Impatience Time Research Articles

Multi-server Markovian heterogeneous arrivals queue with two kinds of working vacations and impatient customers

This paper deals with multi-server queueing system with two kinds of Working Vacations (WVs) and impatient customers. A random timer is started whenever a customer comes into the system. The customer may abandon the system if the service is not completed before the impatience timer expires. Each time after serving all the customers, the system becomes empty and then the server begins 1st kind of vacation. On returning from 1st kind of WV, the server begins 2nd kind of WV whenever a system has no customers. When the server comes back from either 1st kind or 2nd kind of WV, if there is at least one customer in the system, the server switches to busy period. The steady state probabilities have been derived using the Probability Generating Functions (PGFs) method. Various measures of performance are presented and numerical illustrations are also provided.

The Markovian Bernoulli queues with operational server vacation, Bernoulli's weak and strong disasters, and linear impatient customers

This paper studies the stationary analysis of a Markovian queuing system with Bernoulli feedback, interruption vacation, linear impatient customers, strong and weak disaster with the server's repair during the server's operational vacation period. Each customer has its own impatience time and abandons the system as soon as that time ends. When the queue is not empty, the server's operational vacation can be interrupted if the service is completed and the server starts a busy period with a probability q or continues the operational vacation with a probability q. A strong disaster forces simultaneously all present customers (waiting and served) to abandon the system permanently with a probability p but a weak disaster is that all customers decide to be patient by staying in the system, and wait during the repair time with a probability p, where arrival of a new customer can occur. As soon as the repair process of the server is completed, the server remains providing service in the operational vacation period. We analyze this proposed model and derive the probabilities generating functions of the number of customers present in the system together with explicit expressions of some performance measures such as the mean and the variance of the number of customers in the different states, together with the mean sojourn time. Finally, numerical results are presented to show the influence of the system parameters on some studied performance measures.

Asymptotic Diffusion Analysis of Retrial Queueing System M/M/1 with Impatient Customers, Collisions and Unreliable Servers

In this paper, a retrial queueing system of the M/M/1 type with Poisson flows of arrivals, impatient customers, collisions, and an unreliable service device is considered. To make the problem more realistic and, hence, more complicated, we include the breakdowns and repairs of the service in this research study. The retrial times of customers in the orbit, service time, impatience time of customers in the orbit, server’s lifetime (depending on whether it is idle or busy), and server recovery time are supposed to be exponentially distributed. The problem of finding the stationary probability distribution of the number of customers in orbit is solved by using the method of asymptotic diffusion analyses under the condition of a heavy load of the system and the patience of customers in orbit. Numerical results are presented that demonstrate the effectiveness of the obtained theoretical conclusions, and a comparative analysis of the method of asymptotic analysis and the method of asymptotic diffusion analysis for the considered problem is given.

Transient solution of an M/M/∞ queue with system's additional tasks and impatient customers

This paper studies the impatient behaviour in an infinite server queue with additional tasks assigned to the system. Whenever the system becomes empty, the system as a whole is assigned a secondary task of duration U whose distribution is exponential. Any arrival during the period U becomes impatient due to the unavailability of service facility. Each individual waiting customer activates an independent impatience timer of duration T which is exponentially distributed. When the system comes back after the completion of U, before T expires, the waiting customers are simultaneously taken for service and they leave the system after the completion of service. If T expires before the completion of task U, the customers abandon the system and never to return. The transient system size probabilities of this model are derived explicitly for both single and multiple task cases. The time-dependent mean and variance of system size are also derived. Further, numerical simulations are also presented to analyse the effect of system indices.

The MX/M/c\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$M^{X}/M/c$$\\end{document} Bernoulli feedback queue with variant multiple working vacations and impatient customers: performance and economic analysis

The present paper deals with an MX/M/c\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$M^{X}/M/c$$\\end{document} Bernoulli feedback queueing system with variant multiple working vacations and impatience timers which depend on the states of the servers. Whenever a customer arrives at the system, he activates an random impatience timer. If his service has not been completed before his impatience timer expires, the customer may abandon the system. Using certain customer retention mechanism, the impatient customer can be retained in the system. After getting incomplete or unsatisfactory service, with some probability, each customer may comeback to the system as a Bernoulli feedback. Using the probability generating functions (PGFs), we derive the steady-state solution of the model. Then, we obtain useful performance measures. Moreover, we carry out an economic analysis. Finally, numerical study is performed to explore the effects of the model parameters on the behavior of the system.

