# REDUNDANCY AND DIFFERENT TYPES OF REDUN- DANT SYSTEMS

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## CHAPTER 3 RELIABILITY ANALYSIS OF A COMPLEX TWO UNIT STANDBY SYSTEM WITH VARYING REPAIR RATE

INTRODUCTION

Introduction of redundancy, repair maintainance and preventive maintainance are some of the well known methods by which the reliability of a system can be improved. Two unit standby redundant systems have been extensively studied by several authors in the past. A bibliography of the work on two unit systems is given by Osaki and Nakagawa (1976), Kumar and Agarwal (1980). It can be shown that any failure or repair time distribution can be approximated arbitrarily closely by a general Erlang distribution. The most useful of the more general distributions are, however, those that give coeﬃents of variation that cannot be reasonably approximated by a special Erlangian distribution (see Cox, 1970). An attempt is made in this paper to study a two unit cold standby system with generalised Erlang distribu-tion for the repair time. For the sake of simplicity, we consider a generalised Erlang distribution with two stages. Most of the studies on two unit cold standby systems are confined to obtaining expressions for various measures of system performance and do not consider the associated statistical inference problems. Chandrasekhar and Natarajan (1994) have considered a two unit cold standby system and obtained the exact confidence limits for the steady state availability of the system. Similar results were obtained for a parallel system, with preparation time by Yadavalli et al (2002). Chandrasekhar et al (2004) have studied in detail a complex two unit warm standby system assuming that the repair time distribution is a two stage generalized Erlang distribution. Besides obtaining expressions for the system reliability, mean time before failure (MTBF) and steady state availability, an attempt is made in this paper to obtain a consistent asymptotically normal (CAN) estimator and an asymptotic confidence interval for the steady state availability of a two unit cold standby system in which the failure rate of the unit while on-line is a constant and the repair time distribution is a two stage generalized Erlangian. The model and assumptions are given in the next section.

THE MODEL AND ASSUMPTIONS

The system under consideration is a two unit cold standby system with a single repair facility. We have precisely the following assumptions:

1. The units are similar and statistically independent. Each unit has a constant failure rate, say λ.
2. There is only one repair facility and the repair time distribution is a two stage generalized Erlang distribution with probability density function (pdf) given by,
3. Each unit is new after repair.
4. Switch is perfect and the switchover is instantaneous.

Note: The density given in (3.2.1) corresponds to the sum of two independent but not identically distributed exponential variates with the parameters and k (k = 1) respectively.

### ANALYSIS OF THE SYSTEM

To analyse the behaviour of the system, we note that at any time t, the system will be found in any one of the following mutually exclusive and ex-haustive states.
S  :   One unit is operating on line and the other is kept in standby
S1 :    One unit is operating online and the other is in the first stage of repair
S2 :    One unit is operating online and the other is in the second stage of repair
S3 :    One unit is in the first stage of repair and the other is waiting for repair
S4 :    One unit is in the second stage of repair and the other is waiting for repair.
Since, a generalized Elang distribution can be considered as the distribution of the sum of two independent but not identically distributed exponential random variables, the underlying stochastic process describing the behaviour of the system is a Markov process. Let pi(t), i = 0, 1, 2, 3, 4 be the probability that the system is in the state Si at time t.
It may be noted that the states S, S1 and S2 are system upstates, whereas S3 and S4 are system down states. We assume that initially, both the units are operable and obtain the measures of system performance as follows:

RELIABILITY

The system reliability R(t) is the probability of failure free operation of the system in [0,t]. To derive an expression for the reliability of the system, we restrict the transitions of the Markov process to the system upstates namely S, S1 and S2. Using the infinitesimal generator given in (3.3.1), pertaining to these upstates and using standard probabilistic arguments, we derive the following system of diﬀerential – diﬀerence equations:
p(t) = −λp(t) + k p2(t)

MEAN TIME BEFORE FAILURE (MTBF)

The mean time before failure of the system is given by
MTBF = L(0) + L1(0) + L2(0)

The steady state availability A is obtained as follows:
Using the infinitesimal generator given in 3.3.1, we obtain the following sys-tem of diﬀerential – diﬀerence equations: totic confidence interval for the steady state availability of the system and an estimator of the system reliability.

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#### CONFIDENCE INTERVAL FOR STEADY STATE AVAILABILITY OF THE SYSTEM

Let X1, X2, …, Xn be a random sample of size n of times to failure of the unit with pdf given by f (x) = λe−λx,
The object of introducing inspection is two-fold:
(i) To increase the reliability of the system and
(ii) To avoid failure of the operating systems, which may be costly and dangerous. Weiss (1962) was the first to consider a single unit system with inspection.
Many researchers, Mazumdar (1970), Luss (1977), Keller (1982), investigated various types of maintenance policies with inspection under diﬀerent sets of assumptions. In all these studies, the time needed for inspection was assumed to be negligible, but in the actual situation there are many cases where the time needed for inspection is not negligible. Another practical aspect, which is generally left out, is that the repairman employed may not be perfect.
In this paper the concept of inspection with a non-negligible time period, together with two repairmen for a two-unit cold standby system – one regular repairman and one expert repairman is introduced. The regular repairer man is always with the system and has a dual role of inspection facility and repair facility, with the known fact that he might not be able to do some complex repairs within some tolerable (patience) time. The patience time is that for which one can wait while the regular repairman tries to repair a failed unit. The expert repairman is called on to do the job on completion of the patience time or on a system failure, which ever is earlier. We also, study the asymptotic confidence limits for the availability of this system [see, Yadavalli et al (2004)].

