At the beginning of the study, in Chapter 2, there is a survey of the
basic concepts of reliability theory. The pointwise availability of the
system is chosen as the value which measures its reliability and this
is shown to be a good representation of its operational ability.
Chapter 2 goes on to examine the various possible types to describe
the distribution of a random variable and the connections between
these functions. From the point of view of this study the principal
function is seen to be the intensity function or rate function.
Distributions are identified on the basis of the failure, repair or
waiting rate (depending on the random variable in question). At the
end of the chapter there is an account of the most common
distribution types in reliability theory and their principal
characteristics. Finally, by means of individual distributions, a
general distribution is arrived at, which covers (by a suitable rate
function choice) all the known distributions that fulfill the
regularity conditions.
In Chapter 3, a detailed description of the system is made and
assumptions regarding the random variables of the system are
introduced. The system consists of two classes of components, i.e. of
two subsystems S1 and S2. The subsystem S1 contains M identical
redundantly-connected components while S2 is composed of N
independent (,in general, different) components connected in series.
Components in S1 are assumed to fail according to some general
distribution (the failure rate of components is an arbitrary function
of time) while in S2 the components have constant failure rates, i.e.
the time between failures for a S2-component is exponentially
distributed, each component in S2 has a failure rate of its own. A
failed component has to wait for repair facilities, and the waiting
time has some general distribution such that each component has a
distribution of its own. All the repair time distributions in the
system are also governed by general probability laws.
The operational behaviour and reliability properties of the system
are studied under four different conditions. The conditions are
specified by two policy variables: the type of redundancy and the
policy to be followed in the system repair. Two types of redundancy
are considered. Chapter 4 illustrates parallel redundancy in S1: all
the M components in S1 start operating together as soon as the
system is put into operation and the system fails due to S1 only
when all the M components in S1 fail. In Chapter 5 stand-by
redundancy is considered: out of the M components in S1 only one
takes part in the operation and the rest are kept as stand-by
arrangement; the system is assumed to switch over to the next
component automatically when the operating component fails.
After a complete failure of S1 all the M failed components are
repaired; S2 is not touched. With regard to system failure due to the
failure of an S2-component two different repair policies are
considered: 1. Opportunistic policy: along with the repair of the
failed component in S2, repair of all the failures components in S1 is
also completed, and 2. Minimum policy: repair of the failed
component in S2 is carried out and the components in S1 are left
unattended.
Thus the treatment of the system is divided into four sections.
Section 4.1 deals with the system with parallel redundancy in S1 and
with minimum repair policy, in Section 4.2 there is parallel
redundancy in S1 and opportunistic repair policy, in 5.1 stand-by
redundancy and minimum policy, and in 5.2 stand-by redundancy
and opportunistic policy. In each case a method based on
supplementary variables and Laplace transforms is used to
formulate a mathematical model for the system. The first step in the
formulaiton of the model is to specify the set of states for the system.
Then the supplementary variable technique is used to obtain the
model's state equations, which consist of simultaneous partial
differential-difference equations with their boundary conditions.
With the help of Laplace transforms and generating functions both
transient and steady-state solutions for these state equations are
found. From these solutions the steady-state availability and the
Laplace transform of the transient state availability are derived.
Furthermore it is shown that the steady-state solutions (and
therefore the steady-state availability) are independent of the type of
waiting time and repair time distributions and in their expressions
only the expected values of these distributions appear. It is also
shown that the steady-state is achieved under quite general
conditions (the expected values of the distributions must exist) and
that the solutions for the steady-state can be found without any exact
knowledge about the distribution of the system.
In Chapter 6 there is a detailed analysis of the availability expressions
obtained and, in particular, a comparison of the availabilities
obtained in different circumstances. In this way an understanding is
obtained of the influence of, on the one hand, the structure of
subsystems S1 (types of redundancy in components) and, on the
other, the repair policy on the reliability characteristics of the system.
It is shown that in the case of stand-by redundancy availability is
always greater than with parallel redundancy and that the
opportunistic repair policy produces greater availability than the
minimum policy. Exact expressions are calculated for the availability
differences. Finally there is an examination of the influence of the
number of redundant components in subsystem S1 on availability.
(Licentiate thesis. Publications of The Turku School of Economics,
Series A I - 4:1977, 117 p.)