Impatient Customers Research Articles

Fluid approximation analysis of a call center model with time-varying arrivals and after-call work

Important features to be included in queueing-theoretic models of the call center operation are multiple servers, impatient customers, time-varying arrival process, and operator’s after-call work (ACW). We propose a fluid approximation technique for the queueing model with these features by extending the analysis of a similar model without ACW recently developed by Liu and Whitt (2012). Our model assumes that the service for each quantum of fluid consists of a sequence of two stages, the first stage for the conversation with a customer and the second stage for the ACW. When the duration of each stage has exponential, hyperexponential or hypo-exponential distribution, we derive the time-dependent behavior of the content of fluid in each stage of service as well as that in the waiting room. Numerical examples are shown to illustrate the system performance for the cases in which the input rate and/or the number of servers vary in sinusoidal fashion as well as in adaptive ways and in stationary cases.

Transient analysis of an M/M/1 queue with impatient behavior and multiple vacations

In this paper, we carry out an analysis for a single server queue with impatient customers and multiple vacations where customers impatience is due to an absentee of servers upon arrival. Customers arrive at the system according to a Poisson process and exponential service times. Explicit expressions are obtained for the time dependent probabilities, mean and variance of the system size in terms of the modified Bessel functions, by employing the generating functions along with continued fractions and the properties of the confluent hypergeometric function. Finally, some numerical illustrations are provided.

Strategic Pricing of Horizontally Differentiated Services with Switching Costs: A Pricing Model for Cloud Computing

ABSTRACTThe pricing of cloud computing services is a major challenge because the industry is in its infancy and firms employ pricing models developed for conventional electronic commerce. In this study, we offer a new approach to cloud services pricing that considers customer behavior. First, we propose a base model centered around a profit-maximizing duopolistic market serving both rational and time-inconsistent customers. Our results reveal that firms can profit from impatient customers. In addition, we extend the base model with the effect of delayed network externalities because typically information goods strongly exhibit this property. For the latter case, we show that the effect of network externalities reduces the impact of low switching costs and firms benefit from time-inconsistent behavior. This study contributes to theories of pricing information goods. Furthermore, practitioners who make pricing decisions for cloud computing services can benefit from our findings.

Staffing to Maximize Profit for Call Centers with Impatient and Repeat-Calling Customers

Motivated by call center practice, we study the optimal staffing of many-server queues with impatient and repeat-calling customers. A call center is modeled as an M/M/s+M queue, which is developed to a behavioral queuing model in which customers come and go based on their satisfaction with waiting time. We explicitly take into account customer repeat behavior, which implies that satisfied customers might return and have an impact on the arrival rate. Optimality is defined as the number of agents that maximize revenues net of staffing costs, and we account for the characteristic that revenues are a direct function of staffing. Finally, we use numerical experiments to make certain comparisons with traditional models that do not consider customer repeat behavior. Furthermore, we indicate how managers might allocate staffing optimally with various customer behavior mechanisms.

On renewal input state dependent working vacations queue with impatient customers

This paper studies the effect of balking and reneging in a finite buffer renewal input queue with multiple working vacations. Arriving customers balk with a probability and renege according to exponential distribution. As soon as the system becomes empty, the server leaves for a working vacation. During working vacations the server works at a different rate rather than completely stopping service. Service times during a service period, during a working vacation period and vacation times are assumed to be exponentially distributed with state dependent rates. Employing the supplementary variable and recursive techniques, the steady-state system length distributions at pre-arrival and arbitrary epochs have been derived. Some performance measures of the model and cost analysis through simulated annealing have been discussed. Finally, sensitivity analysis is carried out through some numerical experiments.

A single server queue with Markov modulated service rates and impatient customers

We consider a single server queue in which the customers wait for service for a fixed time and leave the system if the service has not begun within that time. The customers arrive according to a Poisson process and each arriving customer brings in a certain amount of phase-type distributed work. The service rate of a server varies according to the underlying continuous time Markov process with finite states. We construct a Markov process by using the age process and then obtain the stationary distribution of the Markov process. By using the results of the stationary distribution of the Markov process, we obtain the loss probability, the waiting time distribution and the system size distribution.

