Impatient Customers Research Articles

Optimum System Capacity for Markovian Queueing System with an Adaptive Balking

An M/M/1queueing system with finite system capacity and the customer impatience in the form of random balking is considered in this paper. Assuming an adaptive balking according to which a new customer will join the queue with probability qn if he finds n (n > 0) customers in the system on his arrival, we consider two models based on the balking nature. First we consider a model with all the arriving customers balk with constant probability. In the second model, balking probabilities of the arriving customer increases as the number of customers in the system increases. We obtain closed form expression for the stationary distribution of the system size and proposed a cost model to determine the system capacity that minimizes the cost function and give some numerical illustrations.

Analysis of a working vacations queue with impatient customers operating under a triadic policy

The objective of this paper is to analyze a finite buffer M/M/2 working vacations queue with balking and reneging wherein the servers operate under a triadic (0,Q,N,M) policy. As soon as the system becomes empty, both the servers leave for working vacations with only one of the two servers being active during the vacation. The service times during the working vacation and the vacation times are assumed to be exponentially distributed. Various performance measures are discussed and cost analysis is carried out using the quadratic fit search method. Finally, using some numerical results we present the parameter effect on the performance measures of the model.

A repairable Geo X /G/1 retrial queue with Bernoulli feedback and impatient customers

This paper deals with a discrete-time batch arrival retrial queue with the server subject to starting failures. Different from standard batch arrival retrial queues with starting failures, we assume that each customer after service either immediately returns to the orbit for another service with probability θ or leaves the system forever with probability 1 − θ (0 ≤ θ < 1). On the other hand, if the server is started unsuccessfully by a customer (external or repeated), the server is sent to repair immediately and the customer either joins the orbit with probability q or leaves the system forever with probability 1 − q (0 ≤ q < 1). Firstly, we introduce an embedded Markov chain and obtain the necessary and sufficient condition for ergodicity of this embedded Markov chain. Secondly, we derive the steady-state joint distribution of the server state and the number of customers in the system/orbit at arbitrary time. We also derive a stochastic decomposition law. In the special case of individual arrivals, we develop recursive formulae for calculating the steady-state distribution of the orbit size. Besides, we investigate the relation between our discrete-time system and its continuous counterpart. Finally, some numerical examples show the influence of the parameters on the mean orbit size.

A batch arrival [formula omitted] queue with impatient customers

This paper considers a multi-server batch arrival MX/M/c queue with impatient customers. We give an exact expression for the loss probability, which can be expressed in terms of the waiting time distribution in the standard MX/M/c queue with no impatience. This solves the problem conjectured by Boots and Tijms (1999).

Effects of state-dependent balking on multi-server non-stationary queueing systems

A combination of discrete-time modelling and theoretical analysis is used to develop an easy-to-use Normal approximation for the time-dependent behaviour of multi-server queueing systems subject to state-dependent balking. Key findings that underpin the approximation are that time lags between peaks in arrival rates and congestion levels can be ignored; queue behaviour is insensitive to distribution of service time beyond its mean; and distribution of number in the system is near Normal. This model is then used to derive valuable management implications for service-orientated systems where delays are frequent and abandonments are an important feature. Key insights are that such systems can often adopt very ‘sub-optimal’ behaviour, and that customer impatience can be very beneficial where management wishes to improve system performance. ‘Optimal’ performance is defined, and practical measures for queue managers to move system performance towards ‘optimality’ are identified.

Diffusion Approximations for G/M/n + GI Queues with State-Dependent Service Rates

We consider a sequence of many-server queueing systems with impatient customers of the type G/M/n + GI in heavy traffic. This sequence is indexed by n, where the parameter n represents the number of servers in the nth system. The state process is considered to be the diffusion-scaled total customer count in the system and the service rate is a state-dependent perturbation of a given basic service rate μ0 &gt; 0. When the system is critically loaded in the Halfin-Whitt heavy traffic regime, we obtain the limiting diffusion for the state processes. We also establish the asymptotic relationships among the diffusion-scaled processes representing the total customer count, virtual waiting time, and the number of customer abandonments. Motivated by the cost structures of telephone call centers, we formulate a cost functional and show that the expected value of this cost functional in the nth system converges to that of the limiting diffusion under mild assumptions.

ANALYSIS OF A QUEUEING MODEL FOR A CALL CENTER WITH IMPATIENT CUSTOMERS AND AFTER-CALL WORK

Two essential features needed for queueing models of call centers are and after-call work (ACW) of operators. Their effects on the performance are comparable to those of service times. We pro- pose and analyze a queueing model of a call center with impatient customers and ACW. We consider a two-dimensional birth-and-death process for the state combining the number of calls waiting or being served in the system and the number of operators working on ACW. The steady-state distribution is ob- tained numerically, from which we compute a variety of performance measures, including the blocking probability, the probability of wait, the mean waiting time, the ratio of getting service and abandonment, and the fraction of opera- tors who are serving customers, working on ACW, or being idle at an arbitrary time. Furthermore, we show a method of finding the mean waiting time for those customers who get served and for those who abandon while waiting. Our analysis is validated by numerical examples for a call center model of realistic size, namely 40 operators and 30 incoming telephone lines.

