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ACCENT JOURNAL OF ECONOMICS ECOLOGY & ENGINEERING Available Online: www.ajeee.co.in Vol. 01, Issue 02, June 2016, ISSN - 2456-1037, (INTERNATIONAL JOURNAL)

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MATHEMATICAL APPROACH AND STUDY BASED ON RELIABILITY THEORY Dr. Prakash Deep Agrawal

Department of Mathematics, Govt. Post Graduate College, Kotdwar (Garhwal) Uttarakhand

Abstract- The work of reliability is becoming more and more important in engineering practice. The details depend on whether mechanical, electrical, chemical, or other systems are being investigated, but the concepts and mathematical foundations of reliability do not match a particular application area. Over the last 50 years, thousands of articles and dozens of books on mathematical reliability models have been published. A comprehensive overview of the current developments in mathematical reliability theory fills an extensive book. Based on its relevance to both theory and application, and taking into account the interests of the author, current research is considered in four main areas of reliability theory: coherent systems, stochastic networks, software trust. Gender and conservative theory.

1. INTRODUCTION

With the exception of computer and environmental technology, no other engineering discipline has been as powerfully developed and advanced as reliability technology in the last 40 years.

This is primarily due to the use of high- risk systems as nuclear power plants, the human step into space, and the development of sensitive weapon systems.

However, modern systems, which are less spectacular in industrial production and transportation and communication technologies, are usually so complex that reliability cannot be predicted and maintained without scientific methods.

Therefore, it is not surprising that the rapid development of reliability technology has provided a decisive impetus for the phenomenal development of mathematical reliability theory over the last 40 years.

According to IEC document, reliability is defined as "the ability of a product, system, or service to perform the expected task under specified conditions of use over a specified period of time." For simplicity, this paper only mentions the reliability of a system or its components (subsystems). In reliability theory, the allotted period is called mission time. If the system is unable to perform a task during its useful life, a system error will occur. This concept of system failure in the more general sense is the basis of reliability research. In particular, the IEG definition of reliability can lead to special mathematically well-defined criteria, the so-called reliability criteria. The most important ones are:

1) Probability of survival: This is the probability that a failure will not occur within a specific time interval [0, t].

2) Availability: Point availability is the probability A (t) that a system can perform a task at a particular point in time t.

Inpatient availability is defined by

Therefore, A is the percentage of time associated with the infinite mission time that the system can perform its task.

Reliability theory deals with measuring, predicting, maintaining, and optimizing the reliability of technical systems.

Reliability refers to one or more reliability criteria that are most relevant and most important to the mission of the system under consideration. The main problems addressed by mathematical reliability theory are:

1) Investigation of the interrelationship between the reliability criteria of a system and its subsystems (components).

2) Modeling the behavior of system (component) failures and aging.

3) Statistical inference of reliability criteria.

4) Development, research, and optimization of measures to maintain or restore a certain level of reliability (maintenance theory).

Recent developments deal with the reliability of software and the reliability of human and mechanical systems. This article cannot provide a complete overview of all the new trends and developments in mathematical reliability theory. Nor can you cite all relevant literature. (Even the excellent volume of over 700 pages, MISRA [1] is not perfect.)

In view of the author's interests, the choices are presented here according to the importance of both theory and

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2 application. Binary coherent system Consider a system S consisting of the components e1, e2... en. Two possible states of the system and its components are available and not available. Let's take a closer look at the (0, 1) indicator variables for the state of the system and its components zs and z1, z2... Zn is:

The dependence of zs on the zi is given by the structure function cjl of the system:

The system is called coherent if its structure function has the following properties:

1) ∅(O,O, ... ,O) = 0.

∅ (1,1, ... ,1) = 1 .

2) ∅l is nondecreasing in each zi .

3) For each i = 1, 2, ... , n there is a vector z = (z1,z2, ... ,zn) with the properties

2. STOCHASTIC NETWORKS

Network reliability analysis is required in many important technical areas.

Examples are communication networks, surveillance systems, and transportation and distribution systems. Computer communication networks, in particular, have evolved to meet the enormous demand for the transmission and processing of information. Therefore, it was essential to develop and improve effective tools for performing reliability analysis of complex networks. To simplify the terminology, this section describes communication networks. The basic structure of the network topology under consideration is given by the loop less connected graph G = (V, E). Where V is the set of nodes and E is the set of edges.

Nodes are interpreted as end users and edges are interpreted as connections

between them. G becomes a random graph or stochastic network G by assuming that its edges and nodes exist (exist) or do not exist (not available) according to a given probability distribution. Therefore, node and edge state variables are binary random variables. For simplicity, the following assumes that the node is always available.