SYSTEM DESCRIPTION

1. The system consists of two units. Initially one unit is operating on line and the other one is kept as a cold standby.
2. Failure of a unit is detected by inspection only but system failure is detected instantaneously without inspection.
3. Inspection is carried out periodically. The interval between two succes-sive inspections is a random variable, which is exponentially distributed with parameter d. If by inspection it is revealed that a unit has failed, it is forthwith taken out of the system and repaired. During the time a repair takes place, inspection is held in a state of temporary suspension. The inspections recommences with the same distribution as above, as soon as the repair is complete.
4. Inspection is of instantaneous duration. The probability of discovering a failure by inspection equals one. Inspection does not degrade a unit (if operating).
5. Time to failure of a unit is exponentially distributed with parameter λ.
6. When failure of a unit is detected repair of the failed unit and switching to the standby unit start simultaneously. Switchover is instantaneously.
7. When both units fail, the system becomes inoperable.
8. When the expert repairman is called on to do the job, it takes negligible time to reach the system.

1 INTRODUCTION
1.1 INTRODUCTION
1.2 FAILURE
1.3 REPAIRABLE SYSTEMS
1.4 REDUNDANCY AND DIFFERENT TYPES OF REDUN- DANT SYSTEMS
1.5 MEASURES OF SYSTEM PERFORMANCE
1.6 COST FUNCTION
1.7 STOCHASTIC PROCESSES USED IN THE ANALYSIS OF REDUNDANT SYSTEMS
2 APLLICATIONS OF BIVARIATE EXPONENTIAL DISTRIBUTION IN RELIABILITY THEORY
2.1 INTRODUCTION
2.2 SYSTEM DESCRIPTION AND ASSUMPTIONS
2.3 OPERATING CHARACTERISTICS OF THE SYSTEM
2.4 CONFIDENCE INTERVAL FOR STEADY STATE AVAILABILITY OF THE SYSTEM
2.5 CONFIDENCE INTERVAL FOR THE STEADY STATE AVAILABILITY OF THE SYSTEM
2.6 SYSTEM DESCRIPTION AND ASSUMPTIONS
2.7 ANALYSIS OF THE SYSTEM
2.8 AN ESTIMATOR OF SYSTEM RELIABILITY BASED ON MOMENTS
3 RELIABILITY ANALYSIS OF A COMPLEX TWO UNIT STANDBY SYSTEM WITH VARYING REPAIR RATE
3.1 INTRODUCTION
3.2 THE MODEL AND ASSUMPTIONS
3.3 ANALYSIS OF THE SYSTEM
3.4 RELIABILITY
3.5 MEAN TIME BEFORE FAILURE (MTBF)
3.7 CONFIDENCE INTERVAL FOR STEADY STATE AVAILABILITY OF THE SYSTEM
4 ASYMPTOTIC CONFIDENCE LIMITS FOR A TWO-UNIT COLD STANDBY SYSTEM WITH ONE REGULAR REPAIRMAN AND EXPERT REPAIRMAN
4.1 INTRODUCTION
4.2 SYSTEM DESCRIPTION
5 CONFIDENCE LIMITS FOR A COMPLEX THREE-UNIT PARALLEL SYSTEM WITH ”PREPARATION TIME” FOR THE REPAIR FACILITY
5.1 INTRODUCTION
5.2 SYSTEM DESCRIPTION AND NOTATION
5.3 AVALABILITY ANALYSIS
5.4 ESTIMATES FOR STEADY-STATE PROBABILITIES AND SYSTEM PERFORMANCE MEASURES
5.5 CONFIDENCE LIMITS FOR AVAILABILITY
5.6 NUMERICAL ILLUSTRATION
6 AN INTERMITTENTLY USED k OUT OF n : F SYSTEM
6.1 INTRODUCTION
6.2 SYSTEM DESCRIPTION AND NOTATION
6.3 AUXILIARY FUNCTIONS
6.4 OPERATING CHARACTERISTICS OF THE SYSTEM
7 APLLICATIONS OF TIME SERIES IN RELIABILITY MODELLING
7.1 INTRODUCTION
7.2 DEVELOPED MODELS IN RELIABILITY USING TIME SERIES
7.3 SOME DEFINITIONS AND FAILURE LAWS
7.4 ESTIMATION OF RELIABILITY .
7.5 STOCASTIC MODELLING OF THE ESTIMATED RELIABILITY OF SYSTEMS
SUMMARY
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