The establishment of the time interval between inspections for a cold standby system with component repair

The time interval between inspections of cold standby systems is a crucial decision to ensure the appropriated system reliability and the lowest costs possible. This paper presents a model developed to establish the optimal time interval between inspections for a two-unit cold standby system with component repair and subject to periodic inspection, considering reliability and costs. A Markov chain is used to define possible states, their transition probabilities and the mean time to system failure, as a function of the time interval between inspections. Given the mean time to system failure, the steady state availability is determined. Finally, the costs related to the system maintenance are established and a cost function is developed and optimized for the time interval between inspections. Numerical examples are presented and results for different system parameters are compared. Besides optimizing the time interval between inspections, the analyses also reveal the effect of repair time on system availability and mean time to system failure

A survey of the machine interference problem

This paper surveys the research published on the machine interference problem since the 1985 review by Stecke & Aronson. After introducing the basic model, we discuss the literature along several dimensions. We then note how research has evolved since the 1985 review, including a trend towards the modelling of stochastic (rather than deterministic) systems and the corresponding use of more advanced queuing methods for analysis. We conclude with some suggestions for areas holding particular promise for future studies.Natural Sciences and Engineering Research Council (NSERC) Discovery Grant 238294-200

Optimisation of inspection policy for multi-line production systems

This paper develops a simulation model to determine the cost-optimum inspection policy for a multi-line production system taking account of simultaneous downtime. The machines in the multi-line system are subject to a two stage failure process that is modelled using the delay-time concept. Our study indicates that: consecutive inspection of lines with priority for failure repair is cost-optimal, with a cost reduction of 61% compared to a ‘run-to-failure’ policy; and maintainers need to be responsive to operational requirements. Our ideas are developed in the context of a case study of a plant with three parallel lines, one of which is on cold-standby. Keywords: maintenance; delay-time model; simulation; production; parallel lines; manufacturing; preventive maintenance

A study on the analysis of two-unit redundant repairable complex systems

Two well-known methods of improving the reliability of a system are (i) provision of redundant units, and (ii) repair maintenance. In a redundant system more units made available for performing the system function when fewer are required actually. There are two major types of redundancy - parallel and standby. In this dissertation we are concerned with both these types. Some of the typical assumptions made in the analysis of redundant systems are (i) the repair facility can take up a failed unit for repair at any time, if no other unit is undergoing repair (ii) the system under consideration is needed all the time However, we frequently come accross systems where one or more assumptions have to be relaxed. This is the motivation for the detailed study of the models presented in this dissertation. In this dissertation we present models of redundant systems relaxing one or more of these assumptions simultaneously. More specifically it is a study of stochastic models of redundant systems with 'vacation period' for the repair facility (both standby and parallel systems), and intermittently used systems. The dissertation contains five chapters. Chapter 1 is introductory in nature and contains a brief description of the mathematical techniques used in the analysis of redundant systems. In Chapter 2 assumption (i) is relaxed while studying a model of cold standby redundant system with 'vacation period' for the repair facility. In this model the repair facility is not available for a random time immediately after each repair completion. Integral equations for the reliability and availability functions of the system are derived under suitable assumptions. In Chapter 3, once again assumption (i) is relaxed while studying a model of parallel redundant systems with the same 'vacation period' for the repair facility, explained in the above paragraph. In Chapter 4, the detailed review of intermittently used systems have been studied. In Chapter 5, assumption (ii) is relaxed. This chapter is devoted to the study of an intermittently used 2-unit cold standby system with a single repair facility. This study was carried out using the 'correlated alternating renewal process' and the joint forward recurrence times. All the above models have been studied, when some of the underlying distributions have a non-Markovian nature. They have been analysed using a regeneration point technique.Mathematical SciencesM. Sc. (Statistics

Laplace Transformation Techniques vs. Extended Semi-Markov Processes Method

Abstract We consider a continuously monitored one-unit KumarGupta and Taneja (1995), In this paper, we attempt to study the repairable model in Bieth, Hong and Sarkar (2010) under arbitrary life and repairable times, by employing the traditional Laplace transformation technique. We demonstrate only a partial success: The Laplace transformation technique derives a formal solution to the limiting availability, though the mathematical manipulation is quite complicated. However, it seems to be a formidable challenge to derive the limiting proportion of times the system spends in various states, which is essential for carrying out a cost analysis. On the other hand, this model can be solved completely using the ESMP method, including determination of the limiting proportion times in each state. Thus, the EMSP method not only yields the limiting availability more conveniently, but also it offers the option to conduct cost analysis. The remaining of this paper are organized as follows. Section 2 introduces the model. Section 3 establishes the renewal-type equations. Section 4 provides a formal solution to the limiting availability using the Laplace transformation technique. Section 5 specializes the formal solution to the case of the exponential life-and repair-times, as an example. Section 6 summarizes the solution using the ESMP method. Section 7 concludes the paper with a brief discussion

