takes a decision on the basis of available information whether to join or to balk, which unifies wish for the service as well as reluctance to wait. We discuss equilibrium threshold strategies on the basis of linear reward-cost structure in the fully observable and fully unobservable cases. We present numerical results that establish the impact of the information level as well as various parameters on the equilibrium balking strategies and social benefits. The research outputs may be useful for decision makers to convey information to customers in net benefit maximization and for examining the corresponding social optimization problems.

A Queueing System with Heterogeneous Impatient Customers and Consumable Additional Items

AbstractA single-server queueing system with a marked Markovian arrival process of heterogeneous customers is considered. Type-1 customers have limited preemptive priority over type-2 customers. There is an infinite buffer for type-2 customers and no buffer for type-1 customers. There is also a finite buffer (stock) for consumable additional items (semi-products, half-stocks, etc.) which arrive according to the Markovian arrival process. Service of a customer requires a fixed number of consumable additional items depending on the type of the customer. The service time has a phase-type distribution depending on the type of the customer. Customers in the buffer are impatient and may leave the system without service after an exponentially distributed amount of waiting time. Aiming to minimize the loss probability of type-1 customers and maximize throughput of the system, a threshold strategy of admission to service of type-2 customers is offered. Service of type-2 customer can start only if the server is idle and the number of consumable additional items in the stock exceeds the fixed threshold. Stationary distributions of the system states and the waiting time are computed. In the numerical example, we show some interesting effects and illustrate a possibility of application of the presented results for solution of optimization problems.

Analysis of a finite-capacity system with working breakdowns and retention of impatient customers

In this paper, we consider a finite-capacity Markovian queueing system with working breakdowns, reneging, and retention of impatient customers, in which the server is subject to breakdowns and repairs while serving a customer. In a working breakdown queue, the server works at a lower service rate rather than stopping service during the breakdown period. An arriving customer finding the server busy must wait in the queue and may leave the system without being served with probability θ or remain in the queue with probability 1−θ within a period of time. We numerically investigate the transient behavior of the queueing model based on the fourth-order Runge–Kutta method. A matrix method is used to compute the steady-state probabilities explicitly. We develop system performance measures for transient and steady states. Numerical examples are given to examine the effects of various system parameters on the system performance measures. A cost optimization problem is formulated to find the optimal service rates, which minimize the expected cost per unit time. Finally, we present an example illustrating the application of the queueing model in an order picking system.

Coordinating Supply and Demand on an On-Demand Service Platform with Impatient Customers

We consider an on-demand service platform using earning-sensitive independent providers with heterogeneous reservation price (for work participation) to serve its time and price-sensitive customers with heterogeneous valuation of the service. As such, the supply and demand are “endogenously” dependent on the price the platform charges its customers and the wage the platform pays its independent providers. We present an analytical model with endogenous supply (number of participating agents) and endogenous demand (customer request rate) to study this on-demand service platform. To coordinate endogenous demand with endogenous supply, we include the steady-state waiting time performance based on a queueing model in the customer utility function to characterize the optimal price and wage rates that maximize the profit of the platform. We first analyze a base model that uses a fixed payout ratio (i.e., the ratio of wage over price), and then extend our model to allow the platform to adopt a time-based payout ratio. We find that it is optimal for the platform to charge a higher price when demand increases; however, the optimal price is not necessarily monotonic when the provider capacity or the waiting cost increases. Furthermore, the platform should offer a higher payout ratio as demand increases, capacity decreases or customers become more sensitive to waiting time. We also find that the platform should lower its payout ratio as it grows with the number of providers and customer demand increasing at about the same rate. We use a set of actual data from a large on-demand ride-hailing platform to calibrate our model parameters in numerical experiments to illustrate some of our main insights.

Linking Delay Announcements, Abandonment, and Staffing: A Behavioral Perspective

Using field data from a bank's call center, we study the behavioral determinants of customers' queue abandonment decisions in the presence of delay announcements. We relax traditional behavioral assumptions on how customers should make abandonment decisions and empirically explore how customers actually behave. Our flexible model can account for multiple delay announcements, reference-dependent behavior, sunk costs in time, and heterogeneity in customer patience. We show that customers exhibit loss aversion with respect to time losses in queue, becoming much more likely to abandon if forced to wait longer than the announced waiting time. We also find that customers may fall for the sunk cost effect, becoming less likely to abandon with more time sunk waiting in queue. Additionally, we find that longer announced wait times increase customer abandonment behavior and that customers are often strategic in their arrival decisions, with impatient customers avoiding the busy time of day. In a counterfactual study, we show that, by accounting for our behavioral insights, firms may significantly reduce their staffing levels without increasing customers' overall waiting time.

