27
CHAPTER-I INTRODUCTION
CHAPTER-I
INTRODUCTION
1
1
Chapter-I
Introduction
With the advancement of modern science and technology there is an urgent call for
drastic reduction of equipments, weight and size of the system in addition to the
improvement in their efficiency, accuracy and durability. As a result new complex and
sophisticated systems are being developed at a drastic pace. During the last three decades
the human hands and brain designed and developed many sophisticated systems that
include software systems, satellite control systems, nuclear weapons control systems,
emergency power supply systems, telecommunication systems etc. The complexity,
sophistication and automation inherent in the systems had lead to reliability conception.
Since almost all systems may breakdown after some time which affect adversely not only
the country but the society as a whole. Therefore, the study of system reliability has
become not only inevitable but also of paramount significance. At present it is a matter of
great importance to produce and provide the cost-effective, efficient, user-friendly systems
along with the criteria of reliability.
The probabilistic theory of reliability has grown out of the demand of modern
technology and particularly out of the experience of the World War-II with complex
military system. Due to the complexity and automation of equipments used during the war
resulted in several problems of maintenance and repair.
It is in the early 1950’s that certain areas of reliability, especially life testing and
electronics and missile reliability problems started to receive a great deal of attention from
mathematicians and engineers. A commercial organization, Aeronautical Radio,
Inc.(ARINC), was set up the airlines of Ohio in 1950 which among several functions,
collected and analysed defective tubes and resulted in a remarkable success in improving
the reliability of different types of tubes. In 1952, the U.S. department of defense has
established the Advisory Group on reliability of Electronic Equipment (AGREE) and got
its first report published in 1957. Epstein (1958) worked in the field of life testing with
assumption of exponential distribution. After this the studies were carried out to evaluate
other measures besides finding reliability of the system.
The concept of availability is widely discussed in literature and the main
contributors are Barlow and Hunter (1960), Graver (1963), Sandler (1963), Myers et. al
(1964), Barlow and Proschan (1965), Rau (1970). Nakagawa (1976) considered the
2
2
replacement of the unit at a certain level of damage while Arora (1977), Mine and Kaiwal
(1979) enhanced the system reliability by assigning priority repair discipline.
Reliability of a system can be improved by using superior components with low
failure rates which will require more time and money for development. If the cost of
producing highly reliable components is very high, then the system reliability can be
improved by introducing the technique of redundancy. Nakagawa (1980) studied an
inspection policy for a standby electric generator as an example. Yamashem (1980)
worked on a multi state system with several failure modes and cold standby unit. Goel, et.
al (1984) studied a two unit cold standby system with two types of repair facilities.
Further, the cost of a system is most vital. There must be an optimum balance
between the reliability aspects of a product and its cost. The major items contributing to
the total cost are research and development, production, spares and maintenance. In order
to increase the reliability of the products, we would require a correspondingly high
investment in the research and development activities. The production cost would also
increase with the requirement of greater reliability whereas the total cost of maintenance
and spares would reduce with an increase in the reliability factor.
Systems were also analyzed with the concept of preventive /corrective maintenance
by various researchers. Preventive maintenance includes actions such as lubrication,
replacement of a nut or a screw or some part of the system, refueling, cleaning etc., while
corrective maintenance involves minor repair that may crop up between inspections. Goel,
et. al (1986), Murari and Al-Ali (1986), Goel and Murari (1990) worked on redundant
system subject to random, preventive and corrective maintenance.
Now it is well known that the performance of a system can be considerably
improved by carrying out preventive maintenance on-line. Further where shut down or off-
line maintenance is uneconomical, the system performance can be increased by restoring
to on-line preventive maintenance. Keeping this important aspect in view Gopalan and
Bhanu (1995) studied the cost analysis of a two unit repairable subject to on-line
preventive maintenance and /or repair.
Some other aspects applicable to the systems were also taken by authors. Kumar, et.
al (1996) carried out a comparative study of the profit of a two server system including
patience time and instructions. Gupta, et. al (1997) studied a system having super-priority,
priority and ordinary units. Taneja and Gupta (1999) studied a reliability model for two–
unit cold standby system with instructions and two types of repair. Gupta, et. al (2000)
studied a two-unit parallel system with repaired machine failure and correlated failure and
3
3
repair times. Kumar, et.al (2001) investigated a two-unit redundant system with the
degradation after first failure and replacement after second failure. Tuteja, et.al (2001)
studied reliability and profit analysis of two-unit cold standby system with partial failure
and two types of repairman. Reliability and profit analysis of some systems with different
types of repairman viz. ordinary and expert repairman, regular and visiting repairman are
studied by Rizwan and Taneja (2001), Sindhu and Gupta (2002) respectively.
Taneja and Nanda (2003) studied probabilistic analysis of a two-unit cold standby system
with resume and repeat repair policies. Singh and Chander (2005) analyzed reliability of
two systems each of which contains non-identical units-an electric transformer and a
generator.
Recently, some authors have carried out analysis of some systems on the basis of
collected real data. Tuteja, et. al (2006) discussed reliability and profit of a two-out of-
three unit system particularly for the case of ash handling plant consisting of three ash
water pumps. Taneja, et.al (2006) presented an economic analysis of a model on the
programmable logic controller (PLC) cold standby system. Parashar and Taneja (2007)
carried out the reliability and profit evaluation of a PLC hot standby system based on a
master-slave concept and two types of repair facility. Goyal, et. al (2009) given the
analysis of a two unit cold standby system working in a sugar mill with operating and rest
period.
Modern systems are becoming more and more compact and complex and the
average user is often unable to judge the quality of the system that he or she is buying. To
alleviate this uncertainty from the minds of system user, manufacturers offer some kind of
warranty with their system at the time of sale. Warranty has become an instrument, similar
to product performance and price, used in competition with other manufacturers in the
marketplace. The nature and extent of the warranty affect the sales, market share, costs
and profits of the business. Spence (1977) proposes that if a manufacturer offers a better
warranty than a competitor, then the reliability of the product should also be better to
reduce costs associated with warranty claims.