The establishment of the time interval between inspections for a cold standby system with component repair

Abstract The time interval between inspections of cold standby systems is a crucial decision to ensure the appropriated system reliability and the lowest costs possible. This paper presents a model developed to establish the optimal time interval between inspections for a two-unit cold standby system with component repair and subject to periodic inspection, considering reliability and costs. A Markov chain is used to define possible states, their transition probabilities and the mean time to system failure, as a function of the time interval between inspections. Given the mean time to system failure, the steady state availability is determined. Finally, the costs related to the system maintenance are established and a cost function is developed and optimized for the time interval between inspections. Numerical examples are presented and results for different system parameters are compared. Besides optimizing the time interval between inspections, the analyses also reveal the effect of repair time on system availability and mean time to system failure

Some stochastic problems in reliability and inventory

An attempt is made in this thesis to study some stochastic models of both reliability and inventory systems with reference to the following aspects: (i) the confidence limits with the introduction of common-cause failures. (ii) the Erlangian repair time distributions. (iii) the product interactions and demand interactions. (iv) the products are perishable. This thesis contains six chapters. Chaper 1 is introductory in nature and gives a review of the literature and the techniques used in the analysis of reliability systems. Chapter 2 is a study of component common-cause failure systems. Such failures may greatly reduce the reliability indices. Two models of such systems (series and parallel) have been studied in this chapter. The expressions such as, reliability, availability and expected number of repairs have been obtained. The confidence limits for the steady state availability of these two systems have also been obtained. A numerical example illustrates the results. A 100 (1 - a) % confidence limit for the steady state availability of a two unit hot and warm standby system has been studied, when the failure of an online unit is constant and the repair time of a failed unit is Erlangian. The general introduction of various inventory systems and the techniques used in the analysis of such systems have been explained in chapter 4. Chapter 5 provides two models of two component continuous review inventory systems. Here we assume that demand occurs according to a poisson process and that a demand can be satisfied only if both the components are available in inventory. Back-orders are not permitted. The two components are bought from outside suppliers and are replenished according to (s, S) policy. In model 1 we assume that the lead-time of the components follow an exponential distribution. By identifying the inventory level as a Markov process, a system of difference-differential equations at any time and the steady-state for the state of inventory level are obtained. Tn model 2 we assume that the lead-time distribution of one product is arbitrary and the other is exponential. Identifying the underlying process as a semi-regenerative process we find the stationary distribution of the inventory level. For both these models, we find out the performance measures such as the mean stationary rate of the number of lost demands, the demands satisfied and the reorders made. Numerical examples for the two models are also considered. Chaper 6 is devoted to the study of a two perishable product inventory model in which the products are substitutable. The perishable rates of product 1 and product 2 are two different constants. Demand for product 1 and product 2 follow two independent Poisson processes. For replenishment of product 1 (s, S) ordering policy is followed and the associated lead-time is arbitrary. Replenishment of product 2 is instantaneous. A demand for product 1 which occurs during its stock-out period can be substituted by product 2 with some probability. Expressions are derived for the stationary distribution of the inventor}' level by identifying the underlying stochastic process as a semi-regenerative process. An expression for the expected profit rate is obtained. A numerical illustration is provided and an optimal reordering level maximising the profit rate is also studied. To sum up, this thesis is an effort to improve the state the of art of (i) complex reliability systems and their estimation study (ii) muitiproduct inventory systems. The salient features of the thesis are: (i) Analysis of a two-component reliability system with common-cause failures. (ii) Estimation study of a complex system in which the repair time for both hot standby and warm standby systems are assumed to be Eriangian. (iii) A multi-product continuous review inventory system with product interaction, with a (s, S) policy. (iv) Introduction of the concept of substitutability for products. (v) Derivation of expressions for various statistical measures. (vi) Effective use of the regeneration point technique in deriving various measures for both reliability and inventory systems. (vii) Illustration of the various results by extensive numerical work. (vii) Consideration of relevant optimization problems.Mathematical SciencesPhD (Statistics