The M/PH/1+D queue with Markov-renewal service interruptions and its application to delayed mobile data offloading

Motivated by delayed mobile data offloading in cellular networks, we consider an M/PH/1 queue with deterministic impatience times (i.e., the M/PH/1+D queue), where the server takes ON and OFF states alternately and does not serve any customers in the OFF states. We assume that the state of the server is governed by an alternating Markov renewal process. For this queue, we analyze various performance measures in steady state. Most of the previous studies on the analysis of delayed mobile data offloading assume that service times, ON- and OFF-period lengths, and impatience times are exponentially distributed, and the effect of their distributions on the queueing performance has not been investigated well. We thus discuss the effects of the variability in the service time distribution, the variability in the ON-period distribution, and the correlation between the ON- and OFF-periods, and we show that those factors, especially the variability in service times, has a great impact on the queueing performance. Furthermore, to demonstrate the modeling capability of Markov-renewal service interruptions, we illustrate such models as the mobility of smartphone users in a two-dimensional area is incorporated, and we show how the queueing performance is affected by the geographical distribution of Wi-Fi access points.

Analysis of an M/M/c queue with heterogeneous servers, balking and reneging

This paper analyses a heterogeneous multi-server queuing system with balking and reneging. In this system, when all the c servers are busy, an arriving customer decides either to join the queue with probability p or balk with probability 1 - p. The customers waiting in the queue become impatient due to the long wait for service. Therefore, each individual waiting customer activates an independent impatience timer such that the customer's service starts before his timer expires, he gets the service and leaves the system after the completion of service. Otherwise, he abandons the system and never returns. The time-dependent system size probabilities are derived explicitly using generating function. Further the time-dependent mean, variance, busy period distribution and steady-state probabilities are obtained. Finally, some numerical illustrations are presented.

Transient solution of an M/M/? queue with system's additional tasks and impatient customers

This paper studies the impatient behaviour in an infinite server queue with additional tasks assigned to the system. Whenever the system becomes empty, the system as a whole is assigned a secondary task of duration U whose distribution is exponential. Any arrival during the period U becomes impatient due to the unavailability of service facility. Each individual waiting customer activates an independent impatience timer of duration T which is exponentially distributed. When the system comes back after the completion of U, before T expires, the waiting customers are simultaneously taken for service and they leave the system after the completion of service. If T expires before the completion of task U, the customers abandon the system and never to return. The transient system size probabilities of this model are derived explicitly for both single and multiple task cases. The time-dependent mean and variance of system size are also derived. Further, numerical simulations are also presented to analyse the effect of system indices.

Performance and economic analysis of Markovian Bernoulli feedback queueing system with vacations, waiting server and impatient customers

Abstract This paper concerns the analysis of a Markovian queueing system with Bernoulli feedback, single vacation, waiting server and impatient customers. We suppose that whenever the system is empty the sever waits for a random amount of time before he leaves for a vacation. Moreover, the customer’s impatience timer depends on the states of the server. If the customer’s service has not been completed before the impatience timer expires, the customer leaves the system, and via certain mechanism, impatient customer may be retained in the system. We obtain explicit expressions for the steady-state probabilities of the queueing model, using the probability generating function (PGF). Further, we obtain some important performance measures of the system and formulate a cost model. Finally, an extensive numerical study is illustrated.

The intergenerational transmission of obesity: The role of time preferences and self-control

Previous research has found that impatient time preferences and self-control problems (present bias) are related to increased obesity risk. However, scant evidence exists pertaining to whether parents’ impatience and self-control problems impact the obesity status of their children, too. Accordingly, we explore this study question among a large national sample of US adults and their children. Study results confirm previous findings indicating that intertemporal preferences are related to adults’ obesity status. Moreover, these results extend the literature by finding that children of impatient or present-biased parents have a significantly higher likelihood of being obese, too. Specifically, parents’ low levels of patience and present bias were each independently related to a five-percentage point increase in the likelihood of obesity of their children. These findings were more pronounced when all children were combined in analyses and for the first child; however, they varied for the second and third child. Thus, findings suggest that parents’ time preferences and self-control problems likely affect not only their own weight status but that of their children.

An M/PH/K queue with constant impatient time

This paper is concerned with an M/PH/K queue with customer abandonment, constant impatient time, and many servers. By combining the method developed in Choi et al. (Math Oper Res 29:309–325, 2004) and Kim and Kim (Perform Eval 83–84:1–15, 2015) and the state space reduction method introduced in Ramaswami (Stoch Models 1:393–417, 1985), the paper develops an efficient algorithm for computing performance measures for the queueing system of interest. The paper shows a number of properties associated with matrices used in the development of the algorithm, which make it possible for the algorithm, under certain conditions, to handle systems with up to one hundred servers. The paper also obtains analytical properties of performance measures that are useful in gaining insight into the queueing system of interest.

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.

Computational analysis of multi-server discrete-time queueing system with balking, reneging and synchronous vacations

This paper proposes a discrete-time multi-server queue with multiple synchronous vacations under balking and reneging. Arriving customers decide whether to join the system or balk on the basis of some state-dependent joining/balking probabilities, and renege according to a geometric distribution when servers are busy. The servers take a vacation together if there are no customers in the system at a service completion instant. When the servers are on vacation, an arriving customer activates an impatience timer which is geometrically distributed. The inter-arrival times, service times and vacation times are assumed to be independent and geometrically distributed. We obtain closed-form expressions and develop a computational algorithm for calculating the steady-state probabilities. Specifically, we establish the application of the proposed framework in analyzing a multi-server queueing system with synchronous vacation under balking and reneging. Applications of such models can be found in a wide variety of real-time systems including call centers, computer and communication systems, cloud computing, quality control and maintenance in industrial establishments. We develop a cost model to determine the optimal service rate. Various performance measures and numerical examples are sketched out to demonstrate the impact of the proposed method. Some special cases of the model have also been discussed. Finally, we show that in the limiting case the results converge to the corresponding continuous-time counterparts.

