Evaluation of Reliability and Availability Characteristics of Two Different Systems Using Linear First Order Differential Equation
This study deals with the reliability and availability characteristics of two different systems, the second system differs from the first system due to the additional feature of preventive maintenance. Reliability and Availability analysis of system having one active unit and one warm stand-by unit with self-reset function and one maintenance facility. The failure unit is repaired through self-reset or maintenance according to different failure model. (Mean Time to System Failure), Steady- State Availability, Busy Period Analysis and Profit Function are derived for the two systems using linear first order differential equations. Two systems were evaluated theoretically and graphically to observe the effect of preventive maintenance on systems performance. The result finally shows that increase in failure rate leads to decrease in MTSF, Steady-State Availability and Profit Function of figure 4.1, 4.2 and 4.3. It was also found that increase in repair rate leads to increase in MTSF, Steady-State Availability and Profit Function of figure 4.4, 4.5, and 4.6. Therefore, the result indicated that second system originate better reliability due to the additional feature of preventive maintenance.
TABLE OF CONTENTS
Title Page i
TABLE OF CONTENTS v
LIST OF FIGURES x
CHAPTER ONE 1
1.0: GENERAL INTRODUCTION 1
1.1: Introduction 1
1.2: Background to the Study 2
1.3: Reliability Measures 3
1.3.1: Reliability 3
1.3.2: Mean Time to Failure 3
1.3.3: Failure Rate Function and Repair Rate Function 4
1.3.4: Maintainability and Availability 4
1.3.5: Mean Time to Failure (MTTF) and Mean Time Between Failure (MTBF) 5
1.3.6: Preventive Maintenance 5
1.4: Aim and Objectives 5
1.5: Scope and Limitation 6
1.6: Suggestions for Further Studies 6
CHAPTER TWO 7
2.0: LITERATURE REVIEW 7
2.1: Relationship between Availability, Reliability and Maintainability 7
2.2: Availability Classification 8
2.3: Standby Classification 9
CHAPTER THREE 18
3.0: RESEARCH METHODOLOGY 18
3.1: Introduction 18
3.2: Model Description and Assumptions 18
3.3: FIRST TRANSITION SYSTEM 19
3.4: SECOND TRANSITION SYSTEM 21
3.4.1: Mean Time to System Failure (MTSF1) 22
3.4.2: Steady-State Availability (
( ) T A ) 25
3.4.3: Busy Period Analysis (BP1) 27
3.4.4: Profit Function (PF1) 28
3.4.5: Mean Time to System Failure 2 (MTSF ) 29
3.4.6: Steady-State Availability
( ) T A 31
3.4.7: Busy period Analysis (BP2) 34
3.4.8: Profit Function (PF2) 36
CHAPTR FOUR 37
4.0: RESULT AND DISCUSSION 37
4.1: Introduction 37
4.1.1: Mean Time to System Failure 1 (MTSF ) 37
4.1.2: Steady-State Availability
( ) T A 38
4.1.3: Busy Period Analysis (BP1) 39
4.1.4: Mean Time to System Failure 2 (MTSF ) 39
4.1.5: Steady –State Availability
( ) T A 40
4.1.6: Busy Period Analysis 2 (BP ) 40
4.2: Discussion of Result 44
CHAPTER FIVE 46
5.0: SUMMARY AND CONCLUSION 46
5.1: Summary 46
5.2: Conclusion 46
1.0 GENERAL INTRODUCTION
The role and importance of reliability have been a core of any engineering industry for the last three decades. Reliability is of importance to both manufacturers and consumers. So, the reliability measure is very important, as the improvement of reliability is achieved through quality. While this measure of reliability assumes great importance in industry, there are many situations where continuous failure free performance of the system, though desirable may not be absolutely necessary, Yadavalli and Vanwyk (2012).
Several authors have studied a two (or more) similar and dissimilar unit standby redundant system. Haggag (2009a), studied the cost analysis of dissimilar-unit cold-standby system with three state and preventive maintenance using linear first order differential equations.
El-sherbeny et al (2009), studied the optimal system for series systems with warm standby components and a repairable service station. Researchers in reliability have shown a keen interest in the analysis of two (or more) component parallel system owing to their practical utility in modern industrial and technological set ups.
Two unit warm standby redundant systems have been investigated extensively in the past. The most general model is the one in which both the life time and repair time distributions of the units are arbitrary. However the study of standby system with more than two units, though very important, has received much less attention, possibly because of the built in difficulties in analyzing them. Such systems have been studied only when either the life time or the repair time is exponentially distributed. When both these are general, the problem appears to be intractable even in the case of cold standby systems. The present contribution is an improvement in the state of art in the sense that a three unit warm standby system is shown to be capable of comprehensive analysis. In particular we show that there are imbedded renewal points that render the analysis possible. Using these imbedded renewal points they obtained the reliability and availability functions, Srinivasan and Subramaniam (2006).
