Exponential Repair Time Research Articles

The effect of information on the strategic behavior in a Markovian queue with catastrophes and working vacations

ABSTRACT A stochastic clearing system is studied from a game-theoretic perspective in the paper where the server is subject to a Poisson-generated catastrophe and a follow-up repair process. Whenever a fatal shock (catastrophe) occurs, all customers are cleared from the system and the server fails. A repair is rendered immediately to fix the server with an exponential repair time. During the repair process, no customers are allowed to enter the system. Customers are strategic and they have the right to decide whether to join the system or balk based on a linear reward-cost structure with two types of rewards: A service reward for those customers that receive service and a compensation for those customers that are forced to abandon the system due to a catastrophe. During the service process, the server takes a working vacation after serving all the customers in the system. Our study is the first attempt to provide models to jointly characterize and analyse the queueing system with working vacation and catastrophes, with an emphasis on game-theoretic modeling of such a service system. The customer’s equilibrium strategy and social benefit of the system under four different information scenarios are obtained. In particular, we find that customers obey the follow-the-crowd (FTC) property in almost observable condition, which provides managerial insight on the operations management perspective. Numerical experiments are presented to show the effects of system parameters and information levels on the equilibrium joining behavior of customers.

System Maintenance Using Several Imperfect Repairs Before a Perfect Repair

Allowing several imperfect repairs before a perfect repair can lead to a highly reliable and efficient system by reducing repair time and repair cost. Assuming exponential lifetime and exponential repair time, we determine the optimal probability $p$ of choosing a perfect repair over an imperfect repair after each failure. Based on either the limiting availability or the limiting average repair cost per unit time, we determine the optimal number of imperfect repairs before conducting a perfect repair.

Cost analysis of 2-out-of-4 system connected to two-unit parallel supporting device for operation

This paper studied cost analysis of a hybrid series–parallel system consisting of two subsystems A and B. Subsystem A has two units in active parallel; while subsystem B consists of 2-out-of-4 units. Both units have exponential failure and repair time. Through transition rate table, system of first-order linear differential equations is developed and solved to obtain the explicit expressions for the steady-state availability and busy period probability of repairman due to partial and complete failure of both supporting devices, and the corresponding profit function is derived and examined. Furthermore, the results of the proposed model are beneficial for system engineers and designers, reliability and maintenance managers.

Equilibrium Customer Strategies in Markovian Queues with Setup Times and Partial Failures

This paper considers the equilibrium behavior of customers in a Markovian queue with setup times and partial failures, where the reactivated server must go through a period of setup time to reach the normal working state and a failure can occur at any time during the normal service. When a partial failure occurs, the server continues to serve the customers on spot at a low rate and does not admit a new arrival. Once the system becomes empty, an exponential repair time starts. Assuming that all the customers have the option of joining or balking based on a linear reward-cost structure, the authors analyze the equilibrium strategies of the customers and the average social benefits of the system in the fully observable case and the partially observable case, respectively. And on this basis, the effect of several parameters on customers’ strategic behavior is presented by some numerical examples.

Machine interference problem involving unsuccessful switchover for a group of repairable servers with vacations

The purpose of this paper is to explore the multiple-server machine interference problem with standby unsuccessful switchover and Bernoulli vacation schedule. Failure times of operating and standby machines are assumed to have exponential distributions and repair times of the failed machines and vacation times of servers are also assumed to have exponential distributions. After the completion of service, the server either goes for a vacation or may continue serving for the next machine. The vacation policy we considered is a single vacation policy. In practice, the switchover may experience a significant failure. The matrix analytical method and recursive method are employed to obtain the steady-state probability vectors, and closed-form expressions of some important system characteristics are obtained. The problem of cost optimization dealt with a number of numerical examples is provided by the Quasi-Newton method, the pattern search method, and the Nelder-Mead simplex direct search method. Expressions of various system characteristics are derived. Sensitivity analysis is performed numerically for system parameters. This paper presents the first time that machine interference problem with unsuccessful switchover for a group of repairable servers with vacations has been obtained, which is quite useful for the decision makers.

The effect of purchasing strategy on the economic production quantity model subject to random machine breakdown

In this article, we develop an economic production quantity (EPQ) model subject to process deterioration, machine breakdown and two types of repair, and study the influence of purchasing strategy on the system's expected total cost, as an alternative option to holding safety stock. It is obvious that the shortages may occur due to the prolonged repair time. To avoid this issue, we assume that the manufacturer could purchase some quantities from an available supplier with fixed non-zero lead-time. This paper addresses the following question: how much production lot size and purchasing lot size are required to minimise the expected total cost? The model is developed under general machine breakdown and repairs time distributions. The optimality of the model under exponential failure and exponential repair times are studied. For the numerical example, we illustrate the outcome of the proposed model and perform a sensitivity analysis on parameters, which directly influence optimal decisions.

