Generating Renewal Functions of Uniform, Gamma, Normal and Weibull Distributions for
Minimal and Non Negligible Repair by Using
Convolutions and Approximation Methods
by
Dilcu Helvaci
A dissertation submitted to the Graduate Faculty of
Auburn University
in partial fulfillment of the
requirements for the Degree of
Doctor of Philosophy
Auburn, Alabama
December 14, 2013
Keywords: Reliability, Renewal Function, Renewal Intensity, Convolutions
Copyright 2013 by Dilcu Helvaci
Approved by
Saeed Maghsoodloo, Chair, Professor Emeritus of Industrial and System Engineering
Alice Smith, Professor of Industrial and System Engineering
Mark Carpenter, Professor of Statistics
Fadel M. Megahed, Professor of Industrial and System Engineering
ii
Abstract
This dissertation explores renewal functions for minimal repair and non-negligible repair
for the most common reliability underlying distributions Weibull, gamma, normal, lognormal,
logistic, loglogistic and the uniform. The normal, gamma and uniform renewal functions and
the renewal intensities are obtained by the convolution method. In the uniform distribution case
complexity becomes immense as the number of convolutions increases. Therefore, after
obtaining twelve convolutions of the uniform distribution, we applied the normal approximation.
The exact Weibull convolutions, except in the case of shape parameter , as far as we know
are not attainable.
Unlike the gamma and the normal underlying failure distributions, the Weibull base-line
distribution does not have a closed-form expression for the n-fold convolution. Since the
Weibull is the most important and common base-line distribution in reliability analyses and its
renewal and intensity functions cannot be obtained analytically, we used the time-discretizing
method. Most calculations have been done with the aid of MATLAB Programming Language.
When MTTR (Mean Time to Repair) is not negligible and that TTR has a pdf denoted as
r(t), the expected number of failures, expected number of cycles and the resulting availability
were obtained by taking the Laplace transforms of renewal functions. Finally, the approximation
method for obtaining the expected number of cycles, number of failures and availability using
raw moments of failure and repair distribution is provided.
Keywords: Reliability, Renewal Function, Renewal Intensity, Convolutions
iii
Acknowledgements
The author is grateful to Dr. Saeed Maghsoodloo for his guidance, encouragement,
meticulous attention to details, insight and agreement to oversee this work. The author owes this
very dissertation effort and Ph.D. completion to Professor Emeritus Saeed Maghsoodloo?s
generosity, motivation, patience and wisdom. Dr. Maghsoodloo has always been more than an
advisor; he is a dear friend and the author is thankful to him for his efforts.
The author must also recognize Dr. Alice Smith for her demanding, but continuous
support for this effort. Dr. Alice Smith has contributed immensely in the academic growth of the
author through the efforts, assignments, mentoring and opportunities that she provided to the
author. The author also wants to thank Dr. Mark Carpenter for his agreement and cooperation on
this research. In addition, the author must recognize Dr. Fadel Megahed for his positive support
and guidance.
This work is dedicated to Filiz Kaya (mother) and Dr. Jeremy L. Barnes (Husband).
Without their support, this work would be more difficult. The author shares his deepest gratitude
to all of these people.
The author wants to send appreciation to the Turk Petrol Foundation and Professor Zikri
Altun for supplying the author financial support that was critical for the completion of this
degree. The author wants to add a debt of gratitude for all the friends and colleagues that
encouraged the continuation of this research, despite all odds and setbacks.
iv
Table of Contents
Acknowledgements ........................................................................................................................ iii
List of Tables ............................................................................................................................... viii
List of Figures ................................................................................................................................ ix
1 Introduction .................................................................................................................................. 1
1.1 History of Reliability ........................................................................................................ 1
1.2 Research Objectives ......................................................................................................... 2
1.3 Research Methods ............................................................................................................ 3
1.4 Dissertation Layout .......................................................................................................... 4
2 Literature Review......................................................................................................................... 6
2.1 Reliability ......................................................................................................................... 6
2.2 Availability and Maintainability ...................................................................................... 7
2.3 Reliability versus Quality ................................................................................................. 9
2.4 Reliability Applications.................................................................................................... 9
2.5 Renewal Processes ......................................................................................................... 10
2.6 Importance of Renewal Functions.................................................................................. 11
2.7 Previous Work on Renewal Functions ........................................................................... 12
2.8 Contribution of this work to the Literature .................................................................... 15
2.9 Literature Review Summary .......................................................................................... 17
3 Methodology .............................................................................................................................. 18
v
3.1 Reliability Measure ........................................................................................................ 18
3.1.1 The Reliability (or Survivor) Function R(t) ................................................................... 18
3.1.2 The Mean Time to Failure .............................................................................................. 20
3.1.3 The Hazard (or the Failure Rate) Function h(t).............................................................. 21
3.2 Renewal Intensity Function ............................................................................................ 23
3.3 Convolutions .................................................................................................................. 24
3.4 The Weibull Distribution ............................................................................................... 25
3.5 The Normal Distribution ................................................................................................ 29
3.6 The Gamma Distribution ................................................................................................ 31
3.7 The Uniform Distribution............................................................................................... 34
3.8 Laplace Transforms ........................................................................................................ 36
3.9 Chapter Summary ........................................................................................................... 36
4 Renewal Processes with Minimal Repair .................................................................................. 38
4.1 The Renewal Function M(t) during [0, t] with Minimal Repair..................................... 40
4.2 The Renewal Functions for Known Convolutions ......................................................... 42
4.2.1 The Renewal Function for a Normal Distribution ......................................................... 42
4.2.2 The Renewal Function for a Gamma Baseline Distribution .......................................... 44
4.3 The Renewal Function when the Underlying TTF Distribution is Uniform .................. 49
4.4 Approximating the Renewal Function with Unknown Convolutions ............................ 55
4.5 Discretizing Time in Order to Approximate the Renewal Equation .............................. 56
4.6 Chapter Summary ........................................................................................................... 62
5 Expected Number of Renewals for Non-Negligible Repair ...................................................... 63
5.1 Expected Number of Failures and Cycles ...................................................................... 63
vi
5.2 Availability ..................................................................................................................... 66
5.3 Markov Analysis ............................................................................................................ 69
5.4 Renewal and Availability Functions when TTF is Gamma and TTR is Exponential .... 71
6 The Approximate Expected Number of Renewals for Non-Negligible Repair ......................... 75
6.1 Weibull TBF and Uniform TTR .................................................................................... 76
6.2 Uniform TBF and Weibull TTR..................................................................................... 78
6.3 Gamma TBF and Uniform TTR ..................................................................................... 79
6.4 Uniform TTF and Gamma TTR ..................................................................................... 80
6.5 Intractable Convolutions of f(t) with r(t) ....................................................................... 80
6.6 Approximation Results ................................................................................................... 88
7 MATLAB Program .................................................................................................................... 91
7.1 Minimal Repair .............................................................................................................. 91
7.1.1 Gamma Distribution ....................................................................................................... 92
7.1.2 Normal Distribution ....................................................................................................... 93
7.1.3 Weibull Distribution ....................................................................................................... 93
7.1.4 Uniform Distribution ...................................................................................................... 95
7.2 Non-Minimal Repair ...................................................................................................... 95
7.2.1 The Exact Non-Minimal Repair ..................................................................................... 96
7.2.2 Approximate Non-Minimal Repair ................................................................................ 96
8 Conclusions and Proposed Future Research .............................................................................. 99
8.1 Summary and Conclusions ............................................................................................. 99
8.2 Future Work ................................................................................................................. 100
Appendix ..................................................................................................................................... 108
vii
Appendix A: Uniform Convolution Equations, where c = b-a > 0....................................... 108
Appendix B: 3rd and 4th Moment of Sums of iid rvs ............................................................. 116
Appendix C: Moments of the Most Common Base-line Distributions in Reliability........... 118
viii
List of Tables
Table 1: Gaps and Contributions .................................................................................................. 16
Table 2: Reliability Measures of Most Common Base-line Distributions .................................... 37
Table 3: Normal Approximation to the 12-fold convolution of U(0,1) ........................................ 54
Table 4: Time Discretizing Approximation Method Results ........................................................ 60
Table 5: Moment Based Approximation Results for M1(t) ........................................................... 88
Table 6: Moment Based Approximation Results for M2(t) ........................................................... 89
Table 7: Moment Based Approximation Results for A(t) ............................................................. 89
Table 8: Parameters and Density Functions of Most Common Baseline Distributions in
Reliability .................................................................................................................................... 118
ix
List of Figures
Figure 1: Weibull Graph Based on Different Beta Values ........................................................... 27
Figure 2: The Graph of Weibull Reliability Function ................................................................. 29
Figure 3: Normal Distribution Graph When R(t0.05) = 0.95 .......................................................... 30
Figure 4: Uniform Distribution Graph .......................................................................................... 35
Figure 5: The Intervening Times of Two Successive Renewals ................................................... 38
Figure 6: Relative Error versus t? ............................................................................................... 61
Figure 7: On & off system ............................................................................................................ 69
Figure 8: Relative Error versus Time Graph for M1(t), M2(t) and A(t) ......................................... 90
Figure 9: MATLAB Based Minimal Repair GUI ......................................................................... 92
Figure 10: MATLAB Based Non Minimal Repair GUI ............................................................... 98
x
List of Acronyms
TTF = T Time to Failure
MTTF Mean Time to Failure
MTTR Mean Time to Repair or to Restore
MTBF Mean Time Between Failures
CDF Cumulative Distribution Function
FR Failure Rate
pdf Probability Density Function
pmf Probability Mass Function
rv Random Variable
HZF Hazard Function
CHF Cumulative Hazard Function
Pr Probability
RHS Right Hand Side
CPU Central Processing Unit
IFR Increasing Failure Rate
xi
CFR Constant Failure Rate
DFR Decreasing Failure Rate
MO Modal Point
NID Normally Independently Distributed
RF Renewal Function
RNIF Renewal Intensity Function
xii
Nomenclature
t A specified value in the range space of T
()Ht Cumulative hazard function
()ET Expected Value of TTF
()ft Probability density function (pdf)
()TQt Unreliability function at time t
()Ft Cumulative distribution function
()ht Instantaneous hazard function
()Rt Reliability function
()sNt Number of survivors at time t
()fNt Number of failed items by time t
? Population standard deviation
? Population mean
2? Population variance
0 t?? Minimum or guaranteed life
? The Exponential failure rate
? Weibull shape parameter (or slope)
xiii
? Weibull characteristic life
()t? Renewal intensity function at time t
()Mt Renewal function within the interval [0, t]
t? Time Increment
()At Availability Function
iTTF Time to the i
th failure
( , )Uab Uniform distributions over the real interval [a, b]
()n? Gamma function at n > 0
? The Gamma distribution shape parameter
? The Gamma distribution scale parameter
()()nft n-fold convolution at time t
L Laplace transformation symbol
1
CHAPTER 1
1 Introduction
This chapter introduces the history of reliability, outlines the objectives that the research
intends to achieve, and then gives the research methods adopted. The introduction provides the
framework for the research that follows. The chapter concludes by describing the layout of the
dissertation.
1.1 History of Reliability
?During the expansion after World War I, the aircraft industry was the first to use
reliability concepts. Initially everything was qualitative. As the number of aircraft grew during
the 1930?s, reliability was slowly being quantified as function of mean failure rate and average
number of failures of an airship or airplane? [1].
