Impatient Customer Research Articles

Cost and revenue analysis of an impatient customer queue with second optional service and working vacations

In this article, we propose a finite buffer impatient customer queue with second optional service (SOS) and working vacations. When the server is busy, an arriving customer either joins the queue or balks on the basis of state-dependent joining/balking probabilities. For each customer, the server provides two phases of service, namely, first essential service (FES) and SOS. All the customers demand FES, whereas only few customers opt for SOS after the completion of FES. At a service completion instant, if the system is empty, the server leaves for working vacation. During working vacations, the waiting customers activate an impatient timer which is exponentially distributed. It is assumed that the interarrival times, vacation times, service times during FES, SOS and during working vacations follow exponential distribution. The steady-state probabilities of the model and various performance measures are derived. In order to optimize the total expected cost of the system, particle swarm optimization technique has been adopted for finding the optimum service rates of the server. Numerical results are sketched out to demonstrate the impact of the system and cost parameters.

Transient analysis of impatient customers in an M/M/1 disasters queue in random environment

Purpose This paper aims to consider a single server queue with system disasters and impatience behavior are evident in our daily life. For this purpose, authors require to know the general behavior of these systems. Transient analysis shows for us how the system will operate up to some time instant t. Design/methodology/approach In this paper, authors consider a single server queue with system disaster and impatient behavior of customers in a multi-phase random environment, in which the system transits to a repair state after each system disaster. When the system is in a failure phase or going through a repair phase, the new arrivals would be impatient. In case the system is not repaired before the customer’s time expires, the customer would leave the queue and never return. Moreover, after repair, the system becomes ready for service in an operative phase with probability $q_{i} \ge 0.$. Using generating functions along with continued fractions and some properties of the confluent hypergeometric function, authors obtained on their own results. Findings Explicit expressions have been obtained for the time-dependent probabilities of the underlying queuing model. Also, time-dependent mean and variance of customers in the system are deduced. Research limitations/implications The system authors are dealing with is somewhat complicated, there are some performance measures that cannot be achieved, but some of them have been obtained, such as the expectation and variance of the number of customers in the system. Practical implications Based on the obtained results, some numerical examples are some numerical examples are presented to illustrate the effect of various parameters on the behavior of the proposed system. Social implications Authors’ studied transient analysis of a single server queue with system disaster and impatient customer system is suitable for behavior interpretation of many systems in our lives, such as telecommunication networks, inventory systems and impatient telephone switchboard customers, manufacturing system and service system. Originality/value To the best of the author’s/authors’ knowledge and according to the literature survey, in a multi-phase random environment, no previous published article is presented for transient analysis of a single server queue with system disaster and impatient customer behavior in a random environment.

Analysis of a M/M/1 Queueing System with Two-Phase, N-Policy, Server Failure and Second Optional Batch Service with Customers impatient behaviour

In this paper an M/M/1 queueing system with server vacation, Startup, breakdowns and second optional service with impatient customer behavior is considered. All the customers will be given unique type of individual service in the first phase and then second phase service will be provided on option of the customer. We obtain some important system characteristics, such as the number of customers in the system, the probability that the server is idle, busy and broken down states, the expected waiting time in the system. Sensitivity analysis is also conducted with various parameters on system’s performance measures.

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.

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.

Performance prediction of a discrete-time batch arrival retrial queue with Bernoulli feedback

This paper examines a discrete-time GeoX/G/1 retrial queue wherein the customer is feedback again to the head of the queue with some probability in case when he is unsatisfied with his service. This phenomenon is called as Bernoulli feedback. We consider that if the server is busy at the arrival epoch, the arriving customer either interrupts the customer in service to receive his own service with probability pα or joins the orbit with probability pα¯=p(1−α) or departs from the system completely with probability p¯=1−p. A customer under the above defined three situations is called as either preferred customer or repeated customer or impatient customer, respectively. On the other hand if the incoming customer finds the server idle, he starts his service immediately. The service time and retrial time are general distributed whereas batch size is geometric distributed. We use generating function method to derive expressions for system size, orbit size and other performance measures. Further we provide the continuous time equivalent of our model. Finally numerical results are given to show the effect of some critical system parameters on various performance measures.

Optimal dynamic pricing problem considering patient and impatient customers’ purchasing behaviour

We divide customers into two types according to their purchasing behaviour: one is impatient and the other is patient. An impatient customer will be immediately lost if his reservation price of the product is below the current price, while a patient customer will wait until the price drops below his reservation price and buy one item. The seller’s objective is to maximise his expected discounted overall revenue by dynamically pricing. We prove that the optimal pricing policy has a control limit structure, examine the impact of the proportion parameter on the seller’s expected discounted overall revenue and show that the limiting behaviour of the optimal pricing policy when there are little patient customers is the same as the optimal static pricing policy. We also show that the seller will make more profit if he partially knows the number of waiting customers by providing subscription service to customers. A set of numerical results illustrates the value of dynamic pricing and information with respect to problem parameters such as the arriving probability and the subscription probability.

