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STANDBY REDUNDANCY
Standby redundancy is one of the basic methods of increasing reliability. Standby redundancy consists of an attachment to an operating unit one or more redundant (standby) units, which, when failure occur, take the place of basic operating unit and fulfil its function. These units may be classified as cold, warm or hot (Gnedenko et al (1969)).
1. A cold standby is not hooked up hence completely inactive, it cannot in (theory) fail until it is put to use by replacing a primary unit. Assume that since it is not in operation its reliability will not change when it is put into operation.
2. A warm standby is when a unit is partially energized hence has a diminished load.
The on-line unit and standby unit are not subject to the same loading conditions. The failure of standby unit is attributable to some extraneous random influence. The probability of failure of the warm standby unit is smaller than the probability of failure of the on-line unit. This is the most general type of standby due to the high failure rate of the hot standby unit and possible lapse before it is operable in the case of the cold standby unit.
3. A hot standby is fully energized and active in the system although redundant and the possibility of failure of a hot standby is the same as that of an operating unit in the standby state. A hot standby’s reliability is independent of the instant at which it takes place in the operable unit.
SERIES SYSTEM
A series system is an arrangement of components such that the failure of any of the system component results the entire system failure. That is, a system in which all the components have to work for the system to work. A simple computer consists of a processer, a bus and a memory, chains made out of links; highways that maybe closed to traffic due to accidents at different locations, the food chains of certain animal species and layered company organizations in which information is passed from one hierarchical level to the next are some examples of a system in series.
AVAILABILITY
Availability is also a measure of system performance, which is the probability that the system will be operational (in operable condition or available for use) at the given time ?. It implies that the system is either in active operation or is able to operate if required and consists of aspects of reliability, maintainability and maintenance support. Availability is applicable only to systems which undergo repair and are restored after failure. In theory availability ?(?) should be 100 % but in practice, even equipment coming directly out of storage may be defective. Availability is very important and high availability may be obtained either by increasing the average operational time until the next failure, or by improving maintainability of the system. Pointwise availability is a point function which describes the probability that a system will be able to operate at a given instant of time (Gnedenko and Ushakov (1995)).
Klaassen and Van Peppen (1989), Beasley (1991).
COST FUNCTION
There are a number of constraints facing the designer of a system. Some consideration has to be made system’s reliability and availability, its usefulness and effectiveness. Due to the complexity of the present-day systems, measures such as reliability, availability etc. alone are not sufficient. In addition, cost and profit have become the guiding principles in every industrial and social management endeavor. Hence cost optimization has become one important criterion for system designers. We shall focus, in this thesis to the construction of comprehensive cost function for each of the models considered. Since they are highly nonlinear, analytical optimization of these functions become impracticable, if not impossible. Hence we resort to numerical optimization; assuming that the control parameters are within certain specific intervals, we obtain numerically their optimal values.
RENEWAL THEORY
In renewal theory we are interested in the lifetime of the unit, there exists times commonly random from which onward the future of the process is probabilistic replica of the original process. At the beginning ? = 0 a repairable unit is put into operation and functioning. The unit is replaced by a new one of the same type and subjected to maintenance that completely restores it to an as good as new condition upon failure.
This process is repeated upon failure and a replacement time is considered negligible. These results in a sequence of life times and this study is restricted to these renewal points. The number of renewals ?? upto some time ? is the probability object in these sums of non-negative i.i.d. random variables. A number of researchers have studied specific reliability problems using renewal processes. The homogeneous Poisson process has received considerable attention and happens to be the simplest renewal process. The time parameter may be taken as either discrete or continuous. A proper lead for the discrete case was conducted by Feller (1950) followed by a very lucid account of Cox (1962) for the continuous case (he provided an introduction to renewal theory in the case of a repair facility not being available and failed units queuing up for repair).
Barlow (1962) applied in his research on repairable systems queuing theory. Some operating characteristics of a one unit system were studied by Srinivasan (1971) while Gnedenko et al (1969) worked out the mean time to system failure of a two unit standby system. Some priority redundant systems were studied by Buzacott (1971).
CHAPTER 1: INTRODUCTION
1.1 Introduction
1.2 Failure
1.3 Repairable Systems
1.4 Redundancy and Different Types of Redundant Systems
1.5 Measures of System Performance
1.6 Cost Function
1.7 Stochastic Processes used in the Analysis of Redundant Systems
CHAPTER 2: MAINTENANCE ANALYSIS OF AN n-UNIT WARM STANDBY SYSTEM WITH VARYING REPAIR RATE AND VACATION PERIOD FOR THE REPAIR FACILITY
2.1 Introduction
2.2 System Description and Assumptions
2.3 The Subsystem
2.4 The Main System
2.5 Reliability of the System
2.6 Availability of the System
2.7 Profit Analysis
2.8 Particular Case
CHAPTER 3: APPLICATION OF QUADRIVARIATE EXPONENTIAL DISTRIBUTION TO A THREE UNIT WARM STANDBY SYSTEM WITH DEPENDENT STRUCTURE
3.1 Introduction
3.2 The Model and Assumptions
3.3 Analysis of the System
3.4 System Reliability
3.5 Mean Time to System Failure (MTSF)
3.6 System Availability
3.7 Steady State Availability
3.8 Confidence Interval for Steady State Availability of the System
3.9 Numerical Illustration
CHAPTER 4: A THREE UNIT SERIES PARALLEL SYSTEM WITH PRE-EMPTIVE PRIORITY REPAIR
4.1 Introduction
4.2 System Description
4.3 Reliability of the System
4.4 Availability Analysis
4.5 Steady State Analysis
4.6 Cost Analysis
4.7 Numerical Illustration
CHAPTER 5: CLASSICAL AND BAYESIAN ESTIMATION STUDY OF STANDBY SYSTEM WITH AN ERLANGIAN REPAIR
5.1 Introduction
5.2 The Model and Assumptions
5.3 Analysis of the System
5.4 System Reliability
5.5 Mean Time before Failure
5.6 System Availability
5.7 Steady State Availability
5.8 ML Estimator of System Reliability
5.9 Confidence Interval for Steady State Availability of the System
5.10 Bayes Estimation of MTSF in a two unit Stand-by System
5.11 Numerical Illustration
CHAPTER 6: THREE UNIT SERIES-PARALLEL SYSTEM WITH PREPARATION TIME FOR THE REPAIR FACILITY
6.1 Introduction
6.2 System Description and Notations
6.3 The Reliability of the System
6.4 The Availability of the System
6.5 Availability
6.6 Steady State Availability
CHAPTER 7: TWO UNIT STANDBY SYSTEMS WITH IMPERFECT SWITCH AND PREPARATION TIME
7.1 Introduction
7.2 Model (G|M|M System with Imperfect Switch and Preparation Time)
7.3 Model 2 (Dual Mode of 1)
BIBLIOGRAPHY