Blischke(1990) was the first to review papers on warranties and it dealt with
mathematical models for warranty cost analysis. Blischke and Murthy (1994) define
several costs of interest to manufacturers and buyers. Rinsaka and Sandoh (2001) deal
with the extension of the warranty period. Chukova, et.al (2004) presented an approach to
modeling imperfect repair in warranty analysis. Recently, Nakai (2007) has given an
4
4
extension of the Murthy (1992) model to the case of three usage states- idle, normal use
and abnormal use where the failure of the item is dependent on the mode of usage.
When a sophisticated system is installed the failure rate of the system is high due to
multiplicity of reasons such as design or manufacturing defects, improper installation,
environmental impact, mishandling, jerks during transportation, etc. However, on
removing all early failures (burn-in process) the failure rate of the system reduces to
almost a constant level then the system reaches its useful life period. On attaining certain
age the failure rate of the system again start to increases due to wear-out or aging of its
components (wear-out phase), e.g. in computer system hardware exhibits relatively high
failure rates often attributed to design or manufacturing defects early in its life, defects are
corrected and then failure rate drops to steady-state level for some period of time. As time
passes, the failure rate rises again as hardware components suffer from the cumulative
effects of dust, vibrations, abuse, temperature extremes and many other environmental
maladies, i.e. hardware begins to wear-out. In other words, a sophisticated system has
different stages of operational life with different failure rates. Further most of these
systems are under some warranty policy for some specified period provided by the system
provider. For instance, when a sophisticated system under failure free warranty policy for
some specified period of time is purchased from some company, service(s) of the system
or its component(s) which includes maintenance, repair or replacement of the system or its
component(s) are provided free of charges during the warranty period but after the expiry
of warranty period, service(s) taken by the user are charged.
A sophisticated system may be mechanical, electro-mechanical, electronic system
consisting of several components, e.g. computers, automobile engines, electrical
appliances, electronic devices, mobiles, combat aircrafts, missiles, satellites etc. Many
such one-unit or two unit systems are being used in day to day life in various industry,
companies and households. The sophisticated systems include emergency power systems
also.
An emergency power system is a type of system, which may include lighting,
generators, fuel cells and other apparatus, to provide back up power resource in a crisis or
when regular systems fail. They find uses in a wide variety of settings from residential
home to hospitals, scientific laboratories, data centers, telecommunication systems, power
plants, nuclear power systems and modern naval ships. In fact emergency power systems
were used as early as World War-II on naval ships. In combat, a ship may lose the
function of its steam engines, which power the steam driven turbines for the generator.
5
5
Mains power can be lost due to downed lines, malfunctions at a sub-station, inclement
weather, planned blackouts or in extreme cases a grid-wide failure.
Many of the problems experienced in the areas of data processing, communication,
closed loop instrumentation and on-line computers are the result of power related
problems such as temporary outages, momentary interruptions, surges, sags or noise.
Electronic devices such as computers, communication networks and other modern
electronic devices need not only power, but also a steady flow of it to continue to operate.
If the source voltage drops out significantly or completely, these devices get failed. To
achieve this, extra equipment such as surge protectors, invertors or sometimes a complete
uninterruptible power supply (UPS) is used.
UPS is an electrical apparatus that provides emergency power to a load when the
input power source, typically the utility mains, fails. A UPS differs from an auxiliary
emergency power system or standby generator in the sense that it provides instantaneous
or near- instantaneous protection from input power interruptions by means of one or more
attached batteries and associated electronic circuitry for low power users. The present
study uses the information/data gathered relating to various structures, functioning,
failures, repairs, etc. of UPS systems.
We now discuss some fundamental concepts related to reliability and to the
performance measures of the systems of interest:
Reliability
Reliability of a unit (or a product) is the probability that it will give satisfactory
performance for specified period under specified operating conditions.
Quantitatively, reliability of a device in time t is the probability that it will not fail
in a given environment before time t. If T is a random variable representing the time till
the failure of the device starting with an initial operable condition at t = 0, then reliability
R (t) of device is given by
R (t) = P[T > t] =1P [T t] =1F(t).
Thus, reliability is always a function of time. It also depends on environmental conditions
which may or may not vary with time. Following assumptions were made:
(i) R (0) = 1 since the device is assumed to be operable at t =0.
(ii) R (∞) = 0 since no device can work forever without failure.
(iii) R (t) is non–increasing function between 0 and 1.
6
6
Failures
It is recognized that there may be many contributing cause to a particular failure
and that, in some cases there may be no completely clear cut distinction between some of
the causes. We can than investigate problems of causes. We can than investigate problems
of prediction, appointment and assessment and present methods for arriving at solutions to
those problems which are both practical from the stand point case utilization and realistic
in the sense that the solutions are valid and useful.
In practice even the best design, manufacturing and maintenance efforts do not
prevent the system failure. In reliability theory there are four types of failures.
(i) Early Failure or Infant Mortality or Initial Failure
Early failures are those which can occur due to some fault in the part of assembly
resulting from a design, manufacturing or inspection deficiency, e.g. poor welds, poor
solder joints, poor connections and incorrect positioning of parts etc. Many of these
failures can be prevented by improving the control over the manufacturing process. These
can be eliminated by debugging burn-in process. The weak components fail in these early
hours of the equipment’s operation and they are replaced by good components.