Performance Analysis of M/M(a,b)/1 Queuing Model with Balking and State-dependent Reneging and Service

The article investigates a M/M[Formula: see text]/1 queuing model with impatient customers. The size of a batch taken up for service depends on the number of customers present in the waiting line. The server serves a minimum of “ a” and a maximum of “ b” customers at a time. The service time is assumed to be exponentially distributed with mean dependent on the batch size. Customers arrive to the system as a Poisson process, and may leave on finding a long queue or may renege after waiting in the queue for an exponentially distributed time. The model is analyzed to find different measures of effectiveness of the system. A new measure of performance is also discussed. The approach adopted for analysis of the model is based on embedded Markov chain.

Markov Models of Queueing–Inventory Systems with Variable Order Size

Markov models of queueing–inventory systems with variable order size are investigated. Two classes of models are considered: with instant service and with nonzero service time. The model with nonzero service time assumes that impatient customers can form a queue of either finite or infinite length. The exact and approximate methods are developed to calculate the characteristics of the systems under proposed restocking policy.

An M/G/1 retrial G-queue with optional re-service, impatient customers, multiple working vacations and vacation interruption

In this work, we consider a single server retrial queueing system with multiple working vacations and vacation interruption, where the regular busy server is subjected to breakdown due to the arrival of negative customers. After the completion of regular service the positive customers may demand re-service of the previous service without joining the orbit or may leave the system. When the orbit becomes empty at service completion instant for a positive customer; the server goes for multiple working vacations. The steady state probability generating function for the system size and orbit size are obtained by using the method of supplementary variable technique. Some important system performance measures and the mean busy period are obtained. The conditional stochastic decomposition law is shown for good for this model. Finally, the effects of various parameters on the system performance are analysed numerically.

An M/G/1 retrial G-queue with optional re-service, impatient customers, multiple working vacations and vacation interruption

In this work, we consider a single server retrial queueing system with multiple working vacations and vacation interruption, where the regular busy server is subjected to breakdown due to the arrival of negative customers. After the completion of regular service the positive customers may demand re-service of the previous service without joining the orbit or may leave the system. When the orbit becomes empty at service completion instant for a positive customer; the server goes for multiple working vacations. The steady state probability generating function for the system size and orbit size are obtained by using the method of supplementary variable technique. Some important system performance measures and the mean busy period are obtained. The conditional stochastic decomposition law is shown for good for this model. Finally, the effects of various parameters on the system performance are analysed numerically.

Analysis of performance measures to handle medical E-commerce shopping cart abandonment in cloud

Abstract The E-commerce zone is crowded with many Internet users. Medical E-commerce has had significant growth in part because of a great deal of growth in the Indian E-commerce field. Medical E-commerce sites use cloud computing to guarantee a high quality of service anywhere and anytime in the world. For online access, the customer's expectations are very high. Medical E-commerce retailers are directed towards cloud service providers based on their quality of service. During online shopping, impatient customers may abandon a specific medical E-commerce shopping cart due to slow response. This is quite difficult to endure for a medical E-commerce firm. The research described herein observed the effect of shopping cart abandonment on medical E-commerce websites deployed in cloud computing. The impact of the idle virtual machine on customer impatience during medical E-commerce shopping was also studied. The ultimate aim of this study was to propose a stochastic queueing model and to yield results through probability generating functions. The results of the model may be highly useful for a medical E-commerce firm facing customer impatience, so as to design its service system to offer satisfactory quality of service.

Analysis of a Constant Retrial Queue with Joining Strategy and Impatient Retrial Customers

This paper treats an M/M/1 retrial queue with constant retrial times. If the server is busy at the arrival epoch, the arriving customer decides to join the retrial orbit with probability q or balk with probability 1-q. Only the customer at the head of the orbit is permitted to access the server. Upon retrial, the customer immediately receives his service if the server is idle; otherwise, he may enter the orbit again or leave the system because of impatience. First, we give the performance analysis for this retrial queue and give some important performance indices. Second, based on a natural reward-cost structure, we analyze the Nash equilibrium customers’ joining strategies and give some numerical examples.

A perishable inventory-queueing model with delayed vacation, negative and impatient customers

A perishable inventory-queueing model with delayed vacation, negative and impatient customers

Transient solution of an M / M /1 vacation queue with a waiting server and impatient customers

Abstract Recently, Ammar [1] has discussed the transient behavior of a multiple vacations queue with impatient customers. In this paper, a similar technique is used to derive a new elegant explicit solution for an M/M/1 vacation queue with impatient customers and a waiting server, where the server is allowed to take a vacation whenever the system is empty after waiting for a random period of time. If the server does not return from the vacation before the expiry of the customer impatience time, the customer abandons the system forever. Moreover, the formulas of mean and variance expressed in terms of the obtained possibilities for this model.

