To improve the reliability of a system, the following two well-known methods are
1. Provision of redundant units, and
2. Repair maintenance
In a redundant system more units are made available for performing the system
function when fewer are required actually. The provision of redundant units could be
performed mainly in three ways, namely, series, parallel and standby. This thesis deals
with these three types.
Following are some classical assumptions that are made in the analysis of redundant
1. The life time and the repair time distributions are assumed to be exponential.
2. The repair rate is assumed to be constant.
3. There is a single repair facility.
4. The repair facility will continuously available.
5. The system under consideration is needed all the time.
6. The lifetime or repair time of the units are assumed to be independent.
7. Usage of only conventional methods for the analysis of the estimated reliability
8. Switch is perfect in the sense that the switching device does not fail.
9. The switchover time required to transfer a unit from the standby state to the
online state is negligible.
10. There is no human error when we handle the machines and no common cause
of failures. 11. The repair rate is independent of the number of failed units.
We frequently come across systems where one or more of these assumptions have
This is the motivation of the detailed study of the models presented in this thesis.
We present several models of redundant repairable systems relaxing one or more
assumptions (1-11) simultaneously. More specifically, it is a study of stochastic
models of redundant repairable systems with a single repair facility.
The estimation study of the system measures is focused in some chapters. Imperfect
switch, non-instantaneous switchover, variating repair rate and common cause of
failure with human errors, etc. are some of the aspects focused in the thesis.
Chapter 1 is essentially an introductory in nature and contains a brief description of
the mathematical techniques used in the analysis of redundant systems.
In Chapter 2 assumptions (1), (2) and (4) are relaxed. Here we deal with an n - unit
warm standby system with varying repair rate. We first consider a model in which the
repair rate of a failed unit is constant depending on the number of failed units at the
epoch of commencement of each repair and the vacation period is introduced after
each repair completion. Introducing a profit function, the optimal number of standby
units is also determined. A special case is obtained by suspending the vacation period.
In Chapter 3, we have relaxed an assumption (6). A three unit warm standby system
with dependent structure, wherein the lifetimes of online unit, standby units and the
repair time of failed units are governed by quadrivariate exponential law is studied.
Measures of system performance such as, reliability, MTSF, availability and steady
state availability are also obtained. A 100(1- α)% confidence interval for the steady
state availability of the system and an estimator of system reliability based on
moments are obtained. Numerical work is carried out to illustrate the behaviour of the system reliability based on moments by simulating samples from quadrivariate
exponential distribution. Generalization of the above results to a (≥ 4) unit warm
standby system with (≥ 2) repair facilities is investigated.
In Chapter 4, a slight modification of an assumption (4) is studied. This chapter deals
with the study of three unit system where unit 1 is connected in series and the other
two units are connected in parallel. The significant feature of this chapter is
modification of an assumption (4) by assuming the repair facility gives priority to the
repair of the unit 1 in the sense that whenever the unit 1 fails in the operable state, and
at that instant if there is already unit 2 or unit 3 under repair, the repair of unit 1
starts immediately keeping the unit under repair in queue, and the repair of which is
taken afresh immediately after the repair of unit 1 is completed.
In Chapter 5, a two unit cold standby system with constant failure rate and two stage
Erlangian repair is studied. Measures of system performance such as reliability,
MTSF, availability and steady state availability are obtained. Furthermore confidence
limits for the steady availability of the system, ML estimator of system reliability and
Bayes estimator of MTSF are derived. Numerical illustration is carried out to study the
performance of the Bayes estimator of MTSF.
A three unit series-parallel system with preparation time is studied in Chapter 6.
Unit 1 is given a priority over unit 2 and 3 for repair as it is connected to a series
system. The expressions for system measures like availability and reliability are
In Chapter 7, two unit warm standby system with imperfect switch and preparation
time is studied. The switching device will have a head-of-line priority over the units
for repair. Assuming various arbitrary distributions for some of the random variables
involved, MTSF and ∞ are obtained.