(ii) Random Failure or Catastrophic Failure or Chance Failure
TThhiiss ffaaiilluurree ooccccuurrss aatt rraannddoomm iinntteerrvvaallss,, iirrrreegguullaarrllyy aanndd uunneexxppeecctteeddllyy.. TThheessee
ooccccuurrrreenncceess ccaann nnoott bbee pprreeddeeffiinneedd.. TThheessee aarree ssoo ccaalllleedd wwhhiicchh nneeiitthheerr ggoooodd ddeebbuuggggiinngg
tteecchhnniiqquueess nnoorr tthhee bbeesstt mmaaiinntteennaannccee pprraaccttiicceess ccaann eelliimmiinnaattee.. TThheessee aarree ccaauusseedd bbyy ssuuddddeenn
ssttrreessss aaccccuummuullaattiioonnss bbeeyyoonndd tthhee ddeessiiggnn ssttrreennggtthh ooff tthhee ccoommppoonneenntt..
((iiiiii)) WWeeaarr--OOuutt FFaaiilluurree oorr FFaattiigguuee FFaaiilluurree
AAfftteerr rraannddoomm,, aass ttiimmee ppaasssseess oonn,, tthhee uunniitt ggeettss oouuttwwoorrkk aanndd bbeeggiinnss ttoo ddeetteerriioorraattee
dduuee ttoo tthhee ggrraadduuaall cchhaannggee iinn tthhee vvaalluueess ooff tthhee ppaarraammeetteerr.. TThheessee aarree ccaauusseedd bbyy wweeaarr oouutt ooff
ppaarrttss aanndd ooccccuurr iinn tthhee eeqquuiippmmeenntt oonnllyy iiff iitt iiss nnoott pprrooppeerrllyy mmaaiinnttaaiinneedd.. TThhee ddeetteerriioorraattiioonn
rreessuullttss dduuee ttoo ccoorrrroossiioonn,, ffrriiccttiioonnaall wweeaarr aanndd sshhrriinnkkaaggee aanndd ccrraacckkiinngg iinn ppllaassttiiccss..
(( iivv)) OOuutt ooff TToolleerraannccee FFaaiilluurree
A fourth possible class is out of tolerance failure which can occur at any time
during the systems operating life. The associated failure distributions for out of tolerance
are the same as those for wear-out failure and mathematical methods used for evaluation
are similar.
7
7
Instantaneous Hazard Rate (or Failure Rate)
It is defined as the conditional probability that the system fails during the time interval
(t, t+t] given that it was operating during (o, t]. Let r( t )t = probability that the device
has life time between t and t+t , given that it has functioned up to time t.
= Pr[ t < T t + t | T > t]
= P [t<T t +t] P [T >t] = P [T < t+t ] – P [T<t] P[T> t]
=)t(R
)t(R)tt(R
)t(R
)]t(R1[)]tt(R1[
Now, the instantaneous failure rate or hazard rate r(t) at time t is defined by
R (t) =)(
)(
)(
)('
)()(
)()(lim
0 tR
tf
tR
tR
ttR
tRttR
where f(t) is defined as the p. d. f. of the device life time. If F(t) is the c. d. f. of failure
times, we have the following relations:
)t(F =
t
f (u) d u =R (t) = exp. [ t
0
r(u) d u ]
f (t) = r(t) exp. [ t
0
r(u) d u ]
SSyysstteemm CCoonnffiigguurraattiioonnss
By a system, we mean an arbitrary device having several units/subsystem
components which comprise the system assuming that their reliabilities are known which
help to predict the reliability of the whole system.
The units of a system may be connected in different ways to give us different
system configurations as follows:
((ii)) SSeerriieess ccoonnffiigguurraattiioonn
This is simplest configuration of units that form, a system in series combinations. A
series system with n components is a system which requires the functioning of all the n
components. If any one of the n components fails, the system fails. For example, the
aircraft electronic system consists of mainly a sensor subsystem, a guidance subsystem
and computer subsystem. This system can only operate successfully if all its components
operate simultaneously. The block diagram of a series configuration is shown as follows in
fig.1.1
8
8
Fig. 1.1 Series Configuration
Let Ri (t) be the reliability of the ith component, we shall assume that the life times of
the n components are independently distributed with densities fi(t),
i =1, 2,------, n (the corresponding distribution functions are given by F i (t), i = 1 ,2,-------,
n ).Then the system reliability is given by
R(t) = Pr [ T > t] = Pr [min. (T1, T2, T3, -----------, Tn )> t ]
=
n
1i
P [Ti > t] =
n
1i
R i( t ) .
where Ti is the life time of the ith unit of the system. The system hazard rate, therefore, is
r( t ) =
n
1i
r i (t) where , ri ( t ) is the instantaneous failure rate of the ith unit.
(ii) PPaarraalllleell CCoonnffiigguurraattiioonn
In this configuration all the units in a system are connected in parallel. The failure
of the system occurs only when all the units of the system fail. For example, four engine
aircraft which is still able to fly with only two engines working. Block diagram
representing parallel configuration is shown in fig. 1.2
Fig. 1.2 Parallel configuration
Unit 1
Unit 2
Unit n
INPUT OUTPUT
Unit 1 Unit 2 Unit 3 Unit nINPUT OUTPUT
9
9
If Ri(t) and Ti be the reliability of the ith component and the life time of the ith unit
in time t respectively, then the system reliability is given by
R (t) = Pr (T >t) =Pr [max.( T1 , T2 , T3 , --------,T n ) >t ]
= 1P[ T1 t , T2 t , T3 t ,------, T n t ]
If the units function independently, then
R( t ) =1 [1R1( t )] [1R2( t )] [1R3( t )] -----[1R n( t ) ] = 1
n
1i
[1R i( t )]
(iii) Standby Redundant Configuration
A system in which more units are made available, for performing the system
function than are actually required is called a redundant system. A standby redundant
system with n components is a system in which the components operate one at a time and
the other (n-1) units are kept as spares. When the operating unit fails, a unit from the spare
pool is switched on line. This process continues till all (n-1) standby units have been
exhausted. Such a system is also known as sequentially redundant system of order n. The
system functions as long as one of the (suitable) units is available for the specific task in
hand. Thus redundancy is a device to improve reliability of a system. A block diagram of
such system is shown as in fig. 1.3 and 1.4
Fig. 1.3 Mixed parallel configuration
Fig. 1.4 Standby redundant configuration
INPUT
Unit 1 Unit 2 Unit 3
Unit 1 Unit 2 Unit 3
Unit 4
Unit 4
OUTPUT
Unit 1
Unit 2
Unit n
INPUT
OUTPUT
10
10
Gnedenko, classified the redundancy on the basis how the units are loaded in the
standby units in two ways:
(i) Active redundancy and (ii) Passive redundancy
(i) Active Redundancy - In this redundancy the system has a positive probability of failure
even when it is not in operation due to the effect of temperature, environment conditions
etc. This can be further classified as:
(a) Hot standby redundancy and (b) Warm standby redundancy.
