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**INTERMITTENTLY USED SYSTEMS**

**INTERMITTENTLY USED SYSTEMS**

In almost all the models of redundant systems studied so far, it is assumed that the system under consideration is needed all the time. But in some practical situations continuous failure free performance may not be necessary. In such cases we have to take into consideration the fact that the system can be in down state during certain intervals without any real consequence. In this case the probability that the system is in the up state is not an important measure; what is really important is the probability that the system is available when it is needed. Gaver (1964) pointed out that is pessimistic to evaluate the performance of an intermittently used system solely on the basis of the distribution of the time to failure. Srinivasan (1966), Nakagawa et al (1976), Srinivasan and Bhaskar (1979 a, b, c), Kapur and Kapoor (1978, 80) extended Gaver’s results for two-unit systems. Detailed study of an n-unit intermittently used system is made. The statistical inference of some of these models has been studied recently by Yadavalli et al (2000, 2001).

** STOCHASTIC POINT PROCESSES**

Stochastic point processes are more general than those considered in the earlier sections. Since point processes have been studied by many with varying backgrounds, there have been several definitions of the point processes each appearing quite natural from the view point of the particular problem under study. [See for example Bartlett (1966), Bhaba (1950), Harris (1963) and Khinchine (1955)]. A comprehensive definition of point process is due to Moyal (1962) who deals with such processes in a general space which is not necessarily Euclidean.

**REGENERATIVE STOCHASTIC PROCESSES**

**REGENERATIVE STOCHASTIC PROCESSES**

The idea of regeneration point was first introduced by Bellman and Harris (1948) while studying population point processes. Feller (1949), in the theory of recurrent events, dealt with a special case of regeneration points. Later on, Smith (1955) generalized Feller’s results and dealt with more general stochastic point processes possessing such regeneration points, familiarity known as regenerative processes. A formal theory of such processes has been developed by Kingman (1964). A regenerative event R of a stochastic process {X (t)} is an event that is characterized by the property that if it is known that R happens at 1 t = t , then the knowledge of the history of the process prior to 1 t loses its predictive value. In some special cases, the event R is the only characteristic, so that the process regenerates itself with each occurrence of R. In more general cases, in addition to the occurrence of R, knowledge of X (t) is necessary for the prediction of the process. The renewal process can be thought of as a general point process in which each point at which the event R occurs is a regeneration point. The occurrence of an event at University of Pretoria etd – Malada, A (2006) 1 t = t uniquely determines the distribution of events from any collection of segments of points 1 t ³ t .If we further specialize to the case when the intervals between successive events are exponentially distributed, we notice that any point (not necessarily a point where an event occurs) on the time axis is a regeneration point. Gnedenko (1964), Srinivasan and Gopalan (1973 a, b), Birolini (1974, 75), Srinivasan and Subramanian (1977), Hines (1987), Hargreaves (2002).

**CONCLUDING REMARKS AND SCOPE OF**

WORK Reliability theory is a very important branch of systems engineering and operations and deals with general method of evaluating the various measures of performance of a system that may be subject to gradual deterioration. Several models of redundant systems have been studied in the literature and the following are some of the typical assumptions made in analyzing such systems: (i) the repair times are assumed to be exponential (ii) the estimated study of the system measures has not been made. (iii) the system is available continuously (iv) Environmental factors not affecting the system (v) The failures take place in one mode (vi) The switching device is perfect (vii) System reliability evaluated for given chance constraints University of Pretoria etd – Malada, A (2006) (viii) The switchover time required to transfer a unit from the standby state to online stage is negligible. (ix) the failures and repairs are assumed to be independent. However, we frequently come across systems in which one or more of these assumptions have to be dropped and hence there is an increasing need for studying models in which at least some of these assumptions could be relaxed. That is the motivation for the detailed study of the models presented in this thesis. This thesis is a study of some redundant repairable systems with ‘rest period’ for the operator, non-instantaneous switchover, imperfect switch, intermittent use and optimization study.