Before World War II, reliability studies were mostly intuitive, qualitative and subjective
[2]. The early development of mathematical reliability models began in Germany during World
War II, where ?a group led by Wernher Von Braun was developing the V-1 missile? [3].
During the 1950?s, nuclear industry started to develop and to use reliability concepts in
nuclear power plants and control systems [1]. Further, this last decade witnessed the initial
stages of the use of component reliability in terms of ?failure rate, life expectancy, design
adequacy and success prediction? [3].
2
Since the turn of the 20th century, much of the reliability research results started to
transfer to industry and academia. In the past eight decades universities have been teaching
reliability theory and applications. There has also been steady growth on reliability publications.
Furthermore, in order to compete in today?s global economy, manufacturing and other industries
should consider reliability as a primary concern [4].
1.2 Research Objectives
The research objectives are a combination of mathematical methods along with the some
approximations. This work focuses on the renewal function (RF), whose definition will follow,
and has mainly four objectives. The RF is simply the mathematical expectation of number of
renewals in a stochastic process.
The first objective explores the RF,Mt() , for minimal repair by using convolutions of
gamma, normal and uniform distributions. The closed-form expressions for the RFs are
provided in the case of normal, gamma (which includes the exponential) and the uniform
distributions. Both renewal functions and the renewal intensities )(t=? dM t d t( )/ ( ) are
provided for these distributions. Further, the exact uniform convolutions through the eighth have
been known since 1983[5].
The second objective is discretizing time in order to approximate the fundamental
renewal equation. Unlike the gamma and normal underlying failure distributions, the Weibull
base-line distribution (except when the shape parameter 1?? ) does not have a closed form
expression for the n-fold convolution nft()() . Therefore, we cannot obtain the Weibull renewal
3
and intensity functions by using the convolution method. Since the Weibull distribution is the
most important of all underlying distributions in reliability analyses, this dissertation uses an
approximation method by discretizing time in order to estimate the RF, which can be applied to
any baseline distribution [see also E. A. Elsayed, (pp. 428-432)].
The third objective is obtaining renewal and availability functions for some time to
failure and time to repair distributions by using Laplace transforms when repair time is not
negligible.
The fourth objective is obtaining an approximation for expected number of failures,
number of cycles and availability when repair is not negligible for some common reliability
distributions by using the first four raw moments of failure and repair distributions.
1.3 Research Methods
The dissertation presented so far generates the renewal functions for some common
distributions under the case of minimal repair and non-minimal repair. It develops and enhances
an overarching process based on multiple mathematical and statistical theories that will generate
renewal and intensity functions, and will also provide approximation methods.
For the exponential distribution obtaining a closed-form expression for the RF was
documented for well over 100 years ago. Unfortunately, for other distributions such as Weibull
and uniform this is very difficult. We used the convolution method for the uniform distribution
in order to obtain its renewal and intensity functions. For the uniform distribution, as the number
of convolutions increases the problem becomes more complex both geometrically and
4
mathematically. Therefore, after convoluting twelve uniforms we applied the normal
approximation.
Unlike the gamma and the normal underlying failure distributions, the Weibull base-line
distribution (except when the shape parameter 1?? ) does not have a closed-form expression for
the n-fold cumulative convolution nFt() () , and hence
nn1M t E N t F t
?
??? ? ()( ) [ ( )] ( )
cannot be
used to obtain the value of the RFMt() . Since the Weibull distribution is the most important
baseline distribution in reliability analyses, we used the time-discretizing approximation method
[6] described in Chapter 4. Further, most calculations are done with the aid of MATLAB
Programming Language. We also used moment based method and Laplace transforms method
for non-negligible repair.
1.4 Dissertation Layout
The dissertation is divided into eight chapters including this first chapter entitled
?Introduction?. The layout and organization of remaining chapters are as follows.
Chapter 2 presents the literature review on reliability. This includes the concept of
reliability in mathematical details. It also explains the terms availability and maintainability and
underlines the differences of these two concepts. Moreover, it identifies the differences between
quality and reliability. Chapter 2 continues on renewal stochastic processes and it discusses
reliability applications. Finally, it explains the importance of renewal functions, contributions to
the literature and the general previous works that have been done.
5
In Chapter 3, the reliability methodology is presented. The principles, mathematical
development, and structure are identified and introduced. Reliability measures like, MTTF
(Mean Time to Failure), hazard function are explained both mathematically and conceptually. It
explains the mathematical concepts of convolution method. Chapter 3 continues by some
distribution functions such as uniform, gamma, normal and Weibull that was studied in this
work. Chapter 3 concludes with a brief summary table.
Chapter 4 describes the renewal processes for minimal repair in detail. It gives the
renewal and intensity functions for the normal and gamma distributions. It also provides
convolutions of the uniform through order n = 12 for the case of minimal repair. By using these
convolutions it calculates the renewal and intensity functions for the interval [0, t] when the
underlying distribution is uniform. Chapter 4 continues by explaining the time-discretizing
method in order to approximate the fundamental renewal equation, which can be applied to any
base-line distribution. It concludes with approximating the Weibull distribution?s RF.
In Chapters 5 and 6, unlike the previous chapters, we assume that MTTR (Mean Time to
Repair) is not negligible and that TTR has a pdf denoted as r(t). Chapter 5 gives the expected
number of failures, expected number of cycles and availability by taking the Laplace transforms
of renewal functions, and Chapter 6 gives the approximate number of renewals for most common
distributions by using raw moments of failure and repair distributions.
Chapter 7 introduces the Matlab program, describes the inputs and outputs of the program
for minimal and non-minimal repair cases. Finally, Chapter 8 gives the conclusion and possible
future work.
6
CHAPTER 2
2 Literature Review
This chapter introduces the literature review on reliability. It defines reliability,
availability and maintainability. It also discusses differences between quality and reliability.
Furthermore, it briefly discusses the applications of reliability which abound. Then, it continues
with importance of renewal functions and previous work on renewal functions. Finally, it
summarizes the contribution of this study.
2.1 Reliability
The concept of reliability is not new. Both manufacturers and customers have long been
concerned with the reliability of products they produce and use [7]. Basically the general
perception about reliability is functioning without any problem. Stating something is reliable
implies it can be depended on to work satisfactorily. However, the real definition of reliability
involves quantifying measures.
?The reliability of an item is the probability that it will adequately perform its specific
purpose for a given period of time under specified stressed conditions? [8]. Another definition is
as follows: ?Reliability is defined to be the probability that a component or system will perform a
required function for a given period of time when used under stated operating conditions? [4].
As it is seen from the two above definitions, reliability is a probability. Therefore, this implies
reliability can never be negative or greater than one. Since reliability is a probability, probability
7
axiom results are used in reliability [8]. Reliability studies deal with different complexity levels
of units or systems [9].
Unreliability induces heavy losses to organizations. It also causes them to lose
reputation. Failure of products before reaching their warranty period is costly. Therefore, today
we have reached a point where reliability is considered as a major performance measure [10].
In summary, reliability is the survival probability of an item or system beyond a certain
point in time. Since the cost of unreliability is high, reliability is considered as a major
performance parameter. Unlike most classical statistical parameters, reliability is always a
function of time (t). Although in cases of very short mission time, reliability can be considered
as static merely as an approximation.
2.2 Availability and Maintainability
Availability includes both failure and repair rates of a system, and therefore, it is
considered to be one of the most important reliability performance measures [6]. ?Availability is
the probability that a system or component is performing its required function at a given point in
time or over a stated period of time when operated and maintained in a prescribed manner? [4].
There are generally four types of availability measures [8]. These are:
1) Point (or instantaneous) availability ()At : Instantaneous availability at time t is the
probability that a repairable unit is functioning reliably at time t. Therefore, if there is no
repair, the availability, ()At , is equal to the reliability function Rt() .
8
2) Limiting availability: As time increases, instantaneous availability will approach a
constant value, once an item has stationary times to failure Ti, stationary times to repair
TTRi, and this value is known as the limiting availability [8].
3) Average availability on (0, t]: This can be explained as the ?expected fraction of time?
that a system is operational during (0, t], [8].
4) Limiting average availability on (0, t]: This is also known as ?steady-state availability?.
It is the availability of a component or system when the time interval is very large [6].
Maintainability is the probability that a failed item can be repaired or restored to become
operationally effective within a specified period of time when repair action is performed in
accordance with prescribed procedures [11]. Maintainability is important for eliminating
defects, correcting defects and their causes, meet new requirements and adapt to a changing
world. A design process should start with defining system maintainability objectives.
Both maintainability and availability are very important measures in reliability theory and
have a wide range of applicability. However, in the context of this work the brief amount of
information given above on maintainability and availability should perhaps be satisfactory.
Maintainability directly affects availability because time for repairs or preventive
maintenance can change a system from available to unavailable state. Therefore, there is a close
relationship between reliability and maintainability. Reliability affects maintainability,
maintainability affects reliability and they both impact availability [12]. Further, like reliability,
availability and maintainability are also probabilities. Therefore, probability theory rules can be
applied to both availability and maintainability.
9
2.3 Reliability versus Quality
Quality may be defined in many different ways. It can be conceptual, perceptual or
conditional and it may be interpreted differently. Basically quality means ?fitness to use? and it
has several dimensions such as ?performance?, ?reliability?, ?durability?, ?serviceability?,
?conformance to standards?, etc. [13], [14]. Therefore, reliability which is closely associated
with product quality is one of several quality dimensions. Quality can be defined qualitatively
and can be achieved through a satisfactory quality assurance program [4]. ?Quality assurance is
the set of activities that ensures quality levels of products and services properly maintained and
that supplier and customer quality issues are properly resolved? [14] .
On the other hand, reliability is largely concerned with how long a product can continue
to operate under specified conditions once it becomes functional. ?Reliability may be viewed as
the quality of a product?s operational performance over time, and as such it extends quality into
the time domain? [4]. Improving reliability is an important part of improving product quality
[15].
2.4 Reliability Applications
Examples of high-reliability systems abound worldwide, such as aircraft systems([16],
[17] etc.), electric power generating stations ([18], [19] etc.), chemical plants ([20] etc.), power
systems to telephone and communication systems ([21], [22], [23], [24], etc.), computer systems
and networks ([25] etc.) [1].
Reliability studies are conducted at either the component or system level. Generally,
reliability calculations are easier at the component than system level. If it is the system level,
10
then there are two different classifications. A system can be static or dynamic. Further, in both
cases it can be serial, parallel or mixed. Explanations of each follow.
In static reliability component or subsystem reliabilities are considered to be
approximately constant for a specified duration of time [26]. In this case, the mission time is
sufficiently short so that the assumption of constant reliability is almost tenable. Whereas, in
dynamic models there will be continual reliability degradation of subcomponents with respect to
time.
In series system-models all subsystems must operate reliably in order for the system to
function properly. As soon as one subsystem fails, the system fails [27]. There is also another
model type called ?chain model? or ?weakest link model? that is described in the literature under
the title of series systems [26]. Based on this model, a system will fail as soon as the weakest
component (or link) fails. Therefore, system reliability is equal to reliability of the weakest
among all components or subsystems [26].