(a) If the units are loaded exactly the same way as the unit operating on line, they are
termed as hot standby. In other words the probability of failing of a hot standby is the same
as that of the unit operating on line.
(b) If the off-line unit can fail and can diminish the load, it is called warm standby
unit. The probability of failure for a warm standby unit is less than that of a unit
operating on line.
(ii) Passive or Cold Standby Redundancy - If the off-line unit can not fail and is
completely unloaded, it is called cold standby unit. It can never fail while in the standby
state.
Reliability R(t) of an n-unit standby system at any time instant t is given by
R(t) = P[
n
1i
Ti > t ]
where Ti is the life time of the ith unit and all the n-units are independent.
SSttoocchhaassttiicc PPrroocceesssseess
Families of random variables which are functions of say, time are known as
stochastic processes. The state space of a stochastic process is a list of possible outcomes
at any time point. At every time point, the process is in exactly one of its possible states.
Thus a stochastic process has a state space and a time structure.
If X denotes the state of the system at time t0 and the set E describes certain
collection of states in the system, then the probability that the system which is at time t0 is
in the state X will pass into one of the states of E at time t, is denoted as P{ t0 ; t , E} .
Then the process of changing from one state to another with time having the probability
P{t0; t, E} is known as stochastic process.
Markov Process
A stochastic process is said to be Markov process if the future state is completely
determined by the present state and is independent of the way in which the present state
11
11
has developed. In this process, the state at time tn is only influenced by the state of the
process at time tn-1.
Thus a sequence of trials X1, X2, X3, ------, is said to constitute a Markov
process if the following properties are satisfy.
(i) Each outcome belongs to a finite set of outcomes {E1, E2, ------, E m}, known as
state space of the system and the outcome Ej at the j th trial is defined as the state
Ej of the system at time j.
(ii) The outcome of any trial depend at most upon the outcome of the immediately
preceding trial and not upon any previous outcomes, i.e. with each pair of states
(Ei, Ej), there correspond a probability pij that outcome Ej happens immediately after Ei.
The values of pij for different combinations of i, j = 1, 2 ,-------, m are known as ‘transition
probabilities’ in a single step. These can be arranged in a matrix form, known as single
step transition matrix P .
(iii) The process has a set of initial probabilities.
Initial state Final state after one step
1 2 3 -------------------- m
1 p11 p12 p13 ------------------p1m
P = 2 p21 p22 p23------------------p2m
3 p31 p32 p33 ------------------p3m
- - - - -------------------
m p m1 p m2 p m3----------------pm m.
MMaarrkkoovv CChhaaiinn
A Markov process with discrete state space is said to be a Markov chain. It is a
special case of Markov process. It is used to study the short and long behaviour of certain
stochastic system.
Mathematically, let E1, E2, -------Ej (j = 1, 2, ------) represent the exhaustive and
mutually exclusive outcomes (states) of a system at any time. Initially, at any time t0, the
system may be in any of these states. The Stochastic process {Xn, n = 0,1,2,-------} is
called a Markov Chain if
P{Xn = k / X n-1= j, X n-2 = j1, 2,------, X0 = j n-1}
= P{X n = k / X n-1 = j} = pjk, when ever the first member is defined.
12
12
The transition probabilities pjk are the basic to the study of the structure of the
Markov chain. The transition probability may or may not be independent of ‘n’. If the
transition probability pjk is independent of n the Markov chain is said to be homogeneous
(or to have stationary transition probabilities). If it is dependent on ‘n’ the Markov chain is
said to be non-homogeneous.
The transition probability pjk refer to the state (j, k) at two successive trials (say, nth
and (n+1)th trials). The transition is one step and pjk is called one–step transition
probability. In more general case, we are concerned with the pair of states (j, k) at two
non–successive trials, say, state j at the nth trial and state k at the (n+m)th trial. The
corresponding transition probability is then called m-step transition probability, denoted
by pjk(m), i .e.
pjk(m) = P{ Xn + m = k / Xn = j }.
RReenneewwaall PPrroocceessss
Suppose we have a repairable system which starts operation at t = 0. If X1 denotes
the time to first failure and Y1 denotes the time from first failure to next system operation
(after repair) then t1= X1+Y1 denotes the time of first renewal. Similarly, if X2 denotes the
time from first renewal to second failure and Y2 denotes the time from second failure to
second renewal then t2 = X2+ Y2 and the time of second renewal is t1 + t2.
In general t i = Xi + Yi (inter arrival time between the ( i -1)th and ith renewal ) for
i = 1, 2, -------, if we define S0 = 0, Sn = t1 + t2 + t3 +------+ t n
= epoch of the nth renewal
and N( t ) = number of renewal during (o , t ] then the process N(t), t ≥ 0 is called renewal
process.