**General Algorithm**

**General Algorithm**

1. Convert the deterministic form of chance constraint into a linear constraint, adopting the technique of sequential linear programming (Rao 2000, Jeeva et al 2002,2004 , Charles and Dutta, 2003). 2. Code any one of the system (5.2) – (5.4) along with respective linearized constraint in MATLAB or LINGO and generate optimal solutions by inputting initial values using random function (in later stages one can use the existing real solution to generate integer solution using the step below given). 3. Apply the branch and bound algorithm given below to get integer solutions.

**Branch-and-bound (B&B) technique**

The B&B technique for CCRSO for stochastic optimization is given below: University of Pretoria etd – Malada, A (2006) 1. Solve the problem as if all the variables were real numbers i.e. not integers, using the general algorithm given above. This solution is the upper bound (for maximization problem) of the CCRSO problem. 2. Choose one variable at a time that has a non-integer value, say xj and branch that variable to the next higher integer value for one problem and to the next lower integer value for the other. The real valued solution of the variable j can be expressed as xj = [xj] + xj * , where [xj] is the integer part of xj and xj * is the fractional part of xj, 0< xj * <1. The lower bound and upper bound constraints of the two mutually exclusive problems are xj = [xj] and xj = [xj] + 1, respectively. Add these two constraints to both branched problems. 3. Now the variable xj is an integer in either branch. Fix the integer of xj for the following steps of branch-and-bound. Select the branch that yields the maximum objective function with all constraints satisfied. Then repeat step 2 on another variable xk ≠ xj for each of the new sub problems until all variables become integers. 4. Stop the particular branch if the solution is not satisfying the constraints of the original problem else stop the branch when all the desired integer values are obtained.

**A COMPLEX SYSTEM WITH CORRELATED FAILURES**

Most reliability models assume the continuous operation of the unit ( or system) until a failure occurs. However, situations may arise where the unit (or system) needs rest after its operation for some time [Muller, 2005]. Very few attempts have been made in this direction. Murari and Muruthachalan (1981), Sarma (1982), Botha (2002), Hargreaves (2003) considered a two-unit system with a provision for rest for the system. The working and the rest are assumed to be random variables with negative exponential distributions. However, the idea of preparation time for the system may prove expensive as no output is obtained from the system during rest. This situation can be avoided in a two-unit cold standby system by providing rest to each unit alternately and operating the other unit when one requires rest. Further, in repairable systems, the dependence of repair time on the failure time of unit is a common experience of systems engineers, but this fact has also been ignored so far by reliability researchers. Keeping these factors in view, we analyse in this chapter a two-unit cold standby system with independent failure and repair times, with provision for the rest of the operative unit.

**CONTENTS :**

- SUMMARY
- CHAPTER 1: INTRODUCTION
- 1.1 INTRODUCTION
- 1.2 FAILURE
- 1.3 REDUNDANCY AND DIFFERENT TYPES OF REDUNDANT SYSTEMS
- 1.3.1 Parallel systems
- 1.3.2 k-out-of-n systems: F system
- 1.3.3 k-out-of-n: G-system
- 1.3.4 n-out-of-n: G system

- 1.4 REPAIRABLE SYSTEM
- 1.5 SYSTEMS WITH NON-INSTANTANEOUS SWITCHOVER
- 1.6 SYSTEMS WITH IMPERFECT SWITCH
- 1.7 INTERMITTENTLY USED SYSTEMS
- 1.8 MEASURES OF SYSTEM PERFORMANCE
- (a) Reliability
- (b) Pointwise availability
- (c) Asymptotic or Steady-State Availability
- (d) Interval Reliability
- (e) Limiting Interval reliability
- (f) Mean number of events in (0, t)
- 1.9 TECHNIQUES USED IN THE ANALYSIS OF REDUNDAT SYSTEMS
- 1.9.1 Renewal theory
- 1.9.2 Semi-Markov and Markov Renewal process
- 1.9.3 Stochastic point process
- 1.9.4 Product densities
- 1.9.5 Regenerative stochastic process
- 1.9.6 Concluding remarks and Scope of the work

- CHAPTER 2: A STUDY OF A TWO UNIT PARALLEL SYSTEM WITH ERLANGIAN REPAIR TIME
- 2.1 INTRODUCTION
- 2.2 MODEL AND ASSUMPTIONS
- 2.3 ANALYSIS OF THE SYSTEM
- 2.3.1 Reliability
- 2.3.2 Mean Time Before Failure (MTBF)
- 2.3.3 System Availability
- 2.3.4 Steady State Availability