The truncated normal distribution: Applications to queues with impatient customers

Motivated by heavy traffic approximations for single server queues with abandonment, we provide an exact expression for the moments of the truncated normal distribution using Stein’s lemma. Consequently, our moment expressions provide insight into the steady state skewness and kurtosis dynamics of single server queues with impatient customers. Moreover, based on the truncated normal distribution, we develop a new approximation for single server queues with abandonment in the nonstationary setting. Numerical examples illustrate that our approximation performs quite well.

One for one period policy for perishable inventory

Recently, for zero ordering cost a new ordering policy named (1, T), in which the time interval between two consecutive orders and the value of the order size are both constant, have been developed for nonperishable products. In this paper, the (1, T) policy is developed for perishable products. Using an analogy among this inventory model, a queueing model with impatient customers, and a finite dam model, the long-run average total cost function of the inventory system is derived. It is observed that the total cost rate is independent from the lead time as is for nonperishable products. Since analyzing the convexity of the model is extremely complicated, a proposition is proved to define a domain for the optimal solution, and then a search algorithm is presented to obtain the optimal solution. Furthermore, a numerical analysis is carried out to examine the sensitivity of optimal T with respect to system parameters and to compare the performance of (1, T) policy with the well known (S−1, S) policy. This analysis shows that for fixed values of system parameters, there is a fixed value of lead time for which the performance of (1, T) policy is better than (S−1, S) policy. Further as the lead time increases this superiority is more pronounced.

The M/M/1/K interdependent retrial queueing model with impatient customers and controllable arrival rates

In this paper an M/M/1/K interdependent retrial queueing model with controllable arrival rates and impatient customers is considered. The steady state solutions and the system characteristics are derived for this model. The analytical results are numerically illustrated and the effect of the nodal parameters on the system characteristics are studied and relevant conclusion is presented. Keywords: finite capacity, interdependent primary arrival and service processes, impatient customers, retrial queue, single server.

On a doubly dynamically controlled supermarket model with impatient customers

In this paper, we provide a key generalization of the supermarket model both from the impatient customers and from a doubly dynamic control, which may also be related to the size-based scheduling through the centered management of the customer resource as well as the total service ability. We first use an infinite-dimensional Markov process to express the states of this supermarket model, and set up an infinite-dimensional system of differential equations satisfied by the expected fraction vector. Then we use the operator semigroup to provide a mean-field limit for the sequence of infinite-dimensional Markov processes, which asymptotically approaches a single trajectory identified by the unique and global solution to the infinite-dimensional system of limiting differential equations. Finally, we provide an effective and efficient algorithm for computing the fixed point of the infinite-dimensional system of limiting differential equations, and use the fixed point to give performance analysis of this supermarket model. Also, some numerical examples are given to demonstrate how the performance measures depend on some crucial parameters of this supermarket model. We believe that the mean-field method developed in this paper will be useful and effective for analyzing more complicated supermarket models in many practical areas.

Analysis of Impatient Customers in Queues with Bernoulli Schedule Working Vacations and Vacation Interruption

This paper analyzes customers’ impatience in Markovian queueing system with multiple working vacations and Bernoulli schedule vacation interruption, where customers’ impatience is due to the servers’ vacation. During the working vacation period, if there are customers in the queue, the vacation can be interrupted at a service completion instant and the server begins a regular busy period with probability 1-q or continues the vacation with probability q. We obtain the probability generating functions of the stationary state probabilities and deduce the explicit expressions of the system sizes when the server is in a normal service period and in a Bernoulli schedule vacation interruption, respectively. Various performance measures such as the mean system size, the proportion of customers served, the rate of abandonment due to impatience, and the mean sojourn time of a customer served are derived. We obtain the stochastic decomposition structures of the queue length and waiting time. Finally, some numerical results to show the impact of model parameters on performance measures of the system are presented.