MMaarrkkoovv--RReenneewwaall PPrroocceessss
Let the state of a process be denoted by the set E = { 0,1,2,------}and let the
transitions of the process occur at epochs t0 (= 0), t1 , t2 , -----, ( t n < t n+1 ). If
P{Xn+1 = k, tn+1– t n t / X0 = i0,----,X n = in ; to , t1 , -----, tn }
= P{Xn+1 = k, t n+1–t n t / Xn = in}
then {Xn , tn} , n = 0 , 1 , 2 , -------- is said to constitute a Markov renewal process with
state space.
13
13
Semi-Markov Process
In the above, assume that the process is time homogeneous i.e.
P{Xn+1 = j, tn-1 – tn t / Xn = i} = Qij, i, j S is independent of n , then there exists limiting
transition probabilities
Pij = limt® ¥
Q ij(t) = P {Xn+1 = j / Xn = i}.
Then {Xn, n = 0, 1 ,2 ,--------} constitute a Markov chain with state space E and
transition probability matrix (t. p .m.) is given by P = ( pi j ).
The continuous parameter Stochastic process Y (t) with state space E defined by
Y(t) = Xn, tn < t < tn+1
is called a semi-Markov process.
In other words, we define the semi-Markov process as a process in which the
transition from one state to another is governed by the transition probabilities of a Markov
process but the time spent in each state before the transition occurs is a random variable
depending upon the last transition made. Thus at transition instants the semi-Markov
process behaves just like a Markov process. However the time at which transition occurs
are governed by a different probability mechanism.
Regenerative Process
A time point at which the system history prior to the time point is irrelevant to the
system conditions is called a regenerative point. Regenerative Stochastic processes were
introduced by Smith (1955) and has been crucial in the analysis of complex system. Let
X(t) be the state of the system at epochs. If t1, t2, -------- are the epochs at which the
process probabilistically restarts, then these epochs are called regenerative epochs and the
process {X(t) , t = t1 , t2 , ----- } is called regenerative process.
FFiirrsstt PPaassssaaggee TTiimmee
Suppose that a system starts with the state j, then time taken to reach a given state k
for the first time from state j is called first passage time. In general, first passage time is a
measure of how long it takes to reach a given state from another state.
MMeeaann SSoojjoouurrnn TTiimmee iinn aa SSttaattee
The expected time taken by the system in a particular state before transiting to any
other state is known as mean sojourn time or mean survival time in that state. If Ti be the
sojourn time in state i, then mean sojourn time in state i is
14
14
i =
0
P (Ti > t)d t
MMeeaann TTiimmee ttoo SSyysstteemm FFaaiilluurree ((MMTTSSFF))
Components and systems (simple or complex) do not operate in the same manner
in all condition. They can not operate for an infinitely long time due to aging of
components or some other reasons (initial failure due to manufacturing defects, random
failures due to change in working stresses or environment conditions, etc.).To avoid the
sudden failure, one must be interested in measure representing the life time of the system.
This measure is aptly described as the Mean Time to System Failure (MTSF), as it
corresponds to the average duration between successive system failures. This measure is
defined as the expected time for which the system is in operation before it completely
fails.
Suppose ‘T’ be life time of the system then the reliability function for the system is
given by R(t) =1 F(t), where F(t) is the failure time distribution function and f(t) =
dt
)t(dFis the failure time density function. The mean time to system failure is given by
MTSF =
0
t f (t) d t
=
0
dtdt
)t(dRt
=
0
R (t) d t =0s
lim
R* (s)
where R*(s) is the Laplace transform of the reliability function R(t).
Let 0(t) be the cumulative distribution function of the first passage time from initial
state to a failed state, then
R*(s) =s
)]s*(*1[ 0
Where 0**(s) is the Laplace Stieltjes transform of the 0(t)
Thus, we have
MTSF =0s
lim s
)]s*(*1[ 0 .
15
15
MMeeaann IInnssttaallllaattiioonn TTiimmee ((MMIITT))
Mean installation time is defined as the expected time required for proper
installation of the system. Proper installation of the system means screening out
completely all early failures of the system which may be due to manufacturing defects,
environmental effects, defects in the design of the system, improper assembly, jerks during
transportation, mishandling etc. This is a part of burn-in process and after this useful life
period of the system starts. The presence of an expert of the system is desirable during
installation period so that the system may start working properly at the earliest.
Let I0(t) be the cumulative distribution function of the first passage time from
initial installation state to after proper installation state. If I0 denotes the mean installation
time (MIT) of the system then in the steady state it is given by
I0 =0s
lim s
)]s*(*I1[ 0
AAvvaaiillaabbiilliittyy
The availability A(t) is the probability that the system is operating satisfactorily at
time t. When a system is often unavailable due to break downs, the concerning department
becomes interested to put it back into operation after each break down with proper repairs.
In fact, it is concerned with availability equally as it does with reliability because of
additional costs and inconvenience incurred when the system is not available. The
differences between the measures of reliability and availability are as follows:
(a) The reliability is an interval function while the availability is a point function
describing the behaviour of the system at a specified epoch.
(b) The reliability function precludes the failure of the system during the interval under
consideration, while availability function does not impose any such restrictions on the
behaviour of the system.
Availability may be categorized as:
(i) IInnssttaannttaanneeoouuss ((ppooiinntt wwiissee)) AAvvaaiillaabbiilliittyy
This is the probability that the system will be able to operate within the tolerance at a
given instant of time.
Let X(t) = 1, if the system is operable at time t, and X(t) = 0, when it is not operable.
The availability A(t) of the system at time t is given by
A (t) = P[X(t) = 1/ X(0) = 1].
16
16
(ii) AAvveerraaggee ((IInntteerrvvaall)) AAvvaaiillaabbiilliittyy
It is the expected fraction of a given interval of time that the system will be able to
operate within tolerances.
Suppose the given interval of time is (0, T]. Then the interval availability
H (0,T] = A(t) for this interval is given by
A (t) =T
1T
0
A ( t ) d t.