Train rescheduling model with train delay and passenger impatience time in urban subway network

SummaryThis paper considers the train rescheduling problem with train delay in urban subway network. With the objective of minimizing the negative effect of train delay to passengers, which is quantified with a weighted combination of travel time cost and the cost of giving up the planned trips, train rescheduling model is proposed to jointly synchronize both train delay operation constraints and passenger behavior choices. Space–time network is proposed to describe passenger schedule‐based path choices and obtain the shortest travel times. Impatience time is defined to describe the intolerance of passengers to train delay. By comparing the increased travel time due to train delay with the passenger impatience time, a binary variable is defined to represent whether the passenger will give up their planned trips or not. The proposed train rescheduling model is implemented using genetic algorithm, and the model effectiveness is further examined through numerical experiments of real‐world urban subway train timetabling test. Duration effects of the train delay to the optimization results are analyzed. Copyright © 2017 John Wiley & Sons, Ltd.

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.

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.

Multi-class M/PH/1 queues with deterministic impatience times

ABSTRACTThis article considers computational procedures for the waiting time and queue length distributions in stationary multi-class first-come, first-served single-server queues with deterministic impatience times. There are several classes of customers, which are distinguished by deterministic impatience times (i.e., maximum allowable waiting times). We assume that customers in each class arrive according to an independent Poisson process and a single server serves customers on a first-come, first-served basis. Service times of customers in each class are independent and identically distributed according to a phase-type distribution that may differ for different classes. We first consider the stationary distribution of the virtual waiting time and then derive numerically feasible formulas for the actual waiting time distribution and loss probability. We also analyze the joint queue length distribution and provide an algorithmic procedure for computing the probability mass function of the stationary joint queue length.

Analysis and computation of the stationary distribution in a special class of Markov chains of level-dependent M/G/1-type and its application to BMAP/M/ $$\infty $$ ∞ and BMAP/M/c+M queues

This paper considers a special class of continuous-time Markov chains of level-dependent M/G/1-type, where block matrices representing downward jumps in the infinitesimal generator are nonsingular. This special class naturally arises in the analysis of BMAP/M/$$\infty $$ź queues and BMAP/M/c queues with exponential impatience times (BMAP/M/c+M). We first formulate the boundary probability vector in terms of a solution of a system of infinitely many linear inequalities. We then reveal that in the above special class, this infinite system is regarded as a nested sequence of simplices, and we identify their vertices. Based on these results, we develop a simple yet efficient computational algorithm for the stationary distribution conditioned that the level is not greater than a predefined N. Note that for a large N, the conditional distribution will provide a good approximation to the stationary distribution. Some numerical examples for BMAP/M/$$\infty $$ź and BMAP/M/c+M queues are shown.

Algorithmic computation of steady-state probabilities in an almost observable GI/M/c queue with or without vacations under state dependent balking and reneging

We consider an almost observable GI/M/c/N queue with customer impatience with or without multiple synchronous vacations under state-dependent balking. Upon arriving, a customer joins or refuses to join the system based on certain state dependent joining/balking probabilities. Once an arriving customer joins the system and finds all the servers busy, it initiates an exponentially distributed impatience timer with random duration T. An equilibrium balking strategy is discussed for constant balking as a special case of state-dependent balking. In the constant balking case, the waiting time of a customer in a queueing system has been associated with a linear cost-reward structure for estimating the net benefit if a customer chooses to participate in the system. A steady-state system of equations is obtained using the supplementary variable technique. After that, a recursive algorithm is proposed to obtain the stationary system-length distribution at pre-arrival and arbitrary epochs using those steady-state equations, from which the mean system sojourn time, the average reneging rate and the average balking rate are derived. Finally, we produce numerical results relating to system performance and its net benefit when investigated for different model parameters. The proposed model has applications in the modeling of balking and impatient behavior of incoming calls in a call center, multi-core computing, multi-path routing in delay sensitive communications networks.

The M/D/1+D queue has the minimum loss probability among M/G/1+G queues

We consider the loss probability Ploss in the stationary M/G/1 queue with generally distributed impatience times (M/G/1+G queue). Recently, it was shown that Ploss increases with service times in the convex order. In this paper, we show that Ploss also increases with impatience times in the excess wealth order. With these results, we show that Ploss in the M/D/1+D queue is smallest among all M/G/1+G queues with the same and finite arrival rate, mean service time, and mean impatience time.