Customer Differentiation in Ancillary Services? Free Service, Prioritization, and Strategic Delay

A service provider/retailer offers ancillary service (e.g., shipping, cloud computing) to two types of customers, impatient and patient, who may be heterogeneous both in their delay sensitivities and service valuations. She can use prioritization and strategic delay to differentiate them. She offers at most two service classes and can price them so that a customer type splits between them. Her objective is to minimize cost while satisfying individual rationality and incentive compatibility conditions. We consider the retailer's problem and characterize the optimal solution when capacity is both exogenous and endogenous. We show that (i) the retailer prioritizes whenever two service classes are offered and (ii) splitting of impatient customers can be optimal. We also find when free service provision is optimal; it is due to two reasons: better customer differentiation and low capacity. We study numerically how optimal service delivery and cost change with different parameters, and derive more insights.

Perishable Inventory System at Service Facilities with Multiple Server Vacations and Impatient Customers

This article presents a perishable inventory system under continuous review at a service facility in which a waiting area for customers is of finite size M. We assume that the replenishment of inventory is instantaneous. The items of inventory have exponential life times. It is assumed that demand for the commodity is of unit size. The arrivals of customers to the service station fo rm a Poisson process. The server goes for a vacation of an exponentially distributed duration whenever the waiting area is zero. If th e server finds the customer level is zero when he returns to the system, he immediately takes another vacation. The individual customer is issued a demanded item after a random service time, which is distributed as negative exponential. Also the waiting customer independently reneges the system after an exponentially distributed amount of time. The joint probability distribution of the number customers in the system and the inventory levels is obtained in steady state case. Some measures of system performance in the steady state are derived and the total expected cost is also considered.

Routing and Staffing in Customer Service Chat Systems with Impatient Customers

We consider customer service chat (CSC) systems where customers can receive real time service from agents using an instant messaging (IM) application over the Internet. A unique feature of these systems is that agents can serve multiple customers simultaneously. The number of customers that an agent is serving determines the rate at which each customer assigned to that agent receives service. We consider the staffing problem in CSC systems with impatient customers where the objective is to minimize the number of agents while providing a certain service level. The service level is measured in terms of the proportion of customers who abandon the system in the long run. First we propose effective routing policies based on a static planning LP, both for the cases when the arrival rate is observable and for when the rate is unobservable. We show that these routing policies minimize the proportion of abandoning customers in the long run asymptotically for large systems. We also prove that the staffing solution obtained from a staffing LP, when used with the proposed routing policies, is asymptotically optimal. We illustrate the effectiveness of our solution procedure in systems with small to large sizes via numerical and simulation experiments.

Reducing session establishment delay using timed out packets in SIP signaling network

SummarySession Initiation Protocol (SIP) has a retransmission mechanism to maintain the reliability for its real time transmission. But these real time transmissions cause overload in the server and creates redundant messages. SIP does not offer sufficient mechanisms for handling overload situations. In this paper, we study the SIP system behavior by separating signaling traffic in two different classes (Invites and Non‐invites) and creating a cut‐off priority queueing model. SIP retransmission mechanism with timeout is modeled as a queueing system with impatient customers. Using this model, the effect of unnecessary retransmissions is studied, and delay distribution and loss probability (Pb_loss) are calculated. The proposed analytical model is verified with simulations that demonstrate that the inclusion of timeout gives better delay performance. Using Pb_loss, an algorithm is developed to control the overload in hop‐by‐hop transaction as described in RFC 6357. The simulation results for the proposed model with overload control and the standard SIP as per RFC 3261 are compared. The results demonstrate that the server utilization factor of an overloaded server always remains less than one and hence avoids system collapse. Further, the redundant messages in the system are reduced by 30% as compared with the standard SIP network. Copyright © 2014 John Wiley & Sons, Ltd.