But in this research, the reliability and availability characteristics of two different systems are study, where the second system differs from the first system due to the additional feature of preventive maintenance. Each system consisting of one active unit and one warm standby unit with self-reset function and maintenance facility. The failure unit is repaired through self reset or maintenance according to different failure models.
1.2 Background to the Study
Reliability and availability are very important indices in a substation control protection system. The Station Computer (SC) has important role in the system. Its function is to maintain the central system data base and provide interfaces to the outsider world-locally to station operators through the local Man Machine Interfaces (MMI) subsystem and remotely to system operators and protection engineers through Supervisory Control and Data Acquisition system (SCADA) communication interfaces. So, its reliability directly influences the reliability of Station Computer (SC). Two units warm standby redundancy is taken. Redundancy is one of the ways of improving the reliability of system when the individual unit of the system remains unchanged. Warm standby is essential for two units to switch within the shortest time. So, the active unit and warm standby unit run in different states, which makes their failure rate different. Commonly, the failure rate of the warm standby unit is smaller than that of the active one. So, compared with a hot standby system, the reliability of the warm standby system is increased. Second, each unit has a self reset function. Each unit performs automatic error detection through self – checking and recovers from some failures, El-Said and El-Hamid (2006).
1.3: Reliability Measures
Reliability is the analysis of failures, their causes and consequences. It is the most important characteristics of product quality as things have to be working satisfactorily before considering other quality attributes. Usually, specific performance measures can be embedded in to reliability analysis by the fact that if the performance is bellow a certain level, a failure can be said to have occurred.
1.3.1: Reliability is the probability that the system will perform its intended function under specified working condition for a specified period of time. Mathematically, the reliability function R(t) is the probability that a system will be successfully operating without failure in the interval from time zero to time t.
R(t) = P(T > t), t ≥ o where T is a random variable representing the failure time or time – to failure. The failure probability, or unreliability is then F(t) = 1- R(t), = P(T≤ t) which is known as the distribution function of T.
1.3.2: Mean Time to Failure the mean time to failure (MTTF) is defined as the expected value of the lifetime before a failure occurs.
1.3.3: Failure Rate Function and Repair Rate Function
The failure rate function, or hazard function, is very important in reliability analysis because it specifies the rate of the system aging.
The Failure Rate Function: Is defined as the quantity representing the probability that a device of age t will fail in the small interval from time t, to t + dt. The importance of failure rate function is that it indicates the changing rate in the aging behavior over the life of a population component.
Repair Rate Function: Is the expected time to repair the system from failure. This include the time it takes to diagnose the problem, the time it takes to get a repair technician on site, and the time it takes to physically repair the system, Pham (2003).
1.3.4: Maintainability and Availability
When a system fails to perform satisfactorily, repair is normally carried out to locate and correct the fault. The system is restored to operational effectiveness by making an adjustment or by replacing a component.
Maintainability: Is defined as the probability that a failed system will be restored to a functioning state within a given period of time when maintenance is performed according to prescribed procedures and resources. Generally, maintainability is the probability of isolating and repairing a fault in a system within a given time. Maintenance personnel have to work with system designers to ensure that the system product can be maintained cost effectively.
The Availability Function of a system, denoted by A(t) is defined as the probability that the system is available at time t. Different from the reliability that focuses on a period of time when the system is free of failures, availability concerns at a time point at which the system does not stay at the failed state.
Mathematically, A(t) = Pr(system is up or available at time instant t).
1.3.5: Mean Time to Failure (MTTF), and Mean Time Between Failure (MTBF) it is important to distinguish between the concepts mean time to failure and mean time between failures (MTBF). The MTTF is the expected time to failure of a component or system. That is, the mean of the time to failure (TTF) for that component or system. The MTBF is the
expected time to failure after a failure and repair of the component or system.
1.3.6: Preventive Maintenance the maintenance carried out at predetermined intervals or corresponding to prescribed criteria and intended to reduce the probability of failure or the performance degradation of an item. Hoyland and Naws (1994).
1.4: Aim and Objectives
The main aim of this research work is to investigate and improve upon the existing methodologies for the reliability and availability characteristics of two different systems, where the second system differs from the first system due to the additional feature of preventive maintenance. To achieve the above aim the following objectives are derived.
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