An improved truncation technique to analyze a Geo/PH/1 retrial queue with impatient customers

This paper considers a discrete-time retrial queue with impatient customers. We establish the global balance equations of the Markov chain describing the system evolution and prove that this queueing system is stable as long as the customers are strict impatient and the mean retrial time is finite. Direct truncation with matrix decomposition is used to approximate the steady-state distribution of the system state and hence derive a set of performance measures. The proposed matrix decomposition scheme is presented in a general form which is applicable to any finite Markov chain of the GI/M/1-type. It represents a generalization of the Gaver–Jacobs–Latouche's algorithm that deals with QBD process. Different sets of numerical results are presented to test the efficiency of this technique compared to the generalized truncation one. Moreover, an emphasis is put on the effect of impatience on the main performance measures.

Transient and steady-state analysis of queueing systems with catastrophes and impatient customers

Sudhesh (2010) has discussed the transient probabilities for queueing systems subject to catastrophic failures and impatience of customers. However, it contains some errors concerning the terminology and the final forms of the transient probabilities of the system size. In this paper, we correct the errors and obtain the correct results in the work carried out by Sudhesh. In addition, the steady state system size probabilities and performance measures are provided. Finally, two special cases are deduced from the more general queueing model under study.

Transient solution of a multi-server queue with catastrophes and impatient customers when system is down

We consider an M/M/c queuing system, which occasionally suffers disastrous failure and all the customers are lost. The repair mechanism starts immediately. When the system is down, the stream of arrivals continue. However the new arrivals become impatient and activate their own timer. If the system is still down when the timer expires, the customer abandons the system never to return. We obtain the exact transient solution for the system by a direct approach. Mathematics Subject Classification: Primary 60K25, Secondary 68M20

A Markovian inventory system with retrial and impatient customers

We consider an inventory system in which the customers who arrive during the stock out period join an orbit of infinite size. The customers in the orbit are assumed not only to retry for their demand, may renege from the orbit at random time. This situation is modelled in this work by assuming Poisson arrivals for customers, exponential lead time for orders placed under (s, S) ordering policy, exponential time between successive attempts of retrial of a customer and exponential times for reneging by customers in the orbit. The condition for ergodicity of the system which is modelled as Markov processes is obtained. The joint probability distribution of the number of customers in the system and the inventory level is obtained in steady state case. The measures of system performance in the steady state are derived and the long run total expected cost rate is derived. The results are illustrated with numerical examples.

Analysis of an M/M/cQueueing System with Impatient Customers and Synchronous Vacations

We consider an M/M/cqueueing system with impatient customers and a synchronous vacation policy, where customer impatience is due to the servers’ vacation. Whenever a system becomes empty, all the servers take a vacation. If the system is still empty, when the vacation ends, all the servers take another vacation; otherwise, they return to serve the queue. We develop the balance equations for the steady-state probabilities and solve the equations by using the probability generating function method. We obtain explicit expressions of some important performance measures by means of the two indexes. Based on these, we obtain some results about limiting behavior for some performance measures. We derive closed-form expressions of some important performance measures for two special cases. Finally, some numerical results are also presented.

A single-server Markovian queuing system with discouraged arrivals and retention of reneged customers

Customer impatience has a very negative impact on the queuing system under investigation. If we talk from business point of view, the firms lose their potential customers due to customer impatience, which affects their business as a whole. If the firms employ certain customer retention strategies, then there are chances that a certain fraction of impatient customers can be retained in the queuing system. A reneged customer may be convinced to stay in the queuing system for his further service with some probability, say q and he may abandon the queue without receiving the service with a probability p(=1? q). A finite waiting space Markovian single-server queuing model with discouraged arrivals, reneging and retention of reneged customers is studied. The steady state solution of the model is derived iteratively. The measures of effectiveness of the queuing model are also obtained. Some important queuing models are derived as special cases of this model.

Optimizing Service Rate and the Capacity of an M/M/1/N Queuing System with Retention of Reneged Customers

In this paper, we present analysis of a single-server finite capacity Markovian queuing system with reneging and retention of impatient customers. It is assumed that arriving customers renege according to an exponential distribution and the reneged customers may be retained with some probability. The cost model of the system is made. The optimal values of service rate and the system capacity are obtained by using search and classical optimization algorithms. Further, the expected system size is compared with and without the optimization of the system. Some important comparisons are done to show the impact of optimization of parameters like system capacity, mean service rate, and the probability of retaining the reneged customers.