Impatient customer queue with Bernoulli schedule vacation interruption

This paper deals with an infinite buffer M/M/1 queue with working vacations and Bernoulli schedule vacation interruption wherein the customers balk with a probability. Whenever the system becomes empty, the server takes a working vacation during which service is provided with a lower rate and if there are customers at a service completion instant, vacation is interrupted and the server resumes a normal working period with probability q or continues the vacation with probability 1−q. The service times during working vacation and vacation times are assumed to be exponentially distributed. During a working vacation customers may renege due to impatience. The closed form expressions of the steady-state probabilities and the performance measures of the model are obtained using generating functions. Various numerical results are presented to show the effect of the model parameters on the system performance measures.

A burst loss probability model with impatient customer feature for optical burst switching networks

SummaryWe consider optical delay line buffer as a solution to reduce the number of lost burst in optical burst switching, one of the promising candidates for future networks. Such network takes burst loss as an important performance criteria in the design step. Network performance, however, cannot be captured efficiently using traditional queueing models, because they often ignore the impatience of messages traveling through optical switches which is one of the popular issues in communication networks. In this paper, we develop an analytic model for this system using queueing theory and considering special impatience features. Simulation results show that (i) the developed model with impatience features can decrease burst loss probability ( ≃ 10%) compared with other approaches, and (ii) applying that model, we demonstrate that shared buffer architecture in optical burst switching network with optical buffer often achieves lower burst loss probability than dedicated buffer way in several different scenarios. Copyright © 2014 John Wiley & Sons, Ltd.

Profit Optimization in SLA-Aware Cloud Services with a Finite Capacity Queuing Model

Cloud computing realizes a utility computing paradigm by offering shared resources through an Internet-based computing. However, how a system control can enhance profit and simultaneously satisfy the service level agreements (SLAs) has become one of the major interests for cloud providers. In this paper, a cloud server farm provided with finite capacity is modeled as anM/M/R/Kqueuing system. Revenue losses are estimated according to the system controls and impatient customer behaviors. Three important issues are solved in this paper. First, a profit function is developed in which both the system blocking loss and the user abandonment loss are evaluated in total revenue. A tradeoff between meeting system performances and reducing operating costs is conducted. Second, the effects of system capacity control and utilization on various performances of waiting time, loss probability, and final arrival rate are demonstrated. Finally, the proposed optimal profit control (OPC) policy allows a cloud provider to make the optimal decision in the number of servers and system capacity, so as to maximize profit. As compared to a system without applying the OPC policy, enhancing providers’ profit and improving system performances can be obtained.

Pricing and Optimal Liquidation of Immediacy

A liquidity provider is approached by an impatient customer wishing to trade a fixed number of shares immediately. In determining whether to enter into the trade, the liquidity provider must consider how to price the initial trade and what to do with the position once it forms part of their inventory.Using techniques from two distinct areas of Mathematical Finance, namely option pricing and portfolio optimisation, we develop a model which addresses this problem. We introduce formulae to price the initial trade with the impatient customer and then develop trading strategies which maximise the mean of overall expected profits for a fixed level of variance. We go on to show that there is a minimum length of time, Tα, that the liquidity provider expects to bear a large proportion of risk associated with buying a fixed number of shares and then selling them optimally over time, that is unaffected by market or trade inputs. By extending Tα, expected profits rise at the expense of the associated variance. We construct an efficient frontier of all the attainable trading strategies, which are unique in their choice of Tα, where the curve defining the frontier maps levels of variance to the corresponding maximum expected overall profit of the trade described above.

An M<SUP align=right>X</SUP>/G/1 retrial queue with two-phase service subject to active server breakdowns and two types of repair

This paper analyses a single server batch arrival retrial queue with active breakdowns, two types of repair and second optional service (SOS). The server provides preliminary first essential service (FES) to the primary arriving customers or customers from retrial group. The server is subject to breakdowns. The customer under service during the failure decides, with probability q, to join the orbit (impatient customer) and, with complementary probability 'p', to remain in the server for repair in order to conclude his remaining service (patient customer). It is considered that the repair time of server during the presence of patient customer and the repair time of the server while the customer (impatient customer) joining the orbit due to failure are different. On completion of FES successfully, the customer may opt for SOS with probability α. Reliability of the proposed model is analysed. Some particular cases are also discussed. Other performance measures are also obtained. The effects of several parameters on the system are analysed numerically.