The Availability of Systems with Bathtub Hazard Rate Function

In our normal life we can see that the most realistic systems possess useful time governed by hazard rateof bathtub shaped. The hazard rate function, however, plays a vital role in the computation of theavailability function. The repair time, however, could be modeled as any statistical distribution. In thispaper I will investigate the nature of availability function and points of availability of systems with bathtubhazard function and exponential distribution repair time using Markovian method.

Comparisons of site-clearing and non-site-clearing Markovian queues with server failures and unreliable repairer

This paper compares the customers’ balking behavior in site-clearing and non-site-clearing Markovian queues with server failures and unreliable repairer. That is, the system is subject to failures (catastrophes or normal failures) that occur when the server is at a functioning state, and it carries out the site-clearing mechanism if catastrophes occur while it carries out the non-site-clearing mechanism if normal failures occur. At a failure epoch, the server is turned off and all present customers are forced to leave the system in a site-clearing queue, whereas they start to experience a repair process in a non-site-clearing queue. After an exponential repair time, the server can be reactivated only with a probability p . Once the server fails to be repaired, all present customers leave the system, and meanwhile the next repair time begins. Comparing the equilibrium and optimal balking strategies of customers for the two types of queues, we find that although customers more possibly face the bad outcome that can not be served eventually, they still prefer to select a site-clearing queue but not a non-site-clearing queue, which coincides with the social planner’s preference. Moreover, the customers’ equilibrium behavior more deviates from their socially optimal behavior in the site-clearing queues.

Analysis of a GI/M/1 queue in a multi-phase service environment with disasters

In this paper, we study a single server GI/M/1 queue in a multi-phase service environment with disasters, where the disasters occur only when the server is busy serving customers. Whenever a disaster occurs in an operative service phase, all present customers are forced to leave the system simultaneously, the server abandons the service and an exponential repair time is set on. After the system is repaired, the server resumes his service and moves to service phase i immediately with probability q i ,i = 1,2, ... ,N . Using the matrix analytic approach and semi-Markov process, we obtain the stationary queue length distribution at both arrival and arbitrary epochs. After introducing tagged customers and the concept of a cycle, we also derive the sojourn time distribution, the duration of a cycle, and the length of the server’s working time in a service cycle. In addition, numerical examples are presented to illustrate the impact of some critical model parameters on performance measures.

Equilibrium strategic behavior of customers in the M/M/1 queue with partial failures and repairs

This paper analyzes the customers’ equilibrium strategic behavior in the M/M/1 queue subject to Poisson generated partial failures and repairs. In such a queueing system, whenever a partial failure occurs, the system does not admit any customer and continues the service at a lower rate instead of stopping working completely. This goes on till the system becomes empty and, at that point, an exponential repair time is sent on. The arriving customer’s dilemma is whether to enter or balk based on a linear reward-cost structure. We discuss the fully observable and fully unobservable queues respectively. For each type of queue, the corresponding equilibrium balking strategies of customers and the expected social benefits per time unit are derived. Finally, we illustrate the theoretical results by presenting some numerical examples.

The GI/M/1 queue in a multi-phase service environment with disasters and working breakdowns

ABSTRACTIn this paper, we consider a GI/M/1 queue in a multi-phase service environment with disasters and working breakdowns. When the server is working in any normal service phases, it may suffer disastrous interruptions, causing all present customers to leave the system. At an exponential failure instant, the system becomes defective, and goes directly to repair phase. During the repair period, instead of stopping service completely, the system is equipped with a substitute server which continues to provide service to the arriving customers. After an exponential repair time, the substitute server stops service and the system moves to normal operative phase i with probability . Using the matrix analytic approach and semi-Markov process, we obtain the stationary queue length distribution at both arrival and arbitrary epochs. We also provide the elaborate analysis of some performance measures and sojourn time distribution of an arbitrary customer. In addition, some numerical examples are presented.

Analysis of the M/G/1 queue in multi-phase random environment with disasters

This paper studies an M/G/1 queue in a multi-phase random environment. When in operative phase i, i=1,2,…,n, the system is subject to disastrous interruptions, causing all present customers (waiting and served) to leave the system. At an exponential failure instant, the server abandons the service and the system goes directly to repair phase. After an exponential repair time, the system moves to operative phase i with probability qi, i=1,2,…,n. Using the supplementary variable technique, we obtain the distribution for the stationary queue at an arbitrary epoch. We also derive results of the cycle analysis, the sojourn time distribution and the length of the server's working time in a service cycle. In addition, some special cases and numerical examples are presented.

Transient analysis of a machine repair system with standby, two modes of failure, discouragement and switching failure

The present study is concerned with a machine repair problem with mixed standbys, permanent and additional repairmen. To deal with the realistic situation, the concepts of balking and reneging are also included. The operating units as well as standby units are assumed to fail in two modes. The units have exponential life time and repair time distributions. It is also assumed that there is a probability of a switching failure of the standby units to the operating state. The transient state probabilities are obtained by Runge-Kutta method. Some performance indices such as expected number of failed units, expected number of standby units, machine availability, etc., are obtained. To determine the expected total cost per unit time, a cost function has been facilitated. The numerical experiment is carried out to validate the analytical results of the developed model. The sensitivity analysis is also performed to explore the effect of different parameters on the performance indices.