(iii) SStteeaaddyy--SSttaattee ((LLiimmiittiinngg IInntteerrvvaall)) AAvvaaiillaabbiilliittyy
It is the expected fraction of time in the long run that the system operates
satisfactory. To obtain steady state availability we simply compute
tlim H (0, T] =
tlim A (T)
The Average availability is recommended for those systems which are used only for
some time interval and is continuously used in that interval. Instantaneous availability
may be the best measure for the systems which are required to perform a function for a
random time. The steady state availability is used for the systems which operate
continuously, for example, radar system.
Maintainability
Maintainability is an indices associated with a system under repair. It is the
probability that the system will be restored to operational effectiveness within a specified
time when the maintenance is taken in accordance with prescribed conditions.
Maintenance is one of the effective ways of increasing the reliability of a system. It is
considered to be beneficial when the repair cost in terms of time and money spent is
considerably low as compared to the cost of the equipment. A low repair time will
minimize the ill-effects of the failure. Maintenance of a system is of two types:
(i) Preventive Maintenance (P. M.), and
(ii) Corrective Maintenance (C. M.)
(i) Preventive Maintenance
In this category equipment may be maintained by replacing or repairing components
prior to the possible occurrence of a failure. It includes actions such as lubrications,
replacement of a nut or a screw or some part of the system, refueling, cleaning, etc.
(ii) Corrective Maintenance
When a system fails to work, repair and adjustment are started immediately to put it
in operable conditions. When such a failure will occur we can not for see exactly and so this
17
17
is called un-scheduled maintenance. It involves minor repairs that may crop up between
inspections.
On failure of a unit, it is sent to a repair facility if available, other wise it queues up for
repair. There may be two types of repair policies as follows:
(a) Repeat repair policy
Due to certain reasons the repair of a failed unit has to be stopped. When the
repair begins again, it is started all over again.
(b) Resume repair Policy
The repair of a failed component is terminated before completion due to one
reason or the other. When it begins again, it is started from the stage where it was prior to
the termination of repair.
Repairable Systems
If on failure, a unit is replaced by a new one, then the reliability of the system increases.
In a good number of cases this will turn out to be expensive and it will be necessary to
repair the failed units. Thus on the failure of a unit, it is sent to a repair facility. If no repair
facility is free, then the failed units queue up for repair and the repairs are normally
undertaken in First in First Out (FIFO) order. We assume that the life time of an on-line
unit, standby and the repair time of a failed unit are all independent random variables and
that the distribution functions of these random variables are known and that they admit the
probability density functions.
Busy Period
Let B (t) be the probability that a repairman is busy with the system in the interval
(0,t] Then in the long run total fraction of time for which a repairman is busy is given by
B =tlim
t
)t(B
Expected Number of Visits by the Service Engineer/Repairman to the System
Let V(t) be a random variable representing the number of times the repairman has
visited the system in the interval ( 0, t ], then the expected number of visits by the repairman
to the system in (0, t] is E [V(t)] and in the long run the expected number of visits per unit
time is given by
18
18
V =tlim
t
)]t(V[E.
Expected Number of Repairs
Let R (t) be a random variable representing the number of times the units are repaired
in (0,t], then the expected number of repairs in ( 0,t ] is E [R(t)] and in the long run the
expected number of repairs per unit time is given by
R =tlim
t
)]t(R[E.
Expected Number of Replacements
Let RP (t) be a random variable representing the number of times the units are replaced
in (0, t], then the expected number of replacements in (0, t ] is E [RP(t)] and in the long run
the expected number of replacements per unit time is given by
RP =tlim
t
)]t(R[E p .
Expected Number of Preventive/ Corrective Maintenances
Let Ei(t) be a random variable representing the number of times the units are
maintained in (0,t] , then the expected number of maintenances in ( 0, t ] is E[Ei(t)] and in
the long run the expected number of maintenances per unit time is given by
Ei =tlim
t
)]t(E[E i .
Warranty Concept
A warranty can be defined as an assurance from a manufacturer/system provider to
a system user that the system sold is guaranteed to perform satisfactorily up to certain length
of time, which is the warranty period. In other words, warranty is a contract offered by a
manufacturer/system provider to system users/buyers to replace or repair a faulty item, or to
partially or fully reimburse the user in the event of a failure. Warranties are very widespread
and serve many purposes including protection for manufacturer, user and signals of quality
and reliability of the system and element of marketing strategies.
19
19
Role of warranty
(i)System user’s/buyer’s point of view
The main role of a warranty in these transactions is protectional. Specifically, the
warranty assures the user that faulty item will either be repaired or replaced at no cost or at a
reduced cost. A second role of warranty is informational.
(ii) Manufacturer/System provider’s point of view
One of the main roles of warranty from the system provider’s point of view is
protectional. Warranty terms specify the use and conditions of the use for which the system
is intended. They also provide for limited coverage or no coverage at all in the case of
misuse of the system/ product.
A second important purpose of warranty is promotional. As user often infers a more
reliable system when a long warranty is offered, this has been used as an effective
advertising tool.
More recently, warranty has been viewed as both an insurance policy and a repair
contract. This has given rise to a third theory of warranty, the investment theory, under
which the warranty is seen as an investment by the user to reduce the risk of early failure.
Three common warranty policies which are frequently used by manufacturers are as
follows:
(i)Ordinary Failure Free Warranty Policy
Ordinary failure free warranty policy is commonly used for repairable products.
Under this policy, the items are replaced/repaired free of cost to the buyer in the warranty
period. Repair returns the failed item to the average condition of a working unit of its age.
(ii)Unlimited Failure Free Warranty Policy
In unlimited failure free warranty policy, the failed items are replaced/ repaired free
of cost as in case of ordinary failure free warranty. But each replaced / repaired item carries
a warranty identical to new purchased item.
(iii)Pro-Rata Warranty Policy
In pro-rata warranty policy, a failed item is replaced by a new one or is repaired at a cost
prorated to the age of the failed item.