However, a pure parallel system is composed of subsystems or components that the
success of any one of which results in system success [28]. Therefore, such a system is reliable
if at least one component is reliable. There are many studies in the literature that deal with
system reliabilities such as [29], [30], [31], [32], etc.
2.5 Renewal Processes
Renewal processes are stochastic events such that their nth stage value is the sum of n
independent random variables of common distributions with nonnegative ranges [33]. As an
example, consider a machine component that is replaced as soon as it fails with a new one. Let
11
Nt() be the number of replacements during the interval [0, t] of length t. Then,Nt() is called a
renewal counting process. The study of renewal processes focus on the following topics:
(1) The pmf (Pr mass function) of Nt() ,
(2) The expected number of renewals during [0, t] or [t0, t0+t], ENt[ ()] , denoted by
()M t E N t ? [ ( )], M for mean, is called the RF. Henceforth, the symbol E will represent the
Expected-Value operator. (Note that this case also includes the negligible repair-time.),
(3) The occurrence Pr mass or density function of a renewal at specific epochs of time, and
(4) The time needed for the occurrence of n events (such as failures that are followed by a
replacements) to occur [34]. [For more details see U. N. Bhat (1984), Elements of Applied
Stochastic Processes, 2nd Ed., Chapter 8.].
2.6 Importance of Renewal Functions
Renewal functions, gives the expected number of failures of a system or a component
during a time interval and this is used to determine the optimal preventive maintenance schedule
of a system [35]. Renewal functions are quantities that have particular importance in analysis of
warranty ([36], [37], [38], [39] etc.) [40]. Expected cost of warranty estimation is closely
related to the RF estimation [36]. For example, consider the case that cell phone manufacturer
has a two year free replacement warranty which means that if a cell phone fails manufacturer
agrees to replace it with a brand new one without any charge. Then , suppose M(2) is the
expected number of failures(replacements) during two years warranty period and C(2) expected
warranty cost is C(2)= c*M(2), where c is assumed as the fixed cost per replacement [36]. It is
12
obvious that the cost of warranty is greatly affected by the number of replacements [35]. In
today`s competitive environment product`s warranty policy is important to attract customer.
Offering longer warranty terms usually attract more customers but it means more cost [41]. So,
warranty has two important roles protection and promotion [42]. Therefore, it is very important
to determine an optimal warranty time and this means obtaining M(t) with greater accuracy is
very essential especially if manufacturer produces large number of units or very expensive items.
Renewal functions play an important role and have wide variety of applications in
decision making such as inventory theory ([33]), supply chain planning ([43], [44]), continuous
sampling plans ([45],[46]), insurance application and sequential analysis ([47][48]) [36],[43].
2.7 Previous Work on Renewal Functions
As we have seen in the previous section renewal functions play an important role in many
applications. Therefore it is important to obtain renewal functions analytically. Based on
analytic method, M(t) is the inverse Laplace transform of ??Mswhere ? ? f(s)M s =
s[1 f(s)]?
([35]), where Laplace transforms will be defined later. ?The advantage of analytical method is
one can carry out parametric studies of the RF, i.e., the behavior of M(t) as a function of the
parameters of the distribution? [40]. However, for most distribution functions obtaining the RF
analytically is complicated and even impossible [43]. Therefore development of computational
techniques and approximations for renewal functions has attracted researchers [49].
13
One of the well-known approximations is 21 2 1( ) / / 2 1ttM ? ? ?? ? ??? which is generally
known as asymptotic approximation and was generated by Tacklind,S (1945)[50] and also cited
in numerous papers such as [51]. The asymptotic expression has a closed-form expression thus it
is easy to apply optimization problems that involve renewal process [43]. However since
asymptotic expansion is not accurate for small values of t, Parsa&Jin (2013) [43] propose better
approximation by keeping the positive features of asymptotic approximations such as simplicity,
closed-form expression, and independence from the distribution. Jiang (2010) [52], proposes an
approximation for the RF with an IFR which is also useful in areas such as optimization where
renewal function needs to be evaluated.
There are series methods available in the literature to approximate renewal functions such
as, Smith and Leadbetter (1963) [53] who developed a method to compute the RF for Weibull by
using power series expansion of t? where ? is the shape parameter of the Weibull. On the other
hand instead of using power series expansion, Lomnicki (1966) [54] proposes another method by
using the infinite series of appropriate Poissonian functions of t?. There are also many other
approximations methods available such as Xie (1989) [55], Smeiticnk & Dekker (1990) [56],
Baxter et al (1982) [57], Gang&Kalagnaman (1998) [58], From ( 2001) [59]etc. For example Xie
(1989) [55] proposed RS-method for solving renewal-type integral equations based on direct
numerical Riemann-Stieltjes integration. There are usually three criteria: model simplicity,
applicability and approximation accuracy to evaluate the value of the analytical RF
approximation [52]. Increasing the complexity may lead more accurate approximation but may
make the process complicated and difficult to implement in practice [60].
14
Studies on renewal functions for some particular underlying renewal functions such as
Weibull have been done. Jiang (2009) [61], proposes an approximation for the RF of Weibull
distribution with an IFR which is accurate for time t up to a certain value of larger than the
characteristic life. On the other hand, Jin & Gonigunta (2010) [62], proposes an approximation
method for Weibull RF with DFR. Sinha (1985) [63] obtains Bayes estimation of the survivor
function of the s-normal distribution. Papdopoulos &Tsokos (1975) [64] obtain confidence
bounds for the Weibull failure model. Many others are also available like [65], [66], [67], etc.
Furthermore, in the literature bounds on renewal functions have been discussed. Since
they provide upper and lower bounds on warranty costs bounds on M(t) are very useful for many
warranty models [40]. Ross (1996) [68], shows that if a distribution has DFR then the RF is
bounded as ? ? 2
211 1 12?ttMt?? ??? ? ? ??? ?
where 1? and 2?? are the first and second raw moments.
Marshall(1973) [69] provides lower and upper linear bounds on the RF of an ordinary renewal
process. Ayhan et.al. (1999) [70] provide tight lower and upper bounds for the RF which are
based on Riemann-Stieljes integration. There are also many other studies available about bounds
on RF such as [71], [72], [73], [74], [75], [76] and etc.
Finally, simulation can be considered as an alternate approach to estimate the value of
renewal function. Brown et al (1981) [77] use the Monte Carlo simulation to estimate the RF for
a renewal process with known interarrival time distribution. Papadopoulos & Tsokos (1975)
[64], perform Monte Carlo simulation to obtain 90% and 95% Bayes confidence bounds for the
random scale parameter and reliability function to illustrate their results. ?The simulation
15
approach offers an alternate method for obtaining the solution to the problem without the need of
solving complicated mathematical formulations? [40].
2.8 Contribution of this work to the Literature
The renewal and intensity functions with minimal repair for the most common lifetime
underlying distributions normal, gamma, uniform and Weibull are explored. The exact normal,
gamma, and uniform renewal and intensity functions are derived by the convolution method.
Unlike these last three failure distributions, the Weibull distribution, except at shape ? = 1, does
not have a closed-form function for the n-fold convolution. Since the Weibull is the most
important failure distribution in reliability analyses, its approximate renewal and intensity
functions were obtained by the time-discretizing method. And also for non-minimal repair
moment based approximation method was generated. Table 1 summarizes the contributions of
this dissertation.
16
Table 1: Gaps and Contributions
Gaps in the Literature Contributions
Obtaining the renewal functions for the
most baseline distributions is not possible.
Obtaining renewal functions and renewal intensities for normal and
gamma distribution by using convolution method.
Approximating Weibull renewal function and renewal intensity by
using time discretizing method.
?The renewal function of uniform
distribution using the convolution
method is not available.
We used geometrical mathematical statistics method to obtain
uniform convolutions from n= 2 through n = 12 and then applied the
normal approximation for convolutions beyond 12 to obtain renewal
function of uniform distribution.
For the case of non-negligible repair only
the closed-form renewal functions exist in
the case of exponential TTF and TTR.
We obtained the closed form renewal function for gamma TTF and
exponential TTR when ? (shape parameter of gamma) is an exact
positive integer from 2 to 7.
We obtained approximations for expected number of cycles
? When TTF is Weibull and TTR is uniform
? When TTR is Weibull and TTF is Weibull
? When TTF is gamma and TTR is uniform
? When TTR is gamma and TTF is uniform
by first obtaining the convolution density functions and then
using the time discretizing method.
We obtained approximations for two parameter exponential, three
parameter Weibull, gamma, normal, lognormal, logistic and
loglogistic distributions by using the first four moments of failure
and repair distributions.
17
2.9 Literature Review Summary
The literature describes a multitude of research in the area of reliability. Reliability has a
wide variety of applications from aircraft to power systems. The literature supports increasing
applications of reliability in today?s competitive global economy. In this chapter, we explained
some important concepts such as availability and maintainability and described the differences
and association between quality and reliability. This chapter also explains the importance of
renewal functions and the general previous work that has been done about them.
18
CHAPTER 3
3 Methodology
This chapter describes the mathematical concepts and relationships that are needed in
reliability and renewal theory presented throughout this dissertation.
3.1 Reliability Measure
There are three measures of reliability:
(1) The reliability functionRt() ,
(2) The mean time to failure (MTTF), and
(3) The hazard (rate) function ht().
If either function (1) or (3) is known, then the other 2 measures can be uniquely
determined, but the knowledge of (2), i.e., ? = MTTF, is not sufficient to obtain unique functions
forRt() andht() . In fact, for the same MTTF of an underlying distribution, there are
uncountably infinite other base-line distributions that have identically the same MTTF.
3.1.1 The Reliability (or Survivor) Function R(t)
The reliability of a component is the probability (Pr) that the device will perform without
failure during the mission time t, under specified stress conditions. For example,
R(of a new passenger tire for t = 500 interstate miles) ?100% = 1.
However, the reliability of the same passenger tire under racing conditions at Indianapolis 500
would be almost zero. Note that the terminology survivor function, ()St is also used for non-
19
repairable items, such as light bulbs, transistors, or rocket-motor of an unmanned spacecraft.
Let T = the random variable lifetime, or time to failure (TTF), with Pr density function .
Then, the reliability function at time t, or the survival Pr for a mission of length t, is given by (the
Pr that a component lifetime exceeds time t)
( ) ( ) d 1 1 1
TT
t
R t T t f x x P r T t F tt QP r ( ) ( ) ( ) ( )
?
? ? ? ? ? ? ? ? ? ??, (3.1)
where ()Ft = the cdf of T at t, and ( ) ( )?TTQ t F t represents the unreliability function at time t,
or the cumulative failure Pr by time t. The pdf (Pr density function), ft() , is also referred to as
the failure (or mortality) density function. Some authors use the notation ()tS for the reliability
or survivor function at time t to imply survival probability beyond t; however, the notation Rt()
is a bit more prevalent in engineering applications. We now obtain the relationship between
Rt() and by differentiating equation (3.1) with respect to t, recalling that
??TTf t d F t d t d Q t d t( ) ( ) / ( ) / because ? ?() ? ???t F t f T d TT . Due to the fact that time cannot
be negative, and thus the lower limit in this last integral must be zero instead of ? ?, i.e.