3. SOFTWARE RELIABILITY

Computers are used today in many important areas where failures can have catastrophic or at least costly consequences. In process control systems for space travel, nuclear power plants, air traffic systems or ballistic missile defense systems. Due to recent advances in hardware technology, software flaws are a major cause of computer system failures.

Therefore, great efforts have been made in the last 20 years to improve the reliability of so-called software. The currently generally accepted definition of software reliability was given by MUSA and OKUMOTO. Software reliability is the probability that a computer program will work properly in a particular environment for a particular period of time.

(Similarities to the definition of hardware reliability are clear.) Engineer and statistician issues arising from software reliability research are as follows (see BARLOW and SINGPURWALLA [28]).

1) Quantify and measure software reliability.

2) Evaluation of changes in software reliability over time.

3) Analysis of software failure data.

4) Decide whether to continue or stop testing the software.

Software reliability issues differ from hardware reliability issues for four main reasons:

1) The cause of software failure is a human error.

2) Once all the bugs are fixed, the software will be completely reliable.

3) There are no processes that lead to errors when using the software.

4) The software life cycle consists of three main phases with access to mathematical modeling.

The testing and debugging phases, the validation phase, and the operational phase. (Currently, there is no promising

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3 approach to modeling the maintenance phase.) The three early models are often used as benchmarks to compare with new models. These are models by JELINSKY and MORANDA [29], LITTLEWOOD and VERALL [30], GOEL and OKUMOTO. [31].

4. MAINTENANCE THEORY

Very few systems are designed to operate maintenance-free. They work in inaccessible or at least very difficult environments, such as space and high radiation fields. Systems typically undergo both preventive and corrective maintenance. Preventive maintenance is the replacement of parts, the application of lubricants, and adjustments before a failure occurs. In this way, system reliability is maintained or improved by preventing the effects of aging. If a failure occurs, we will carry out corrective maintenance. Mathematical theory of maintenance provides tools for efficiently organizing preventative and corrective maintenance measures. F (t), F (t), A. Let (t) be the system failure probability, survival probability, and failure rate. If X represents a random time to the first failure (life) of the system, then F (t) = P (X to t) is a distribution function and f (t) = F1 (t) is the probability density of X. is.

The failure rate is given by

To have an aging system, A. (t) is assumed to be no decreasing. (Otherwise preventive maintenance would make no sense.) Renewal theory deals with the simplest model of corrective maintenance:

On failure the system is replaced by an identical new one. The corresponding maintenance action is called replacement or renewal of the system. (After a replacement the system is "as good as new".) Rate on repair has the same value as immediately before the failure. (After a minimal repair the system is "as bad as old".) More exactly, if a system failure occurring at system age x is removed by a minimal repair, and then the "residual lifetime" of the system has the distribution function

Maintenance policies that were under consideration until the 1970s voluntarily

planned replacements and minimal repairs. That is, regardless of the nature of the failure that caused the maintenance effort (BARLOW and PROSCHAN [38, 39]. The work of BElCHELT [40] caused a breakthrough (see). Also, BEICHELT and FISCHER) two types of system failures are shown here.

Type 1 error: Eliminate with minimal repair. Type 2 error: Must be eliminated by replacement. Therefore, failures with minor consequences are eliminated with minimal repairs and with minimal failures that cause serious damage to the system during replacement. The probabilities p (t) and 1 p (t) are given that the error that occurs at time t is type 2 or type 1. When using only corrective maintenance, that is, maintenance according to the type of failure, cm indicates the estimated cost of repair; cr indicates the expected cost of replacement, and the expected long-term per unit time. The total maintenance cost is given by case p = p (t)

This criterion serves as a benchmark against which to compare more sophisticated maintenance policies. As a by-product, the 2-type-failure model allows the mathematically exact treatment of the so-called repair-limit-replacement- policy. Under this policy a system will be replaced if the repair cost exceeds a given limit L. Otherwise a minimal repair is carried out. Hence, the two failure types specified above are now generated by the random repair cost C. Let R(x) is the distribution function of C. Then p = 1 - R(L).

Substituting this p in (6) yields the expected total maintenance cost rate per unit time:

The significance of the paper BElCH ELT [40] only became then obvious when the same results were published some years later by BROWN and PROSCHAN [43] for constant p and by BLOCK, BORGES and SAVITS [44] for

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4 time-dependent p, see also BElCH ELT [45], BElCH ELT and FRANKEN [46].

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