Availability of a repairable retrial system with warm standby components

In this paper, we study a repairable K-out-of-(M+W) retrial system with M identical primary components, W standby components and one repair facility. The time-to-failure and time-to-repair of the primary and standby components are assumed to be exponential and general distributions, respectively. The failed components are immediately for repair if the server is idle, otherwise the failed machines would enter an orbit. It is assumed that the retrial times are exponentially distributed. We present a recursive method using the supplementary variable technique and treating the supplementary variable as the remaining repair time to obtain the steady-state probabilities of down components at arbitrary epoch. Then, a unified and efficient algorithm is developed to compute the steady-state availability. The method is illustrated analytically for the exponential repair time distribution. Sensitivity analysis of the steady-state availability with respect to system parameters for a variety of repair time distributions is also investigated.

The effect of catastrophes on the strategic customer behavior in queueing systems

We consider the single server Markovian queue subject to Poisson generated catastrophes. Whenever a catastrophe occurs, all customers are forced to abandon the system, the server is rendered inoperative and an exponential repair time is set on. During the repair time new arrivals are allowed to join the system. We assume that the arriving customers decide whether to join the system or balk, based on a natural linear reward-cost structure with two types of rewards: A (usual) service reward for those customers that receive service and a (compensation) failure reward for those customers that are forced to abandon the system due to a catastrophe. We study the strategic behavior of the customers regarding balking and derive the corresponding (Nash) equilibrium strategies for the observable and unobservable cases. We show that both types of strategic behavior may be optimal: to avoid the crowd or to follow it. The crucial factor that determines the type of customer behavior is the relative value of the service reward to the failure compensation. © 2013 Wiley Periodicals, Inc. Naval Research Logistics, 2013

Optimal and equilibrium balking strategies in the single server Markovian queue with catastrophes

We consider a Markovian queue subject to Poisson generated catastrophes. Whenever a catastrophe occurs, all customers are forced to abandon the system, the server is rendered inoperative and an exponential repair time is set on. We assume that the arriving customers decide whether to join the system or balk, based on a natural reward-cost structure. We study the balking behavior of the customers and derive the corresponding Nash equilibrium and social optimal strategies.

Queueing system with a constant retrial rate, non-reliable server and threshold-based recovery

In this paper, we examine a Markovian queueing system which is composed of a retrial queue with constant retrial rate and a non-reliable server. Upon arrival a customer occupies the server if it is idle, otherwise it goes to the retrial queue. The customer at the head of the retrial queue is allowed to retry for service. When the server is busy, the server is subject to breakdowns or failures that occur according to a Poisson process. In the failed state the server can be repaired at a repair facility with exponential repair time with respect to the threshold policy: the repair starts when the number of customers in the system reaches some prespecified threshold level q ⩾ 1. We perform a steady-state analysis of the corresponding continuous-time Markov chain, derive mean performance characteristics and waiting time distribution as well as calculate optimal threshold level to minimize the long-run average losses for the given cost structure.

Development and integration of a simulation-based repair and overhaul execution strategy in ERP systems

We consider the production control of a defence repair and overhaul organisation where planned remanufacturing demands at the end of the equipment expected life and unplanned demands triggered by a major equipment failure are performed with the same resources. The difficulty of controlling this type of operation within an integrated logistics system resides in the variable nature of the remanufacturing process. A suboptimal control policy described by inventory thresholds triggering the use of different execution modes has already been formulated in previous research to address this problem under exponential repair time distribution. The aim of this study is to generalise this control policy and to propose its implementation through a functional architecture integrating optimisation and simulation techniques with Enterprise Resource Planning (ERP) and Manufacturing Execution Systems (MES).

Instantaneous availability assessment of renewable component in exponential distributions

A renewable component with exponential failure and repair times is considered, and its instantaneous availability at time t is denoted by A(t). This paper proposes two methods for constructing lower confidence limits of A(t) with Chebyshev inequation and generalized p-value approach

Optimal design of unreliable production–inventory systems with variable production rate

In this article, we study an economic manufacturing quantity (EMQ) problem for an unreliable production facility where the production rate is treated as a decision variable. As the stress condition of the machine changes with the production rate, the failure rate of the machine is assumed to be dependent on the production rate. The unit production cost is also taken as a function of the production rate, as the machine can be operated at different production rates resulting in different unit production costs. The basic EMQ model is formulated under general failure and general repair time distributions and the optimal production policy is derived for specific failure and repair time distributions viz., exponential failure and exponential repair time distributions. Considering randomness of the time to machine failure and corrective repair time, the model is extended to the case where certain safety stocks in inventory may be useful to improve service level to customers. Optimal production policies of the proposed models are derived numerically and the sensitivity of the optimal results with respect to those parameters which directly influence the machine failure and repair rates is also examined.