0
( ) ( )? ?t F t f T dTT = Failure Probability by time t. The developments below show the
relationship between Rt() and ()f t
20
1??? ? ? ? ? ? ? ? ???d R t d d F tF t f t d R t d F td t d t d t( ) ( )( ) ( ) ( ) ( )
and
? ? ? ? ?f t d R t d t f t d t d R t( ) ( ) / ( ) ( )
.
3.1.2 The Mean Time to Failure
We are now in a position to obtain the relationship betweenRt() and the mean time to
failure (MTTF=?), which is defined as the mathematical expectation of T = TTF. Henceforth,
the symbol V will represent the variance operator, and the reader must be cognizant of the fact
that anytime the operator E or V is applied to any rv (random variable), the end-result will
always be a population parameter. The Mean Time to Failure (MTTF) is given by;
00
M T T F
??
? ? ? ???E T tf t d t t d R( ) ( ) ( )
M TT F
0( ) ( ) 0 ( ) ( )0 0 0
? ? ? ?? ? ? ? ? ?? ? ?tR t R t d t R t d t R t d t] . (3.2)
The above equation clearly shows that the unconditional mean-life, ( ) MTTF?E T , starting at age
zero, of any device or system is given by the integral of its reliability function evaluated always
from zero to infinity so that the MTTF is the total area under Rt() and the abscissa t (i.e., from 0
to ? even if the minimum life is greater than zero).
21
Note that if a component or system is repairable (or renewable, i.e., failed units are
almost immediately replaced), then ET()is called the mean time between failures (MTBF). The
MTBF of any system also can be found by integrating its reliability function from zero to infinity
[78] (a simple technique of obtaining MTBF for complex systems). Again, the lower limit of the
integral must always be zero even if the minimum life t0 = ? > 0.
3.1.3 The Hazard (or the Failure Rate) Function h(t)
By definition the failure rate (FR) of any device is defined as the failure rate during
?t t ?t(, ) given that the device age is t, i.e.,
? ? ? ? ? ?PrP t T t ? t T t P r t T t ?t R t R t ?tF R t ? t T t ? t R t ? t? ? ? ? ? ? ? ??? ? ?? ( ) ( )( ) ( ) (3.3)
This implies that the failure rate of a component at time t is the probability that it will fail in the
interval ?t t ?t(, ) given that its life has exceeded time t (i.e., given that the age of the
component is t). Put differently, the failure rate of a population of identical items at time t is the
proportion of the units failing per unit of time in the interval ?t t ?t(, ) amongst all survivors at
time t. The hazard function (HZF), (),ht is simply the instantaneous FR, i.e.,
? ?
0
1 ][
?
??????? ? ? ? ?
???t
Rt ?t R t d R t d t f tht
Rt ?t R t R t
() ( ) / ( )( ) l im
( ) ( ) ( )
(3.4)
In other words, if we have a system with N0 identical items on test at time 0, ??sNt
survivors at time t and ??fNt failed items by time t, then by the above definition ht() is the rate
22
of failure, fdN t dt( )/ (this derivative is not quite appropriate because ??fNt is discrete), only
amongst the survivors ??sNtbeyond time t, i.e.,
0 0
0
()
() ()
()
()( ) ( ) ( )
s
f s s
ss s s
Nt
ddN
d N N t Nd N t
d R t d t f td t d t d t d tht
NtN t N t N t R t R t
N
??
?????
??? ? ? ???
?? ? ? ? ? ?( ) / ( )
( ) ( )
as before. The quantity
? ? ? ?( t ) ( t ) ( t ) ( t ) |h d t d R R d F R P t T t d t R P t T t d t T t/ / / ,? ? ? ? ? ? ? ? ? ? ? ?
gives the proportion of items that will fail within (t, t dt)? amongst those that are still
functioning at time t. From the cumulative of hazard function,CHF H t? () , we can obtain the
relationship amongst the reliability measures ()ft , Rt() , and ht()as shown below:
t
000( ) ( ) ( )
?? ? ? ?? ? ? ? ? ? ? ? ?? ? ? ??? ()/ l n ( ) l n ( )tt HtH t h x d x d R R R x R t R t e
t
0
x dx
( ) e
??
?
h
R t
()
; ?? ? ? ? Htfth t f t h t R t h t e
Rt
()()( ) ( ) ( ) ( ) ( )
()
(3.5)
Properties of the Hazard function
(a) () 0?ht for all t.
23
(b) 00 0 0
0fh f hR? ? ?()( ) ( ) ( )()
must be finite at t = 0 unless f(0) = ?
(c) ()
?? ??t htlim
iff h(t) is an IFR, simply implying that any man-made system must have a finite
life (or a finite TTF), i.e., no man-made system can last forever!
(d) ()
00
( ) ( ) 1
?? ?
???? Htf t d t h t e d t. Note that the above relationships imply that the assumption of
almost constant reliability during a short interval (0, t) leads to an almost zero HZF value.
3.2 Renewal Intensity Function
The renewal intensity, ()t? , gives the instantaneous renewal rate at time t, i.e.,
? ?
0
()
?t
Mt ? t M t d M tt ? t d t? ()( ) lim ,
?
???? (3.6)
so that ()?t ?t?
gives the unconditional probability element of a renewal during the interval t t ?t?(, ) ,
and in the case of negligible repair time, ()t? also represents the instantaneous failure intensity
function; hence,
0
( ) d()??t xM t x? . Note that nearly authors in Stochastic Processes and some in
reliability literature refer to ()t? as the renewal density because ()?t ?t?
gives the probability element
of a renewal during the interval t t ?t?(, ) ; however ()t? is never a pdf. Hence, we have chosen to refer
to ()t? as the renewal intensity in lieu of renewal density. The renewal intensity, ()t? , should not be
confused with the hazard rate function h(t) [79], because ()?ht?t is the conditional probability of a
failure during time interval (t, t+?t ) given that the unit?s age is t, whereas ()t ?t? ? is the unconditional
24
probability of a failure during ?t . The hazard rate function is a relative rate pertaining only to the first
failure, whereas the intensity function is an absolute rate of failure for also repairable systems [4],
including minimal repair. Only in the case of exponential base-line distribution (CFR) both ht() and
()t? are identically equal to the constant failure rate ?.
3.3 Convolutions
The n-fold convolution of a statistical distribution arises in a wide variety of applications
of probabilistic models such as reliability theory, renewal theory, inventory theory, queuing
theory, continuous sampling plans, insurance risk analysis, and sequential analyses [80].
Mathematically, a convolution of two density functions ??1ft and ??2ft denoted 12 f*f ,
gives the density of sum of two variates 12TT? . It can be proven that the convolution of with
is given by [81],
1 2 2 1
00
12
tt
f t * f t f t u f u d u = f t u f u d u? ? ???( ) ( ) ( ) ( ) ( ) ( ) (3.7)
Note that in general the lower limit would be ?? but in reliability theory, the lower limit
is always zero. In the literature review there are some studies about uniform convolutions [82],
[83], [81], but we used the geometrical approach to obtain the precise uniform convolutions
through order 12 that are presented in Chapter 4. Maghsoodloo and Hool [5] obtained the
uniform convolutions for orders 2, 3, 4, 5, 6 and 8.
25
3.4 The Weibull Distribution
It is well known that the underlying distribution of almost any manufactured dimension
by man can be approximately modeled by a normal (or Laplace-Gaussian) pdf. The Weibull pdf
plays the exact same important role for the underlying distribution of TTF (or lifetime) of most
mechanical and electrical components or systems. The key events in the derivation of what is
now known as the Weibull distribution took place between the years 1922 and 1943. There were
three groups of scientists working independently for different aims. Waloddi Weibull (1887-
1979) was one of the three working on this distribution. The reason that the distribution bears
his name is the fact that he propagated it internationally and interdisciplinary [84]. To arrive at a
Weibull pdf, consider an exponential pdf at the constant failure rate ? = 1 failure per unit of time.
Note that the symbol ? is used throughout this dissertation if and only if ht() is a constant failure
rate (CFR). Clearly,
0
1xe dx .
? ?
?? (3.8)
Then, for convenience letting t0 = ?, we make the transformation 0?t ?x ?
????????????,
. As a
result, 1 1?d x t ??
dt ? ? ? ?
?? ? ? ???
? ? ? ???? ? ? ?, and substitution into (3.8) yields
11
1
??t ? t ???
? ? ? ?
??
t ? d t ? t ?e ? e d t
? ? ? ? ? ? ? ?
? ? ? ???????? ? ? ?
? ? ? ?? ? ? ?????? ? ? ?? ? ? ?
? ? ? ??? .
(3.9)
26
Since the value of the integral in Eq. (3.9) is equal to 1 (or 100%), the integrand
1 ?t ?? ??? t ?
e? ? ? ?
???? ???
???? ?? ??? ?? ???
? ?? ?
must be a probability density function (pdf) over the range [? ? . The pdf,
? ? 1 0
?t ??
??? t ?f t e t ?t? ? ? ?
???? ? ??
???? ? ? ??? ? ?? ? ? ???
? ? ? ? ,
(3.10)
and () 0ft? for 00 t ?t? ? ? , is called the Weibull model, denoted ? ? ? , with
minimum (or guaranteed) life ? (the location parameter), the characteristic life ?, and slope
(or the shape parameter) ?; ? ? ? is called the scaling parameter. Different authors tend to use
different symbols for the three parameters of a Weibull pdf, but ? is the most common symbol
for the slope. Figure 1 shows the Weibull density based on different shape (? ) values.
The reliability function for Weibull distribution is,
? ?
1
()
?
?
x ? ?
?? u
t t ?
??
? x ?R t T t e d x e d u .
? ? ? ?Pr
??? ???? ??
? ???
?????
???
???? ? ? ?
???? ????
? ? 1 , 0
,
t
t
Rt
et
??
??
?
?
???? ??
???
????
? ??
? ? ??
(3.11)
27
Figure 1: Weibull Graph Based on Different Beta Values
The MTTF of Weibull distribution,
00
( ) ( )
????? ? ??
???? ? ?? ? ?
?t ?
? ??
?
E T R t d t d t e d t; letting ?t ?x ?????????
??
in the second integral results in,
? ? ? ?1 111
00
()
??? ?????
? ? ? ???? ?x ?x? ? ? ?E T ? e x d x ? x e d x?? //.
Since by definition, 1()
0
? ??? ? nx ? n x e dx, and ( ) ( 1)?n? n ? n+ we obtain
? ? ? ?1M TTF ( ) 1 1??? ? ? ? ? ? ? ???? ?ET ? ? ? ? ? ? ? ?
?/( ) ( ) /
(3.12)
The above MTTF ?? iff 1?? (i.e., only for the CFR = constant failure rate, and IFR =
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0.0
5 0.3
0.5
5 0.8
1.0
5 1.3
1.5
5 1.8
2.0
5 2.3
2.5
5 2.8
3.0
5 3.3
3.5
5 3.8
4.0
5 4.3
4.5
5 4.8
Beta=1
Beta=2
Beta=3
28
increasing failure rate cases). The modal point of Weibull is given by
11 ?MO ? ? ? ? ???? ? ? ??? /( ) ( ) / (3.13)
The variance of Weibull is given by
? ? 222 2111??? ? ? ?? ? ? ? ? ???? ? ? ?