Reliability indices of a Geo/G/1/1 Erlang loss system with active breakdowns under Bernoulli schedule

This paper deals with a discrete-time Erlang loss system, in which breakdowns occur randomly at any instant while the server is serving the customers. As soon as breakdown occurs, the server is sent to repair directly, the customer being served before server breakdown decides, with probability 1 - q, to depart the system (impatient customer) and, with complementary probability q(0 ≤ q ≤ 1), to wait for the server to complete his remaining service (patient customer). Additionally, Bernoulli vacation schedule is introduced into this model: after completion of each repair without interrupted customer waiting there or after completion of service, the server either goes for a vacation with probability θ(0 ≤ θ ≤ 1) or may wait for serving the next customer with complementary probability 1 - θ. Firstly, we obtain the z-transforms of the probabilities of server state by using a new type discrete supplementary variable technique. Secondly, we present some performance measures of this model, such as the steady-state availability, failure frequency of the system, mean time to the first failure and the probabilities when the server is idle, busy, down or on vacation. Finally, some numerical examples show the influence of the parameters on some crucial performance characteristics of the system.

OPTIMAL ORDERING POLICY FOR A DETERIORATING ITEM WITH IMPERFECT QUALITY AND PARTIAL BACKORDERING

ABSTRACT This study develops a production-inventory model considering a deteriorating item with imperfect quality and partial backordering. Such implicit assumptions are reasonable in view of the fact that poor-quality items do exist during production. Defective items are usually picked up during the screening process. Shortages are partially backordered since not all customers are willing to wait for new replenishments of stocks. The impatient customer will result in lost sale. The analysis shows that our model is a generalization of the models in current literatures. Also, a lower bound on the back-ordered ratio is obtained to ensure concave of the net expected profit function. An algorithm and numerical analysis are used to illustrate the solution procedure. Computational results indicate that our model leads to more realistic results.

Performance analysis of a telephone system with both patient and impatient customers

In this paper, the performance of a telephone system in which there are both patient and impatient customers is studied. Depending on the waiting time in the buffer, a customer may become impatient and therefore the call is incomplete. An impatient customer still consumes real processing time. The impact of impatient customers on the system is much more remarkable than people would expect. We formulate this model as a queueing system with a finite buffer. The status of a customer, i.e. whether patient or impatient, depends on the waiting time in the buffer and the service time is different for the different type of customers. The joint distribution of the numbers of patient and impatient customers in the system is obtained for the system in equilibrium. Expressions of many performance measures, including the average number of patient customers or impatient customers in the system, the average length of the service time, the average arrivals during a service period, and the proportion of the "useful" service time contributing towards patient customers' service, can be expressed in terms of the joint probabilities. The waiting time probabilities and the average waiting time of a customer in the buffer and the probability that a customer will be served as a patient customer are also obtained. Numerical examples are presented.

A Path Construction for the Virtual Waiting Time of an M/G/1 Queue

Consider the M/G/1 queue, the finite dam M/G/1 with capacity T, and the impatient customer M/G/1 model, where customers become lost customers if their waiting time exceeds τ. In this note we prove that for all three models and each xe(0, r) the distribution of the number of downcrossings of the virtual waiting time process with level x during a busy cycle is identical. This implies the weaker statement that on [0, T) the distribution functions of the steady state distributions of the amount of unprocessed work (virtual waiting time) are proportional. A number of applications is given.

Asymptotic Results for Buffer Systems under Heavy Load

This paper considers a dam (or storage) model of the GI/G/I type with a finite capacity K. An arriving input being larger than the unfilled capacity of the dam causes an overflow where the excess amount is lost. Important performance measures for this system are the overflow probability and the long-run fraction of input that is lost. We give asymptotic expansions for these measures for large K both for the case of a load factor less than 1 and for the case of a load factor larger than 1. Also, related results are obtained for the impatient customer model of the M/G/l type.

Use of discrete transforms for the study of aGI/M/S queue with impatient customer phenomena

The problem is to determine an explicit form for the stationary distribution of an imbedded Markov Chain in aGI/M/S queue with balking and reneging.

On a Class of Queuing Problems and Discrete Transforms

The main object of the paper is to demonstrate a method by which a number of problems arising in the study of impatient customers in queues can be solved. The main difficulty in analyzing such queuing systems is the presence of variable coefficients in the difference-differential equations describing the processes; the usual generating function method fails in such cases. When there is a finite number of states, it has been shown that at least formal solutions can be obtained through matrix algebra, which yields what are called “discrete transforms.” The results are explicitly given when the balking and reneging coefficients are linear in n. An alternative method in which the spectral resolution of matrices is used is also given for solution of the set of equations. In the special case of a M/G/1 queue with finite waiting room, both the methods fail (as the underlying matrix is in the Jordan form), and the solution is obtained by a simple manipulation of the underlying matrix. The methods presented here lend themselves readily to computer solutions. This paper demonstrates the generality of the basic M/G/1 model by relating it to problems arising in other branches of operations research, such as reliability theory, inventory control, etc. Finally, the methods used for the M/G/1 case are shown to hold for a similar class of problems in the GI/M/1 case. As very little work has been reported on impatient customer phenomena in GI/M/1, systems, possibly because of the difficulty in solving equations with variable coefficients, this method may prove to be useful.