Warranty cost
For products sold with warranty the manufacturer incurs additional cost resulting
from the servicing of claims under warranty. Warranty claims occur due to item failures
which are influenced by several factors /reasons such as the engineering decision during
20
20
design and manufacturer that determine the inherent reliability of the system, the usage
intensity and environment and the maintenance effort expended by users.
Despite the fact that warranties are so commonly used, the accurate pricing of
warranties in many situations remains an unsolved problem. This may be surprising that the
fulfillment of warranty claims may cost companies large amounts of money. The data
relevant to the warranty costs in a particular industry are usually highly confidential, since
they are commercially sensitive.
Profit Analysis
Any manufacturing industry is basically a profit making organisation and no
organisation can serve for long without minimum financial returns for its investment. There
must be an optimal balance between the reliability aspect of a product and its cost.
Availability of the system leads to the revenue whereas the busy period of the
repairman, expected number of replacements, etc. leads to the cost of maintenance and
spares. Therefore, profit analysis is an important aspect in the field of reliability and
depends upon production cost of maintenance and spares, failure rates, repairman employed,
accidents, cost of calling repairman, etc.
The revenue and cost function leads to the profit function of a firm, as the profit is
excess of revenue over the cost of production. The profit function takes the form
P (t) = Expected revenue in (0, t] expected total cost in (0, t].
Let us consider a system which involves only the following costs:
C0 = revenue per unit up time of the system.
C1 = cost per unit time for which the repairman is busy.
C2 = cost per visit of the repair man.
A = the total fraction of time for which the system is up.
B = the total fraction of time for which the repairman is busy.
V = expected number of visits of the repairman.
Then the expected profit in the steady-state is given by
P = C0 A C1 B C2 V.
When the user invests money for a system, the costs and benefits occuring from the
investment will continue for a number of years. When the system is put to use, the user has
to spend money on the operating cost, failure and associated costs and cost of preventive/
21
21
corrective maintenance during the life-cycle of a system. Then the expected profit (P1) is of
interest to be computed in steady state.
Any effort on the part of manufacturer to increase the reliability of the system will
increase quality costs and internal-failure costs. However, after some time internal costs
namely, investigation cost, testing and measuring costs, training and management cost, etc.
will start decreasing. The external costs, namely, after-sale service costs and maintenance
costs for the warranty period will show a decline with an increase in reliability. Thus the
system provider/manufacturer may be interested to obtain its expected profit (P2) in steady
state.
Transforms and Convolutions
(a) Laplace Transform:
Let f(t) be a function of a positive real variable t. Then the Laplace transform (L.T.) of
f(t) is defined as
L [f(t)] = f *( s ) =
0
ste f (t )d t
for the range of values of ‘s’ for which the integral exists. Here f(t) is called an inverse
Laplace transform of f*(s) and we write f(t) = L -1[f *(s)].The following are some important
properties of Laplace transform:
(i) L[
n
1i
c if i(t)] =
n
1i
c i f*(s)
(ii) L[ t n f(t)] = (1) n d n f *(s) /d s n
(iii) L [ t
0
f(u)d u] = L[F(t)] = f*(s) / s
(iv)0t
lim
f(t) =0s
lim
s f*(s) (initial value theorem )
(v)t
lim F(t) =0s
lim
s f*(s) (final value theorem )
(vi)0s
lim
f* (s) =1 if f*(s) is L. T. of a p. d. f.
(b) Laplace Stieltjes Transform:
Let X be non-negative random variable with distribution function
F(x) = Pr [X x],
22
22
Then the Laplace Stieltjes Transform (L. S. T.) of F(x) is defined , for s > 0 by
F** (s) =
0
ste d F (x).
Therefore, we have
F** (s) =
0
sxe f(x) d x = f*(s), where f(x) =dx
)x(dF.
Convolution
Let f (t) and g(t) be two real valued non-negative continuous functions of t, then the
integral
t
0
f(tu) g(u) d u = t
0
g(tu) f(u) du = f(t) © g(t) = L-1 [ f*(s) . g*(s) ]
is called Laplace convolution of the functions f(t) and g(t).
If F(t) and G(t) be two real valued distribution functions defined for t > 0 the resulting
convolution is again a distribution function and the integral
t
0
F (tu) d G(u) = t
0
G (tu)d F(u) =F(t) G(t)
is known as Stieltjes Convolution of F(t) and G(t).
Distribution Used
We have used exponential distribution in the present work. Exponential distribution
plays an important role in reliability studies. Besides a number of desirable mathematical
properties, it has a very important Memory Less Property i.e. if the life length T of a
structure has the exponential distribution, previous use does not affect its future life length.
For example, an electric fuse (assuming it can not melt partially) whose future life
distribution is practically unchanged as long as it has not yet failed. In most cases, the time
to failure of components obeys exponential distribution. The family of exponential
distribution is the best known and most thoroughly explored, largely through the work of
Epstein (1958) and his associates.
23
23
The exponential distribution is defined as follows:
A continuous random variable having range 0 x < is said to have an exponential
distribution if it has the probability-density function of the form
f (x) = xe , 0 < x <
= 0, x < 0, where is a positive constant.
The corresponding distribution function is
f (x) = 1 xe , 0 x <
= 0, x < 0.
The hazard rate ‘’ is constant. The Laplace Transformation of p. d. f. of exponential
distribution iss
.