Integrated defective production model for impatient customers with price dependent demand under the effect of learning

In this paper, a three-echelon supply chain model has been explored, comprising of a supplier, a producer and a retailer where demand rate is functionally related to the selling price. In this study, it is assumed that customers are impatient in nature so shortages occur at the end of retailer. Production process assumes to be imperfect and defective items reworked and available for sale as good ones. Number of defective items present in supplier end follows the learning curve. The model has been explored numerically as well, an optimal solution has been obtained and the sensitivity of that solution has also been checked with respect to various system parameters. The final outcome shows that the model is not only stable but also economically feasible.

Markovian queue optimisation analysis with an unreliable server subject to working breakdowns and impatient customers

This study investigates an infinite capacity Markovian queue with a single unreliable service station, in which the customers may balk (do not enter) and renege (leave the queue after entering). The unreliable service station can be working breakdowns even if no customers are in the system. The matrix-analytic method is used to compute the steady-state probabilities for the number of customers, rate matrix and stability condition in the system. The single-objective model for cost and bi-objective model for cost and expected waiting time are derived in the system to fit in with practical applications. The particle swarm optimisation algorithm is implemented to find the optimal combinations of parameters in the pursuit of minimum cost. Two different approaches are used to identify the Pareto optimal set and compared: the epsilon-constraint method and non-dominate sorting genetic algorithm. Compared results allow using the traditional optimisation approach epsilon-constraint method, which is computationally faster and permits a direct sensitivity analysis of the solution under constraint or parameter perturbation. The Pareto front and non-dominated solutions set are obtained and illustrated. The decision makers can use these to improve their decision-making quality.

Transient behavior of / /1 [ ] M G X Queueing Model With State Dependent Arrival, Balking and Bernoulli Vacation

This paper analyzes single server queueing system wherein the arrival units are in batches following compound Poisson process with varying arrival rates for dierent states. The arrival rates dier for dierent states like idle state, busy state and vacation period. The server provides service one by one and after completion of a service, the server either takes a vacation of random length with probability p (0 p 1) or continue to serve the next customer with probability 1-p, if any. Both service time and vacation time follow general distribution. In addition to this one of the customer impatience, balking is also discussed. We have obtained state dependent solution for the model the steady state analysis.

Analysis of an M/M/1 queueing system with impatient customers and a variant of multiple vacation policy

In this paper, we consider an M/M/1 queueing system with impatient customers and a variant of multiple vacation policy, where we examine the case that customer impatience is due to the servers' vacation. Whenever a system becomes empty, the server takes a vacation. However, the server is allowed to take a maximum number $K$ of vacations if the system remains empty after the end of a vacation. This vacation policy includes both a single vacation and multiple vacations as special cases. We derive the probability generating functions of the steady-state probabilities and obtain the closed-form expressions of the system sizes when the server is in different states. We further make comparisons between the mean system sizes under the variant vacation policy and the mean system sizes under the single vacation policy or the multiple vacation policy. In addition, we obtain the closed-form expressions for other important performance measures and discuss their monotonicity with respect $K$. Finally, we present some numerical results to show the effects of some parameters on some performance measures.

Introduction: queueing systems special issue on queueing systems with abandonments

The investigation of queueing systems with impatient customers has a long history in queueing theory, with the now standard Erlang-A model going back at least to Palm’s classic 1957 paper. Despite this long record of work, focus on systems with abandonments has been rekindled due to recent interest in analyzing complex service systems, most notably large call centers. Interestingly, this new surge of work has been motivated both by applied and theoretical advances. In the applied realm, very large call centers are now an important component in many industrial sectors and advanced call management software enables the use of sophisticated algorithms in operating these centers. On the more theoretical side, there have been great advances in approximating large queueing systems with abandonments via many-server asymptotics and related diffusion approximations. Furthermore, there have been numerous advances in the understanding of flexible server models with abandonments, which serve as models for systems with “cross-trained” agents. Techniques from revenue management, game theory, and stochastic optimization have also been applied to the operation of such systems. In addition, queueing systems with impatient customers also have a rich connection with perishable inventory and organ transplantation systems. Thus the requirements of practitioners, and the myriad new tools of theoreticians, have motivated this focused issue on stochastic systems with abandonments. This special issue actually consists of nine papers, eight of which appear together here. Due to a publishing error, one of the papers, by Baris Ata and Mustafa H.

Using Estimated Patience Levels to Optimally Schedule Customers

Customer impatience has become an integral component of analyzing services, especially in the context of call centers. Typically, when customers arrive to such systems, they seem identical or homogeneous, however, from the system's perspective, as they wait in the queue, their residual willingness to wait changes. For instance, a customer who has already waited for 10 minutes may have a different residual willingness to wait as compared with a customer who has only waited for 1 minute. In this manner, as time progresses, customers become differentiated on their estimated patience levels. We exploit this dimension of customer heterogeneity to construct scheduling policies in overloaded systems that dynamically prioritize customers based on their time in queue in order to optimize any given system performance metric. Interestingly, the optimal policy has a very simple structure, and we find that implementing it can lead to significant improvements over the first come first serve policy.