Unreliable bulk retrial queues with delayed repairs and modified vacation policy

The present investigation deals with the bulk arrival M/G/1 retrial queue with impatient customers and modified vacation policy. The incoming customers join the virtual pool of customers called orbit if they find the server being busy, on vacation or in broken down state otherwise the service of the customer at the head of the batch is started by the server. The service is provided in k essential phases to all the customers by the single server which may breakdown while rendering service to the customers. The broken down server is sent to a repair facility wherein the repair is performed in d compulsory phases. As soon as the orbit becomes empty, the server goes for vacation and takes at most J vacations until at least one customer is noticed. The incoming customers are impatient and may renege on seeing a long queue of the customers for the service. The probability generating functions and queue length for the number of customers in the orbit and queue have been obtained using supplementary variable technique. Various system characteristics viz. average number of customers in the queue and the orbit, long run probabilities of the system states, etc. are obtained. Furthermore, numerical simulation has been carried out to study the sensitivity of various parameters on the system performance measures by taking an illustration.

Optimal control of service parameter for a perishable inventory system maintained at service facility with impatient customers

This paper deals with the problem of optimally controlling the service rates for the perishable inventory system at service facilities with impatient customers. We consider a finite capacity inventory system with Poisson demands and exponentially distributed service times. The maximum inventory level is fixed at S. The ordering policy is (s, Q) policy, that is, whenever the inventory level drops to s, an order for Q(= S − s) items is placed. The ordered items are received after a random time which is also distributed as exponential. The waiting customer independently reneges the system after an exponentially distributed amount of time. The items in the inventory have shelf life times that are assumed to follow an exponential distribution. Here we determine the service rates to be employed at each instant of time so that the long run total expected cost rate is minimized. The problem is modelled as a semi-Markov decision problem. The stationary optimal policy is computed using linear programming algorithm and the results are illustrated numerically.

On multiple priority multi-server queues with impatience

We consider Markovian multi-server queues with two types of impatient customers: high- and low-priority ones. The first type of customer has a non-preemptive priority over the other type. After entering the queue, a customer will wait a random length of time for service to begin. If service has not begun by this time he or she will abandon and be lost. We consider two cases where the discipline of service within each customer type is first-come first-served (FCFS) or last-come first-served (LCFS). For each type of customer, we focus on various performance measures related to queueing delays: unconditional waiting times, and conditional waiting times given service and given abandonment. The analysis we develop holds also for a priority queue with mixed policies, that is, FCFS for the first type and LCFS for the second one, and vice versa. We explicitly derive the Laplace–Stieltjes transforms of the defined random variables. In addition we show how to extend the analysis to more than two customer types. Finally we compare FCFS and LCFS and gain insights through numerical experiments.

A newsvendor inventory model with an emergency order to supply a non-increasing fraction of shortage

This paper generalizes the newsvendor inventory model when the possibility of an emergency order to satisfy the excess of demand exists. In this situation, we assume that there are impatient customers who do not wait for the emergency order and other customers who are willing to wait to satisfy their demand. We consider that the fraction of shortage that is satisfied with delay through the emergency order is described by a function, which is non-increasing with respect to the amount of shortage. Our objective is to determine the optimal order quantity, which maximizes the relevant expected total profit for the period, when the demand follows a uniform probability distribution. As is well known, this distribution is usually used to denote the results of a demand whose values are unknown, except for the fact that such results belong to an interval. The uniqueness and existence of optimal decisions are proved, and a procedure to determine the optimal policy and the maximum expected profit is developed. Illustrative examples, which help us to understand the theoretical results, are also given.

Paying for Express Checkout: Competition and Price Discrimination in Multi-Server Queuing Systems

We model competition between two firms selling identical goods to customers who arrive in the market stochastically. Shoppers choose where to purchase based upon both price and the time cost associated with waiting for service. One seller provides two separate queues, each with its own server, while the other seller has a single queue and server. We explore the market impact of the multi-server seller engaging in waiting cost-based-price discrimination by charging a premium for express checkout. Specifically, we analyze this situation computationally and through the use of controlled laboratory experiments. We find that this form of price discrimination is harmful to sellers and beneficial to consumers. When the two-queue seller offers express checkout for impatient customers, the single queue seller focuses on the patient shoppers thereby driving down prices and profits while increasing consumer surplus.