- 2.4 CONFIDENCE INTERVAL FOR STEADY-STATE
- AVAILABILITY OF THE SYSTEM
- 2.5 MLE OF SYSTEM RELIABILITY
- 2.6 NUMERICAL ILLUSTRATION

- CHAPTER 3: AN n UNIT SYSTEM OPERATING IN A RANDOM ENVIRONMENT
- 3.1 INTRODUCTION
- 3.2 MODEL AND ASSUMPTIONS
- 3.3 NOTATION
- 3.4 STATE OF THE SYSTEM
- 3.5 STATIONARY DISTRIBUTION
- 3.6 TIME TO THE FIRST DISSAPOINTMENT
- 3.7 MEAN NUMBER OF DISSAPONTMENTS
- 3.8 MEAN STATIONARY RATE OF DISSAPOINTMENTS
- 3.9 CONCLUSIONS
- 3.9.1 Two unit system
- 3.9.2 Intermittently used n-unit standby redundant system

- CHAPTER 4: BUSY PERIOD ANALYSIS OF A TWO UNIT SYSTEM WITH PREVENTITIVE MAINTENANCE AND IMPERFECT SWITCH
- 4.1 INTRODUCTION
- 4.2 SYSTEM DESCRIPTION
- 4.3 NOTATION
- 4.4 SYMBOLS FOR STATES OF THE SYSTEM
- 4.5 RELIABILITY ANALYSIS (MODEL 1)
- 4.6 AVAILABILITY ANALYSIS
- 4.7 BUSY PERIOD ANALYSIS
- 4.8 EXPECTED NUMBER OF VISITS BY REPAIR FACILITY
- 4.9 MODEL
- 4.10 AVAILABILITY ANALYSIS (MODEL 2)
- 4.11 BUSY PERIOD ANALYSIS (MODEL 2)
- 4.12 EXPECTED NUMBER OF VISITS BY REPAIR FACILITY
- 4.13 PROFIT ANALYSIS
- 4.14 NUMERICAL ILLUSTRATION

- CHAPTER 5: RELIABILITY STOCHASTIC OPTIMIZATION WITH BRANCH AND BOUND TECHNIQUE
- 5.1 INTRODUCTION
- 5.1.1 Case 1: b follows Uniform distribution
- 5.1.2 Case 2: b follows normal distribution
- 5.1.3 Case 3: b follows log-normal distribution

- 5.2 GENERAL ALGORITHM
- 5.3 BRANCH-AND-BOUND (B&B) TECHNIQUE
- 5.4 NUMERICAL EXAMPLE
- 5.5 CONCLUSION

- 5.1 INTRODUCTION
- CHAPTER 6: A TWO UNIT COLD STANDBY SYSTEM WITH NON-INSTANTANEOUS SWITCOVER
- 6.1 INTRODUCTION
- 6.2 MODEL ASSUMPTION
- 6.3 NOTATION
- 6.4 TRANSITION PROBABILITIES AND
- SOJOURN TIMES
- 6.5 TIME TO SYSTEM FAILURE
- 6.6 AVAILABILITY ANALYSIS
- 6.7 BUSY PERIOD ANALYSIS
- 6.8 EXPECTED NUMBER OF VISITS
- BY THE REPAIRMAN
- 6.9 SWITCHOVER ANALYSIS
- 6.10 COST ANALYSIS

- CHAPTER 7: A COMPLEX SYSTEM WITH CORRELATED FAILURES
- 7.1 INTRODUCTION
- 7.2 SYSTEM DESCRIPTION
- 7.3 NOTATION
- 7.4 TRANSITION PROBABILITIES AND SOJOURN TIMES
- 7.5 MEANTIME TO SYSTEM FAILURE
- 7.6 AVAILABILITY ANALYSIS
- 7.7 BUSY PERIOD ANALYSIS
- 7.8 PROFIT ANALYSIS
- 7.9 SPECIAL CASES
- 7.10 NUMERICAL ILLUSTRATION
- BIBLIOGRAPHY

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STOCHASTIC RELIABILITY MODELLING FOR COMPLEX SYSTEMS