? ? ? ???T? V T ? ? ????()
(3.14)
Special cases of the Weibull model
(i) When ? , the Weibull reduces to the exponential pdf with minimum life t0 = ?
and mean-life (or characteristic life tc) equal to ?. The Weibull pdf with slope ?
can be used to model the TTF of a component during its useful life (constant failure
rate = CFR).
(ii) When ? and ? , the Weibull becomes the Rayleigh density function
? ? 2 2?22 ?? ???( ) ( ) ( ) t ? tft ? t ? e te/ ///, where 2 ???c?t / .
(iii) When 0 < ? < 1, it can be shown that the hazard function of the Weibull is a
decreasing function of time (DFR = decreasing failure rate) so that the Weibull may
be used to model the TTF during the burn-in (or debugging, or infant-mortality)
period of a component (see Ebeling pp. 31-32).
(iv) When ? > 1, the hazard function is increasing (i.e., increasing failure rate = IFR) and
the Weibull density can be used to model the TTF during the wear-out period of a
component.
29
(v) When 1 < ? < 2, h(t) is an IFR concave function, and for ? > 2, h(t) is an IFR convex
function because 2
2d ()htdt
> 0 when ? > 2.
Figure 2: The Graph of Weibull Reliability Function
3.5 The Normal Distribution
The normal distribution which is also called Laplace-Gaussian is the most commonly
used distribution in the field of statistics [85]. The reasons are related to its mathematical
properties, central limit theorem, and various experimental responses often have distributions
that are approximately normal [86]. The normal density was first discovered by Abraham de
Moivre (a French mathematician) in 1738 as the limiting distribution of the binomial pmf and
T h e W e i b u l l R E f u n c t i o n f o r d e l t a = 3 0 0 0 0 , t h e t a = 6 0 0 0 0 ,
a n d s l o p e = 4
0
0 . 2
0 . 4
0 . 6
0 . 8
1
1 . 2
0 20000 40000 60000 80000 100000 120000
Ti m e i n K m
R
(
t
)
30
was named the ?normal? by Karl Pearson in 1894 [87]. It can be proven that the points of
inflection occur at ? ? ?.
Figure 3: Normal Distribution Graph When R(t0.05) = 0.95
A continues random variable T is said to have a normal distribution with parameters ?
and ? (or ? and ?2), where M T T F??? ? ? ? ? and , iff the pdf of T is [88]
? ? ? ? ? ?2 221
2
t? ?f t ? ? e ????? t
??
??? ?? ? ? ?/;, (3.15)
The reliability function is derived from
? ?
2
2
11 1 ?
22t
x? t?R t e x p d x
??? ?
? ??? ?????
? ? ? ? ????
??
?() , (3.16)
31
where ? represents the cdf of the N(0,1). There is no closed-form solution to the above integral,
and it must be evaluated numerically. The hazard function cannot be written in closed form
either and is provided below.
? ? ? ?2 221
2
1 ?
t? ?e
ft ??ht
Rt t?
?
??
?? ??
??
????
/
()()
() (3.17)
It must be noted that the normal failure law is applicable only if its MTTF = ? is more than 10
standard deviations to the right of the origin (or zero) because ?(?10) < 7.62?10?24.
3.6 The Gamma Distribution
By definition 1x
0
( ) x xn? n e d? ??? ? . Dividing both sides of this last definition we obtain:
1x
0
11 x x()? ??? ? n ed?n , and per force the integrand ? ? 11 ??? ntf t n t e?n(); must be a density
function called the standard gamma pdf. When n is not an integer, the common notation for the
shape parameter n is ?, ?, or ?.
A single integration by parts of ()?n will show that ( ) ( 1 ) ( 1 )? ? ?? n n ? n. After 1?n()
integration by parts, we obtain ? ?? ? ? ?( ) 1 2 1? ? ?? n n n ...?. Inserting 1n= into the definition of
()?n yields 11
00
( 1 ) 1? ? ?? ? ???tt? ? t e d t e d t
??; therefore,
32
? ? ? ? ? ? ? ?( ) 1 2 1 1 1 0 1? ? ? ? ? ? ? ?? n n n n ?!!.... Further, this last result also implies that
?? n+ 1 n ? n( ) ( ) ( ); it can be proven that (1/ 2)? ?? and hence (3 / 2 ) (1 / 2 ) (1 / 2 )??? ,
5 3 3 3 12 2 2 2??? ? ? ? ? ?? ? ?? ? ? ? ? ?? ? ? ? ? ?? , etc. For example, ? ? 9 36288010 ??? ! . To obtain the gamma
pdf, we make the transformation x t?? in the definition of gamma function:
11 ? 1 ?
0 0 0
1() ? ? 1 ? ?
()
n x n t n t? n x e d x ? t e d t ? t e d t
?n
? ? ? ? ? ?? ? ? ?? ? ?? ? ?( ) ( ), or
1 ?
0
1 ? ? 1() ntt e d t?n ?? ??? () . (3.18)
The above equation clearly shows that the integrand must be a pdf over the range [0, ?)
because its integral over [0, ?) yields 100%. The function under the integral is called the gamma
pdf, as shown below, in statistical literature with rate ? (or ? = 1/? the scale parameter) and the
shape parameter n.
1
() ntf ?nt t e? ?????( ) ( ) .
(3.19)
When n is a positive integer the above density is called Erlang and has extensive
applications in Queuing Processes. When n is not a positive integer, the most common notion
for shape is ?; thus, a better representation for the general gamma pdf is 1
() tf t?te? ????? ??( ) ( )
.
33
The meaning of the parameters n and ? will be made clear in the application example
provided below, where 1/? is also called the scale parameter. The major application of the
gamma pdf occurs from the fact that the sum of n independent and identical exponential rvs,
TSystem = T1 + T2 +...+ Tn, has a gamma time to failure density function with parameters n and ?,
where each Ti is distributed according to e??? t . Therefore, an n-unit standby system with
quiescent failure rates of almost zero for standby units and perfect switching, has a lifetime that
has the gamma density with parameters n and ?. Further, the gamma density has applications in
maintenance scheduling where the amount of deterioration during an interval [t1, t2] has a gamma
pdf with scale parameter 1/? and shape 21n ? t -t? (), where ?>0 is a constant of proportionality.
The first four moments of the gamma pdf are given by
? ? ? ?12 ?? ? ? ? ? ?S y s t e m nE T E T T T n /
? ? ? ? 212 ?? ? ? ? ? ?S y s t e m nV T V T T T n /
the standardized third moment 3 2?n? / , and 4 36?n??( / ); hence the kurtosis is equal to
4 6 ? ? n? / , and it can be verified that 1 ?MO n??( )/ . Note that the values of 3 2?n? / and
4 36?n??( / ), clearly show that the limiting distribution (i.e., as n ? ?) of the gamma density
is the Gaussian N(n/?, n/?2). Unfortunately, n must exceed 200 (because the exponential is
highly skewed) before the normal approximation to gamma becomes fairly adequate.
34
To obtain the reliability function for the gamma pdf, we make use of its relationship with
the Poisson pmf as described below, where X(t) describes the number of failures during a
mission time of length t and Tn represents time to the nth failure.
1
0
( ) = 1) ? ??
?
??? ? ? ? ?? ?? ? kn t
k
R t T t P r X t n fa il u r ts ekenP r ( ) ( ) () !( . So far, we know the expressions
for the gamma pdf, the reliability function, and the MTTF ?? /n . The hazard function can be
obtained from f t R t( )/ ( ) .
3.7 The Uniform Distribution
If a < b, the random variable T is said to have a continuous uniform probability density
on the interval [a, b] if and only if the density function of T is given by [89]
? ? 1 / ( ) ,
0, b t be ls e wa h e ra eft ? ? ??? ??
(3.20)
where a is the minimum-life and b is the maximum-life. The graph of the above pdf is shown
atop the next page. Note that all values of t from a to b are equally likely in the sense that the
probability that t lies in an interval of width entirely contained in the interval from a to b is
equal to a ? regardless of the exact location of the interval [90]. Its reliability measures
are listed below.
The MTTF is: ? ?12? b a??. (3.21)
Median of the uniform distribution is: ? ?
05 12t b a??.
. (3.22)
35
Figure 4: Uniform Distribution Graph
Variance of the distribution is: ? ?22 112? b a?? (3.23)
Because 11? ???
? ? ???
b
tt
btd x d xb a b a b a,
the reliability function is given by:
???Rt
1 , 0
,
0
a
bt a
b
t
tb
, b < t
a
???
??
??? ??
?
??
(3.24)
The hazard function is: ? ?? ? ? ?1() b
b t b
aftht
Rt a??
? ?
??/
/()
()
0 , 01
,
a
a
t
b bt t
????
? ??
???
, which is an IFR.
Hence the CHF is given by H(t) =
0
n
0
l
,
,
a
ba a
b tb
b
t
t
,t
???
?????
???
??
???
?
????
(3.25)
?
b
1/(b- )
36
3.8 Laplace Transforms
If ()ft is a known function of for values of , its Laplace transform ? is
defined by the equation,
? ? -s t
0
( ) (t) t
?
? ?f t e f d ,L (3.26)
and abbreviated as,
? ? ? ?f s ( )ft?L (3.27)
Thus, the above operation transforms the function ()ft of the real variable t into a new
function ? of the subsidiary variable s, which may be real or complex [91]. Often, it is easier
to work in the s-space than the t-space to obtain solutions to mathematical problems.
3.9 Chapter Summary
This chapter introduced the mathematical concept of the dissertation. The table below
shows the summary of reliability measures of most important base-line distributions. Only the
Logistic and Loglogistic density reliability measures, both of which have also reliability
applications, are not yet provided.
37
Table 2: Reliability Measures of Most Common Base-line Distributions
Lifetime
Distribution
Failure Density
f(t)
Survival Function R(t) Hazard
Function
MTTF
Exponential te??? ??te ? 1/?
Weibull
()1() tt e ??? ????? ? ? ? ??? ????
()te ?????? ? 1()t ???? ? ? ? ???? ()[(1/ ]1) x ? ? ??????
Gamma
1()() ntten ?? ?? ?? 1
0
()!kn t
k
t ek ??? ?
??
, when n is
a pos. integer
? ? ? ?f t tR/
n/?
Lognormal
2ln()1
2
te
t
??
??
??
ln( )[]?? t? ?
? ? ? ?f t tR/
2/2e???
Beta, t= ? +
(U ? ?) x ,
0 ? x ?1
11() (1 )( ) ( ) abab xx??? ??? ?
Standard Beta pdf
1 , ,tb e ta d is t a bU ?????? ?????
? ? ? ?f t tR/
? ?()Uaab?? ???
Normal ? ? ? ?2 2/21
2 x?e? ? ??? 1 ????? ????t??
? ? ? ?f t tR/ ?
Uniform 1 / ( ),
0, b t belsewheraae? ? ????
1 , 0
,
0
a
bt a
b
t
tb
, b < t
a
????
? ??? ?
??
??
0 , 0
1 ,
a
a
t
b bt t
????
? ??
???
? ?12b??