Analysis of variant working vacations queue with customer impatience

This paper analyses an M / M / 1 queueing system with a variant of working vacations wherein an arriving customer may balk. The service times during regular busy periods, working vacation periods and vacation times are exponentially distributed and are mutually independent. In working vacation periods, the customer may renege due to impatience and the impatient timer follows an exponential distribution. We derive the probability generating function of the steady-state probabilities and obtain the closed form expressions of the system size. In addition, some performance measures and the sojourn time of a tagged customer are derived.

A computational algorithm for the loss probability in the M/G/1+PH queue

ABSTRACTThis article develops a computational algorithm for the loss probability in the stationary M/G/1 queue with impatient customers whose impatience times follow a phase-type distribution (M/G/1+PH). The algorithm outputs the loss probability, along with an upper-bound of its numerical error due to truncation, and it is readily applicable to the M/D/1+PH, M/PH/1+PH, and M/Pareto/1+PH queues.

Dynamic Scheduling in a Many-Server Multi-Class System: The Role of Customer Impatience in Large Systems

We study optimal scheduling of customers in service systems, such as call centers. In such systems, customers typically hang up and abandon the system after waiting for a long time for their service to commence. Such abandonments are detrimental for the system, and so managers typically use scheduling as a tool to mitigate it. In this paper we study the interplay between customer impatience and scheduling decisions when managing heterogeneous customer classes. Specifically, our focus is on general patience distributions, in which a customer's instantaneous propensity to abandon may change over time. We find that incorporating these temporal changes into the scheduling decisions can lead to a significant improvement in the system performance. Formally, we solve the underlying Diffusion Control Problem for these systems, and characterize near-optimal scheduling policies. One of our main results is that for a class of parameters, we establish sufficient conditions for both the optimality and non-optimality of threshold policies, which are widely used due to their ease of implementation.

A Unified Approach to Diffusion Analysis of Queues with General Patience-Time Distributions

We propose a unified approach to establishing diffusion approximations for queues with impatient customers within a general framework of scaling customer patience time. The approach consists of two steps. The first step is to show that the diffusion-scaled abandonment process is asymptotically close to a function of the diffusion-scaled queue length process under appropriate conditions. The second step is to construct a continuous mapping not only to characterize the system dynamics using the system primitives, but also to help verify the conditions needed in the first step. The diffusion approximations can then be obtained by applying the continuous mapping theorem. The approach has two advantages: (i) it provides a unified procedure to establish the diffusion approximations regardless of the structure of the queueing model or the type of patience-time scaling; and (ii) it makes the diffusion analysis of queues with customer abandonment essentially the same as the diffusion analysis of queues without customer abandonment. We demonstrate the application of this approach via the single-server system with Markov-modulated service speeds in the traditional heavy-traffic regime and the many-server system in the Halfin-Whitt regime and the nondegenerate slowdown regime.

Optimal Price/Lead-Time Menus for Queues with Customer Choice: Segmentation, Pooling, and Strategic Delay

How should a firm design a price/lead-time menu and scheduling policy to maximize revenues from heterogeneous time-sensitive customers with private information about their preferences? We consider a queueing system with multiple customer types that differ in their valuations for instant delivery and their delay costs. The distinctive feature of our model is that the ranking of customer preferences depends on lead times: patient customers are willing to pay more than impatient customers for long lead times, and vice versa for speedier service. We provide necessary and sufficient conditions, in terms of the capacity, the market size, and the properties of the valuation-delay cost distribution, for three features of the optimal menu and segmentation: pricing out the middle of the delay cost spectrum while serving both ends, pooling types with different delay costs into a single class, and strategic delay to deliberately inflate lead times. This paper was accepted by Assaf Zeevi, stochastic models and simulation.

Mx/G/1 retrial vacation queue for multi-optional services, phase repair and reneging

The present paper deals with a batch arrival general retrial queue with multi-optional services, vacation and impatient customers undergoing reneging. The incoming customer (or a batch of customers) on finding the server in busy or in non-working condition forms a virtual pool of customers called orbit otherwise undergoes service if the server is free. The service is provided at two levels; first essential service or second optional service depending upon customer’s choice to opt for it or not. Moreover, the server may go for vacation if he finds no customer waiting for service and returns back to the service center again as soon as the customer approaches for the service. The server is unreliable as such subject to breakdown during the service; as soon as the server fails, it is immediately sent for repair so as to restore its functionality as before failure. The repairman assigned for the repair of the server also takes some setup time before commencing the repair process. By using supplementary variables technique and probability generating function method, the explicit expressions for the queue size distribution and other performance measures have been derived. The sensitivity of different parameters on various performance indices has been examined numerically by taking an illustration. Also, the neuro fuzzy approach has been used to approximate the analytical results.