Thesis at a Glance
The present thesis entitled “Reliability and Profit Analysis of some Models on
Sophisticated systems” is an attempt to analyse the reliability and profit of some modern
sophisticated systems by computing various measures of system effectiveness. A
sophisticated system may be mechanical, electro-mechanical, electronic system e.g.
computer, automobile engine, electric motor, electric meter, electric boilers, refrigerators,
washing machine, UPS, PLC system, combat aircrafts, missiles, satellites etc. Many such
one-unit or two unit systems are being used in day to day life in various industry,
companies and households. These systems may have different operational stages, different
types of failures and repairs. Further, most of the systems are under some warranty policy
for some specified period provided by the system provider. The present study investigates
various stochastic models corresponding to such one-unit and two- unit cold standby
systems considering different operational stages with different failure rates. The main
emphasis is on cost optimization and reliability. The work done is useful to study life
times, improvement in reliability, minimization of operational costs, reliability and profit
analysis of various systems viz. emergency power systems, computer systems, nuclear
power plants, communication network, combat aircraft, missiles and satellites etc. The
thesis consists of eight chapters. Here is a brief summary of these chapters:
Chapter I includes the fundamental concepts and definitions related to the work
done in the thesis to make it sufficient in itself.
24
24
Chapter II deals with the information/data collected related to a sophisticated
system-UPS (uninterrupted power supplier) system to develop models for such systems.
For the purpose, a manufacturing plant of some electrical appliances such as UPS system,
electric meters etc. was visited and information/data was collected regarding structure,
functioning, failure, repair and various operational stages of UPS system from the plant,
company area sales manager, system distributor/supplier and the service engineer.
Chapter III analyses a single unit sophisticated system under warranty having
different operational modes. It is assume that the failure rate of the unit during its
installation is high (burn-in period) and after the proper installation the failure rate
decreases, i.e. the unit improves and reaches to its useful life period. After acquiring
certain age the failure rate of the unit starts increasing, i.e. the unit deteriorates, due to
wear out of its components (wear-out period). It is assumed that the unit is under ordinary
failure free warranty policy up to its useful life period. Under the warranty policy, during
the warranty period, the system provider through its service engineer provides service(s)
free of charges to the system user that includes maintenance; repair or replacement of the
unit or its component(s). But on expiry of the warranty period, service(s) taken by the
system user is charged by him. Two models are analysed here. In model-I, it is assumed
that the unit on deterioration fails completely whereas in model-II the unit works with
higher failure rate. Moreover, while analysing model-I, two operational modes of the
system viz. the system during warranty period and after the expiry of the warranty period
or wear-out period are considered whereas in the model-II, three operational modes/stages
viz. during proper installation or burn-in process (Stage-I), useful life period (Stage-II) and
wear-out period (Stage-III) are taken.
Chapter IV investigates a single unit sophisticated system under ordinary failure
warranty policy having three operational stages with the possibilities of occurrence of two
types of faults viz. minor faults and major faults during each operational stage of the
system. It is assumed that an occurrence of the minor fault leads the system either to a
down state or to a degraded state whereas a major fault leads to complete failure of the
system. Further, on the occurrence of minor faults, on-line/ off-line repairs whereas on the
occurrence of major faults replacements of the unit is carried out by the service engineer.
During each of its operational stages the unit or its component(s) may requires a number
of repairs/replacements and the unit requires re-installation process after each replacement.
Two reliability models have been taken up. In model-I, the occurrence of a minor fault
lead the system to a down state and online repair is done whereas in model-II the
25
25
occurrence of a minor fault degrades the system and the possibilities of both online as well
as offline repairs are considered.
Chapter V also analyze a single unit warranted sophisticated system having above-
mentioned three operational stages considering two types of service facilities, viz. service
engineer during warranty period (i.e. Stage-I and Stage-II) and a repairman immediately
available on payment basis (called available repairman) after the expiry of the warranty
period (i.e. Stage-III). The available repairman takes negligible time to reach the system.
Here also two reliability models for the system have been studied. In model-I, on-line
repair on occurrence of a minor fault and off-line replacement of the unit on occurrence of
a major fault are taken into consideration. In model-II, it is assumed that on failure of the
system due to occurrence of a major fault the service engineer/available repairman first
inspects the system to detect whether the fault(s) occurred in the component(s) is
repairable/ ir-repairable and then repair/replacement of the component(s) or
replacement/re-installation of the complete unit is done.
Chapter VI deals with the analysis of a two-unit cold standby system wherein units
have three operational stages i.e. proper installation or burn in period (Stage-I), useful life
period (Stage-II) and wear out period (Stage-III). It is assumed that upon failure of an
operative unit, the cold standby unit becomes operative instantaneously and the failed unit
goes under repair in any of its operational stages. There is a single service facility that
carries out the service/repair of the unit on first come first served (FCFS) pattern. It is also
assumed that upon failure, the unit is repairable during its first two operational stages and
is irreparable at the third operational stage.
Chapter VII investigates a two-unit cold standby sophisticated system wherein
each unit undergoes three stages of its operation as in previous chapter with priority of
operation to a unit operating at its second stage. As in the system the failure rate of the unit
operating during its second stage, i.e. during useful life period, is lower in comparison to
the failure rate at the other two operational stages. It, therefore, seems logically better to
give priority to a unit operating in second stage over the unit working in any other stage.
Chapter VIII gives the comparative analysis of models (taken two at a time)
already studied in the preceding chapters with respect to their profits evaluated in these
chapters. Some conclusions regarding the models have been drawn on the basis of
graphical study. Graphs have been plotted for difference of the profits of the model with
respect to various rates/costs/revenue keeping fixed the other rates/ costs/revenue/
probabilities.
26
26
The various models discussed in the thesis have been analysed by making use of
semi-Markov processes and regenerative point technique. Various measures of the system
effectiveness such as mean installation time, mean time to system failure(MTSF), steady-
state availability, busy period analysis of service engineer/repairman, expected number of
maintenance, expected number of repairs, replacements, inspections and expected number
of visits by the service facilities have been obtained. Profits for both the system user and
system provider are evaluated for each model discussed in chapter III to V. Graphical
study is also made for the reliability and profit analysis of each of the models in the
chapters from III to VII.
----- o -----