38
CHAPTER 4
4 Renewal Processes with Minimal Repair
Suppose that failures occur at times Tn (n = 1, 2, 3, 4, ?) measured from zero and
assuming that replacement (or restoration time) is negligible relative to operational time, then Tn
represents the operating time (measured from zero) until the nth failure, where T0=0 . Because
the pdf of T1 may be different from the intervening times X2 = (T2 ? T1), X3 = (T3 ? T2), X4 = (T4
? T3), ..., we consider only the simpler case of probability density function (pdf ) of time to first
failure f1(t) being identical to those of intervening times X2, X3, X4, ? as depicted below.
Figure 5: The Intervening Times of Two Successive Renewals
Note that X1, X2, X3 ? represent intervening times between failures, while Ti represents
time to the ith renewal measured from zero. Further, all Xi?s are assumed iid (independently and
identically distributed). The above figure clearly shows that Tn (Time to the nth renewal) =
ii1X??
n = sum of the times to the 1st failure from zero plus the intervening times of 2nd failure until
the nth failure. If n > 60, then the Central Limit Theorem (CLT) states that the distribution of Tn
0
X 1
T 1 T 2
X 2
T 2
T 3
X 3
T ime
39
approaches normality with mean n?, where ? = E(Xi) = the mean time between successive
renewals, i = 1, 2, 3, 4, ? and with variance 2n? , where ?2 = V(Xi). However, if the pdf of Xi,
f(x), X being the parent variable, is highly skewed and/or n is not sufficiently large, then the
exact pdf of Tn =
ii1X??
n is given by the n-fold convolution of f(x) with itself denoted as
? ? ? ? ? ? ? ?1 1*nT nf t f t f t?? , where ? ???1nft? is the pdf of the sum X2 +X3 +?+X n, or the (n ? 1)
convolution of f(t) with itself. Note that Figure 5 is also approximately valid for the case of
Minimal-Repair, i.e., the case when MTTR ? 0, or MTTR = Mean Time to Repair is negligible
relative to MTTF. Therefore, in this section by renewal we mean either the replacement of a
failed component with a brand-new one, or the case when the failed component can almost
immediately be repaired and consequently be put back on-line.
The simplest and most common renewal process is the homogeneous Poisson process
(HPP), where the intervening times are exponentially distributed at the constant inter-renewal (or
failure) rate ?. Because ? is a constant and intervening times are iid, a Poisson process is also
referred to as a homogeneous renewal process. Throughout this dissertation, we will establish
that only in the case of exponential failure Pr (Probability) law with CFR (Constant Failure-
Rate), the RNIF is identical to the constant instantaneous hazard function h(t) = ?. Further, it is
also well-known that for a HPP the ? ?V tNt ???? ?? , and hence the coefficient variation of ()Nt is
given by () 1/Nt tCV ??.
40
4.1 The Renewal Function M(t) during [0, t] with Minimal Repair
Because the two events ? ? ( ) N t n? and ? ? nTt? are equivalent, it follows that
? ? ? ? ? ? ( )n nP N t n P T t F t? ? ? ????? , where ? ? ? ? ? ? ? ? ? ?11( ) *nnF t F t F t? is the n-fold convolution
representing the cdf of
ii1X
n
nT ???
. Thus,
? ? ? ? ? ? ? ? ? ? ? ? ? ?1 ( ) 1 nnP N t n P N t n P N t n F t F t?? ? ? ? ? ? ? ?? ? ? ?? ? ? ?
It has been proven by many authors both in Stochastic Processes and Reliability
Engineering that the RF for the duration [0, t] is given by
n1 ()( ) [ ( ) ] ( )
?
??? ? nM t E N t F t
, (4.1a)
and ? ? ? ?
()1 2( 2 1 ) ( )nn nV FN t M tt
?
?? ? ?? ? ? ?? ? ? ??
(4.1b)
where the random variable N(t) represents the number of renewals that occur during the time
interval [0, t], and ()()nFt is the cdf (cumulative distribution function) of the n-fold convolution
of f(t) with itself, i.e.,
( ) ( )0( ) ( )
t
nnF t f d? ? ??
, where ()()nft is the n-fold convolution density of f(t).
It is also widely known that the RNI (Renewal Intensity) of Ft() by definition is given by
)( ( ) /t=? dM t dt. Authors in Stochastic Processes refer to )(t? as the renewal density, while
some authors in Reliability Engineering refer to )(t? as the RNIF (Renewal Intensity Function);
41
because it is also well known that )(t? is never a pdf, throughout this dissertation we refer to it
as the RNIF. The reader should distinguish between the RNIF )(t? and the generally constant
and unit-less traffic intensity parameter in Queuing Theory defined as ? = (average arrival rate
of customers)/ (average service rate) = expected fraction of time a single server is busy. Further,
an approximate expression for the third raw moment of N(t) is given by Kambo et al (2012) [92].
Their expression for E[N(t)3] can be used to approximate the skewness of N(t).
It is also well-known that for a homogeneous renewal process the pdf of interarrivals Xi?s
is given by () ???? tf t e , and that of the time to nth failure (or arrival, or renewal) is given by
? ? ? ? ? ? ? ? 1 ?() ?n n tTnf t f t t e?n ? ???? (the gamma density with shape n and scale ? = 1/ ?). As a
result, the use of Eq. (4.1a) for the interval [0, t] leads to the RF ? ?()M t E N t??????
()=1 ()
??
nn Ft
=
1
1 0 ()()
t nx
n x x e d xn
? ? ??
? ?
? ??? ? = 1x
n1x0
( x )e d x( 1) !????
??
?? ???t nn= x
0x0
( x )e d x!???
??
?? ??t n
n n
=
x0
dx
?
??t =?t , a fact
that has been known for more than 100 years. Further, the RNIF for a HPP is a constant and is
given by )(t=? ( ) / /??d M t d t d ( t) d t= ?. It has also been proven in the theory of stochastic
processes [see D. R. Cox and H. D. Miller (1968), pp. 340-347][93] that the ? ?lim = /
??t M tt ?
,
where ? = MTTF. In the case of CFR (Constant Failure-Rate), because the mean of the
exponential base-line distribution is ? = 1/?, then it follows that this last limiting result is an
exact identity only for the exponential density ??? te , 0 ? t < ?. Further, when MTTR (Mean
42
Time-to-Restore) is almost zero, then the renewal intensity is practically the same as the failure
intensity. In the case of exponential base-line distribution, with minimal repair, the point
availability function is simply equal to its reliability function ? ? ??? tR t e .
4.2 The Renewal Functions for Known Convolutions
Unfortunately, obtaining a closed-form expression for the RF for all distributions is not as
simple as the case of exponential interarrival times. This is due to the fact that the n-fold
convolution of most baseline distributions used in reliability analyses is not either known or
attainable.
4.2.1 The Renewal Function for a Normal Distribution
For the sake of illustration, suppose that the time between failures, Xi , i = 1,2, 3, 4, ?,
are NID(? = MTBF, ?2), i.e., normally & independently distributed with MTBF (Mean Time
Between Failures), and process variance ?2. Then, Statistical Theory dictates that time to the nth
failure (measured from zero) is the n-fold convolution of N(?, ?2) with itself, i.e., Time-to-the-
nth-Failure Tn = TTFn ~ N(n?, n?2). Hence, in the case of minimal-repair, from Eq. (4.1a) the RF
is given by
()11( ) ( ) ()
??
??
??? ???n
nn
tnt F t nM ?? (4.2)
where ? universally stands for the cdf (cumulative distribution function) of the standardized
normal deviate N(0, 1), and ()()nFt gives the Pr of at least n renewals by time t. Xie et al (2003)
43
[94] gives the same exact expression for the normal RF as in Eq. (4.2), which they used as an
approximation for the Weibull renewal with shape ? ? 3. It should be born in mind that the
normal failure law is approximately applicable in reliability analyses only if the coefficient
variation of T, denoted CVT, ? 0.15625 = 15.625% because the support for the normal density is
(? ?, ?), while TTF can never be negative (this assures that the size of left-tail below zero is
less than 1?10?10). If the CV is not sufficiently small, then the truncated-normal can qualify as a
failure distribution; From (2001) [59] discusses the RF for the truncated-normal. As an example,
suppose a cutting tool?s TTF has the lifetime distribution N(? = MTBF = 15 operating hours, ?2
= 2.25) with minimal-repair (or replacement-time), where CVT = 0.10. Then, Eq. (4.2) shows
that the expected number of renewals (or replacements) during 42 hours of use is given by
1
4 2 1 542 15 2 1 2 4 1 0 7( ) ). .(?
?
?????
nM
nn while M(62 hours) = 3.747561 expected renewals. Using
the limiting result ()Mt ? /t ? , we obtain M(42 hours) ? 42/15 = 2.80 (% relative-error = 31.82,
while M(62) ? 62/15 = 4.133333 with % relative-error = 10.29).
There is a more accurate approximation for ()Mt , through the second moment, given by
2 2 2( ) / + 2?? ? ? ?M tt ( ) / ( )= 2/ 0 .5 0(C V 1)Tt ? ?? [51]. For the above normal
nonhomogeneous Process, M(62 hours) = 4.1333333 + (2.25?225)/450 = 3.638333 (a much
closer approximation). We attempted to obtain a more accurate approximation through the third
moment for RF M(t), but the corresponding approximate Laplace transform had complex roots,
44
consistent with other findings. Further, for a Laplace-Gaussian process, the renewal intensity by
direct differentiation of (4.2) is given by
2222
11
1
2
?? ??
??
? ? ??? ( ) / /[]()( ) ( , ) tn ? n ?
nn
d M t? ? n ? n ? e
dt ?t n ?
(4.3)
where the symbol ? stands for the standard normal density. The value of renewal intensity at 42
hours for the N(15, 2.25) baseline distribution, from Eq. (4.3), is ?(at 42 hours) = 0.07883674
failures/hour. Note that the value of the hazard-rate at 42 hours is given by h(42) = f(42)/R(42) =
10.358138 failures/hour, where R(t) is the reliability function at time t. Because the normal
failure Pr law always has an IFR (Increasing Failure-Rate) h(t), then h(t) > ?(t). (Sheldon M.
Ross, 1996, pp.426-427) [68] proves that / ( )??t M t? / ?t ? 20.50(CV 1)T ? if h(t) is a DFR
(Decreasing Failure-Rate), and he further proves when h(t) is a DFR, then h(t) ? ?(t) for all t ?
0, and as a result R(t) ? ()eMt? , and we add that equalities can occur only at t = 0.
4.2.2 The Renewal Function for a Gamma Baseline Distribution
Suppose that the TTF of a hot-water heater has a gamma failure density with shape
parameter ? = 1.5 and scale ? = 1/? = 3.5 years [this is quite similar to the Example 9.6 on p.
226 of Ebeling (2010) [4]]. When the heater fails, it is replaced with a new one with the same
identical shape and scale (i.e., minimal replacement-time). Our objective is to obtain the RF
M(t) for t years of operation. In order to obtain the RF and RNIF of a gamma NHPP, we first
resort to Laplace-transforms (LTs).
45
It has been proven in theory of stochastic processes that the LTs of ()t? and M(t) are,
respectively, given by [for a proof see (Bhat,1984) [34], pp. 277-280)]
L{?(t)} = s
0
f ( s )() 1 f ( s )? (s) te t d t? ??? ?? ? , s > 0, (4.4a)
and
M()s = L{M(t)} = s
0
f ( s ) se ( ) d s [1 f ( s ) ] s? ( )t M t t? ? ???? (4.4b)
Further, it is also widely known that the LT of the gamma density ? ? 1( ) e
() tt tf ???? ?? ???
is given by f(s) = ?
(?+s)
?
?
, ?(s) =L{?(t)} = ?
(?+s) ??
?
??
, and the gamma M(s) = L{M(t)}=
?s[(? +s) ? ]???? . We used Matlab?s ilaplace at ? = 1/3 and ? =1.5, but Matlab(R2012a,64bit)
could not invert ?(s) to the t-space at ? = 1.5. It seems that when the shape parameter ? is not a
positive integer, there exists no closed-form inverse-Laplace transform for the gamma density;
however, when the underlying failure distribution is Erlang (i.e., gamma with positive integer
shape), then there exists a closed-form inverse Laplace-transform for ? = 2, 3, and 4; for positive
integers beyond 4 there does exist complicated closed-form expressions. In fact, for the specific
Erlang density with shape ? ? 2 and scale ? = 1/?, it is well known that M(s) = st
0
e (t)dtM? ?? =
46
2
22? 1 14 s 4 ( s 2 )s ( s 2 ? ) 2 s??? ? ? ???
=
2114s 4 (s 2 )2s???? ??
. Upon inversion to the t-space, we
obtain the well-known M(t) = E[N(t)] =L? 1{M(s) } = L? 1
2114 s 4 (s 2 )2s????????????
=
21 ?1e4 2 4 tt+ ???? . Hence, at ? ? 2, the RNIF is given by 2?22 t?? ? ? ?() ?() d M t?edtt , which is
quite different from the corresponding gamma (at ? ? 2) IFR HZF
? ? 10 ()( ) e ! k tkt ttekht ?? ? ??? ?? ? ? ??= ( ) / (1 )tt? ? ?? > ()?t for t > 0. We used Matlab?s ilaplace
function as an aid in order to obtain the RF and RNIF for the Erlang at shapes ? ? 3 and 4
which are, respectively, given below.
?()Mt L?1 333?s[(? + s) ? ]???????
??
= ( 3 / 2 )e c o s ( 3 / 4 ) s i n ( 3 / 4 ) / 3 / 3 1 / 3[]tt t t? ?? ? ? ? ? ? ? ?,
??t() L?1 333?(? + ) ?s????????? = ( 3 / 2 )1 e [ c o s ( 3 / 4 ) 3 s i n ( 3 / 4 ]3 {}? ?? ? ? ? ?t tt,
?()Mt 2 /8e / 8 e c o s ( ) s i n( ) / 4 3[]ttt t t????? ? ? ? ? ? ? ? ?,
??t() L?1 444?(?+s) ????????
??
= 21 e 2 e s in ( )4 []????? ? ? ?tt t
The gamma HZF at ? ? 3 is given by h(t) = 2 2 2( ) / ( 3 ) ] / (1 / 2 )t t t??? ? ? ? ? ?, which is not the
same function as the first ()?t above. At ? = 5, 6, 7 ? Matlab(R2012a) provides an expression
47
for ()?t only in terms of roots of a polynomial of at least order 5. The user has to find the roots
in order to obtain ()?t .
Referring back to our example where ? = 1/3 and ? =1.5, we use the cdf F(n)(t) =
Pr(Tn ? t) in order to obtain the RF directly from Eq. (4.1a):
1
11 0
?() ?( ) ( ) ( ?)
(
?? ??
??
?? ??? ? n ??
t x
nnnF t x eM nt dx?
1
110
( ? , )( ? ??????
??
??????
?t n u
nn
u e du ? t n
?n
, (4.5)
where ( , )?tn?? = Matlab?s gammainc 1
0
1( , ) () nutt n u e d un ?? ?? ????? ? represents the
incomplete-gamma function at point ?t and shape n?. In fact, ( , )?tn?? gives the cdf of the
standard gamma density at point ?t and shape n?. Thus, for the Water-heater example
? ?12M t years? =
1 (12 / 3.5, 1.5 )n n?
?
??
= 2.11934672 expected failures. Using the 2nd-order
approximation we obtain ? ? 2 2 2 2/ ( / ) ( / / ) / [ 2 ( / ) ]? ? ? ? ? ?? ? ? ? ?tM t =
/ 0 .5 0 (1 ) /t ??? ? ? ? = 2.1190476 (which yields a ?0.0141% relative error). We next directly
differentiate Eq. (4.5) in order to obtain the gamma RNIF. That is,
??()() dM t?
dtt 11 0 ()(
? ? ? ?
?
?? ???? ?t nx
n x e d xtn
??? = 1
1 0 ()(
? ? ? ?
?
?? ???? ?t nx
n x e d xtn
???
= 1
1 ()(
? ? ??
?
? ??? nt
n ten
???
(4.6)
48
However, the function under the summation in Eq. (4.6) is simply the gamma density
with shape n? and scale ? = 1/?. Then, we used our Matlab program, to obtain the value of Eq.
(4.6) at t = 12 years, at shape ? = 1.5 and scale ? = 3.5 years, which yielded ?(t =12) = 0.190348
renewals/ year. The value of the hazard-function at 12 years is h(12) = f(12)/R(12) =
0.0193613/0.0765932 = 0.252781 failures/year. Because the gamma density is an IFR model iff
(if and only if) the shape ? > 1, then ?(t) < h (t) for t > 0. Only at ? = 1, the gamma baseline
failure distribution reduces to the exponential with CFR, the only case for which ?(t) ? h (t) = ?.
In order to check the validity of ?(t = 12) = 0.190348, we resort to the limiting form
? ? 1 1 M T B Ftlim / /t ???? ??. Because the expected TTF of the gamma density is ? =???, then
for the Water-heater example ? = MTBF = 1.5?3.5 = 5.25, which yields ?(12 years) ? 1/5.25 =
0.190476/year. Note that since the renewal-type equation for the RNIF ?(t) is given by
? ? ? ? 0 ( x ) ( x ) d x ,t tft f t ?? ?? ?? this last equation clearly shows that ?(0) = f(0); further h(t) =
f(t)/R(t) for certain yields h(0) = f(0), and hence ?(0) = f(0) = h(0) for all baseline failure
distributions. Moreover, if the minimum-life ? > 0, then ?(?) = f(?) = h(?).
Some authors in Reliability Engineering, such as Ebeling (2010) [4], use the expression
0 ()(0, )ee
t x dx
Mt ??? ?? to represent the reliability function R(t) for a NHPP. Clearly, for the case of
gamma baseline distribution, which represents a NHPP, the above Ebeling?s expression is an
approximation because the exact unconditional reliability is always given by
49
0 ( x ) d x ()( ) e e
t h
HtRt ? ????, H(t) being the CHF (Cumulative Hazard Function). Leemis (2009) [8]
defines the RF as the cumulative intensity function using his notation ?
0
( ) ( )dtt? ? ? ? ?? ?, which is
identical to the RF M(t), where ?(t) is his notation for the RNIF. It seems that he is also using,
atop p. 146, the notation ?(t) as the hazard function for the Weibull. In section 4.3 it will also be
established that the HZF h(t) and the RNIF ?(t) of the Weibull are not the same, except at t = 0.
It is well known from statistical theory that the skewness of gamma density is given by ?3
= 2/ ? and its kurtosis is ?4 = 6/?, both of these clearly showing that their limiting values, in
terms of shape ?, is zero, which are those of Laplace-Gaussian N(?/?, ?/?2). We compared our
gamma program at ? = 70, ? = 15, and t = 5000 which yielded M(5000) ? 4.186155, while the
corresponding normal program yielded M(5000) ? 4.185793 expected renewals.
4.3 The Renewal Function when the Underlying TTF Distribution is Uniform
?Uniform distribution is used to model the time of occurrence of events that are equally
likely to occur at any time during an interval? [95]. Kececioglu (2002)[95] states that ?the most
frequently used distributions in Reliability Engineering are exponential, Weibull, normal,
lognormal, extreme value, Rayleigh (the Rayleigh being a special case of the Weibull with
minimum-life ? = 0 and shape ? = 2), and uniform?. Electrical bulbs, stress of mechanical
component which has lower and upper psi limits, network systems are some examples in which
uniform distribution is used in reliability models. Zhao and Duan [96] propose a reliability
50
estimation model of IC`s interconnect based on uniform distribution of defects on a chip. And
also ?the uniform distribution is used in Bayesian estimation as a prior reliability distribution?
([97], [95]) as an example please see [64].
Accordingly, suppose the TTF of a component or system (such as a network) is
uniformly distributed over the real interval [a, b weeks]; then ? ? 1/f t c? , a ? 0, b > a, c = b?a >
0, and the cdf is F(t) = (t?a )/c, a ? t < b weeks. Further, succeeding failures have identical
failure distributions as U(a, b). From a practical standpoint, the common value of minimum-life
a = 0. Then, the fundamental renewal equation is given by ? ? ? ?
1 0 ( ) ( ) d
t MtM t F t f??? ? ? ? ?
[[34], pp.277-280]. Since we are considering the simpler case of time to first failure distribution
being identical to those of succeeding times to failure, then
? ? ? ?
0
( x ) (x )d x ,? ?? ?tM t F t M t f (4.7a)
whereas before F(t) represents the cdf of T = TTF. However, Hildebrand (1962) [98] proves that
f(s)M(s)? = L
0
( x ) (x ) xt M t f d?????????? = L
0
( x) (x)t M t dF?????????? , the integral inside the first brackets
representing the convolution of M(t) with f(t). Conversely we can conclude that f(s)?M(s) =
51
L
0
( x) (x)t F t dM?????????? . Upon inversion of this last LT we obtain L
?1{ f(s) M(s)? } =
0
( x) (x)t F t dM?? =
0
( x) (x ) (x )t F t d??? . Hence, Eq. (4.7a) can also be represented as
? ? ? ? ? ?
00
( x ) ( x ) ( x ) ( x ) xttF t d M F tt t F t dMF ?? ? ????? ? (4.7b)
The renewal-equation of the type (4.7b) has been given by many authors such as [55], [99], [8],
and other notables.
In order to obtain the RF for the uniform density, we substitute into Eq. (4.7a), for the
specific uniform U(0, b) baseline distribution, for which a = 0, in order to obtain
? ? ? ?
0
t ( x ) d x/???????? ?tM t E N t M t bb; letting t? x= ? in this last equation yields
M(t) = 01 ( )( d )
t
tb Mb??? ??=
0
1 ( )d .ttb Mb? ? ??
(4.8)
The above Eq. (4.8) shows that the RNIF is given by ( ) 1 ( )() ? ? ?d M t M tt d t b b? ?
d ( ) ( )dtM t M tb? = 1b ? ttd ( ) ( )eedt bbM t M tb//??? = t1e b/? ? /d [ ( ) ]dt tbM t e? = t1e ba /? ?
/t() bMt e? = t/1 e d t Cbb ? ?? = t/eCb???, where C is the constant of integration. Applying the
boundary condition M(t = 0) = 0, we obtain ? ? ? ? = e 1bM t E N t ????? ? t/, where time must start at
zero, i.e., this last expression is valid only for 0 ? t ? b, b > 0. Note that Ross (1996) [68] gives,
52
without proof, the same identical ??Mt only for the standard U(0, 1) underlying failure density.
For example, if the time to down-state of a network is U(0, 8 weeks), then the expected number
of down-states during the interval (0, 8 weeks) is given by ()M 8weeks = 8/8e ? 1 = 1.718282.
In order to calculate the RF for the same uniform distribution during the interval
(2, 4) we may use the above result:
4
2
44 / 8 / 8
22
1( ) ] 0 . 3 6 4 6 9 68() ttt d t e d t eM 2 , 4 w e e k s = ? ? ? ???
Bartholomew (1963) [100] describes ()t? ??t as the (unconditional) Pr element of a
renewal during the interval (t, t+?t), and in the case of negligible repair-time,?t() also
represents the instantaneous failure intensity function. However, as described by nearly every
author in Reliability Engineering, the HZF h(t) gives the instantaneous conditional hazard-rate at
time t only amongst survivors of age t, i.e., ()ht ??t = Pr(t ? T ? t +?t)/R(t). The hazard function
for the U(0, b) baseline distribution is given by h(t) = 1bt? , 0 ? t ? b, b > 0, which is infinite at
the end of life-interval b, as expected. Because the uniform HZF is an IFR, then for the uniform
density it can be proven, using the infinite series for 1/1/btb? and the Maclaurin series for ebt/ ,
that h(t) >?t()for all 0 < t ? b.
Next in order to obtain the RF for the U(a, b), we transform the origin from zero to
minimum-life = a > 0 by letting ? = a + (b?a)t/b = a + ct/b in the RF ? ? ? ? =M t E N t?????
53
e1b?t/ . This yields, ()M? = ? ?e 1ac?? ? , and hence ? ?( ) = [ ( ) ] e 1t a cM t E N t ???, 0 ? a ? t < b,
and c = b ? a > 0. The corresponding RNIF is given by ? ?( ) e t a ctc? ???? ??, 0 ? a ? t < b.
Because the uniform renewal function is valid only for the interval [a, b], we will obtain
the n-fold convolution of the U(a, b)-distribution which in turn will enable us to obtain M(t) for t
> b by making use of Eq. (4.1a) that uses the infinite-sum of convolution cdfs, F(n)(t). As stated
by Olds (1952) [101], the convolutions of uniform density of equal bases, c, have been known
since Laplace. The specific convolutions of the uniform density with itself over the interval
[?1/2, 1/2] were obtained in [5] only for n = 2-fold, 3, 4, 5, 6, and 8-folds. There are other
articles on the uniform convolutions such as [102], [103], and [81]. We used the procedure in
Maghsoodloo & Hool (1983) [5] but re-developed each of the n = 2 through n = 8 convolutions
of U(a, b) by a geometrical mathematical statistics method. Further, we programmed this last
geometric method in Matlab in order to obtain the exact 9 through 12-fold convolutions of the
U(a, b) with itself. Convolution-densities of U(a, b) are given at the appendix.
The Matlab program, uses the exact n = 2 through 12 convolutions F(n)(t) and then applies
the normal approximation for convolutions beyond 12. The question now arises how accurate is
the normal approximation to F(n)(t), n = 13, 14, 15, ?? We used our 12 -fold convolution of the
standard uniform U(0, 1) to determine the accuracy. Clearly, the partial sum T12 = 12
ii1X??
, each
Xi ~ U(0, 1) and mutually independent, has a mean of 6 and variance 12(b?a)2/12 = 1, where a =
0, and maximum-life b = 1. Table 3 shows the normal approximation to F(12)(t) for intervals of
54
0.50-Stdev. The table clearly shows that the worst relative-error occurs at ? StDev, and that the
normal approximation improves as Z moves toward the right-tail. The accuracy is within 2
decimals up to one StDev and 3 decimals beyond 1.49 StDevs. Therefore, we conclude that the
normal approximation to each of F(n)(t), n = 13, 14, 15, ? , due to the CLT , should not have a
relative error at Z = 0.50 exceeding 0.002960.
Table 3: Normal Approximation to the 12-fold convolution of U(0,1)
It should also be noted that the normal approximation to the uniform F(n)(t), n = 13, 14,
15, ? must be very accurate from the standpoint of the first 4 moments. Because the skewness
of n-fold convolution of U(a, b) with itself is identically zero, which is identical to the Laplace-
Gaussian N(n(a+b)/2, n(b?a)2/12), and hence a perfect match between the first 3 moments of
? ? ? ?? ?nT nf t f t? with those of normal. It can be proven (the proof is at the appendix) that the
skewness of the partial-sum
ii1 Xn
nT
???
, Xi?s being iid like X, is given by
3 / 2 2 3 / 2 33 3 2 3 3 3[ X ) ( X (( ) ( ) ( ) ] ( ( ))( X))? ? ???nn nT ? ? n ? n ? ? ? nTT n = (4.9a)
Further, the kurtosis of Tn is given by
? ? ? ? ? ?4 4 444 33 X /? ( X ) / + 3 ( 1 ) ( X )/ [ ] /? ? ??? ? ? ? ? ?nn=T T nn n n n, (4.9b)
Z 0.5 1 1.5 2 2.5 3 3.5 4
F(12)(z) 0.689422 0.839273 0.932553 0.977724 0.994421 0.998993 0.999879 0.999991
Normal
Approx.
0.691462 0.841345 0.933193 0.97725 0.99379 0.99865 0.999767 0.999968
Rel-Error 0.002960 0.002469 0.000686 -0.00049 -0.00063 -0.00034 -0.00011 -2.3E-05
55
where ?i ( i = 2, 3, 4) are the universal notation for the ith central moments. Eq. (4.9b) clearly
shows that the kurtosis of the uniform ?? )(
nft
, for n = 13, 14, and 15, respectively, are 4? =
?1.20/13 = ?0.09231, ?1.20/14 = ?0.08571, and ?1.20/15 = ?0.08000, the amount ?1.20 being
the kurtosis of a U(a, b) underlying failure density. Thus, an n = 120 is needed in order for the
kurtosis of ? ???
nft
to be within 0.01 of Laplace-Gaussian N(n(a+b)/2, n(b?a)2/12). Fortunately,
the previous summary table clearly indicates that the normal approximation is superior at the
tails, where kurtosis plays a more important role, than the middle of ? ???
nft
density.
In order to compute the RNIF ?t() for t > b, we used two different approximate
procedures, one of which will be detailed in section 4.5. Eq. (4.1a) clearly shows that
nnn 1 n 1dt F t f tdt
??
??????? ( ) ( )( ) ( ) ( )
; the Matlab program knows the exact 1()()ft= f(t) and uniform
convolutions nft()() , for n = 2 , 3, ?, 12. For n > 12, it uses the ordinate of normal density N(n?,
n?2) approximation, where ? = (a+b)/2 and ?2 = (b?a)2/12.
4.4 Approximating the Renewal Function with Unknown Convolutions
Unlike the gamma, normal and uniform underlying failure distributions, the Weibull
base-line distribution (except when the shape parameter ? = 1) does not have a closed-form
expression for the n-fold cdf convolution F(n)(t), and hence Eq. (4.1a) cannot directly be used to
calculate the renewal function M(t) for all ? > 0. When minimum-life = 0 and shape ? = 2, the
Weibull specifically is called the Rayleigh pdf; we do have a closed-form function for the
Rayleigh ?(s) but it cannot be inverted to yield a closed-form expression for its ?(t). Because the
56
Weibull is the most important underlying mortality density in reliability analyses, we will use the
discretizing method to approximate the renewal function for three parameter Weibull
distribution. The parameter ? is the minimum-life or threshold (a location parameter), ? is
called the characteristic-life (? = ? ?? being the scale parameter), and ? is the slope (or shape)
parameter. Further, it is well-known that the Weibull?s HZF
1() () ?
0 , t < ?
ht ? t ? ,t ??
??
? ?? ??
???
is
an IFR (with CV < 100%) iff the slope ? > 1, h(t) = ? = 1/? is a CFR iff ? ? 1 (with CV =
100%), and it is a DFR iff 0 < ? < 1 (with CV > 100%). It must be highlighted that there
have been many articles on approximating the Weibull RF such as Jiang (2007)[66], From
(2001) [59], and other notables. Note that Murthy et al (2004) [99] provide an extensive treatise
on Weibull Models, referring to the Weibull with zero minimum-life as the standard model.
These last three authors also highlight the confusion and misconception resulting from the
terminologies of intensity and hazard function for the Weibull. Jin & Gonigunta (2008) [60] first
approximated the cdf of the 2-parameter Weibull (i.e., threshold ? = 0) by an optimum
generalized exponential function; then they obtained the LT of the corresponding generalized
exponential, which could be inverted to yield their actual Weibull RF.
4.5 Discretizing Time in Order to Approximate the Renewal Equation
Because M (t) = F(t) +
0
( ) ( )dt M t f? ? ??? and the underling distributions are herein
specified, the first term on the RHS (Right-Hand Side) of M (t), F(t), can be easily computed.
57
However, the convolution integral on the RHS,
0 ( ) ( )d
t M t f? ? ??? , except for rare cases, cannot
in general be computed and has to be approximated. The discretization method was first applied
by Xie [55], where he called his procedure ?THE RS-METHOD?, RS for Riemann-Stieltjes.
However, Xie [55] used renewal-type Eq. (7b) in his RS-METHOD.
The first step in the discretization is to divide the specified interval (0, t) into equal-length
subintervals, and only for the sake of illustration we consider the interval (0, t = 5 weeks) and
divide it into 10 subintervals (0, 0.50), (0.50, 1), ? , (4.5, 5). Note that Xie?s method does not
require equal-length subintervals. Thus, the length of each subinterval in this example is
0.50?t? weeks. As a result, 5
0
(5 ) ( )dMf? ? ??? ? /210
i = 1 ( 1 ) / 2
(5 ) ( )di
i
Mf? ? ?
?
?? ? , where the index i
= 1 pertains to the subinterval (0, 0.50), and i = 10 pertains to the last subinterval (4.5, 5). We
now make use of the Mean-value Theorem for Integrals, which states: if a function f(x) is
continuous over the real closed interval [a, b], then for certain there exists a real number x0 such
that b
a
f(x)dx? ? f(x0)?(b?a), a ? x0 ? b, f(x0) being the ordinate of the integrand at x0. Because
both M(t??) and the density f(t) are continuous, applying the above Mean-value Theorem for
Integrals to the 4th subinterval, there exists for certain a real number ?4 such that
2
3 / 2
(5 ) ( )dMf? ? ??? ? 44( 5 ) ( ) ( 2 3 / 2 )Mf????, 3/2 ? ?4 ? 2. As a result, 5
0
(5 ) ( )dMf? ? ??? ?
58
10
=1 (5 ) ( )(1 / 2 )iii Mf????
, where 0 ? ?1 ? 0.50, 0.5 ? ?2 ? 1, ?, 4.5 ? ?10 ? 5. Clearly the exact
values of ? ?5 iM ?? , i = 1, ?, 10 cannot in general be determined, and because in this example
( )(1/2)if ? = /2
( 1)/2
( )di
i
f ??
??
= Pr[(i?1)/2 ? TTF ? i/2], it follows that 5
0
(5 ) ( )dMf? ? ??? ?
/210
=1 ( 1 ) / 2
(5 ) ( )dii
i i
Mf? ? ?
?
?? ?. As proposed by Elsayed [35] who used the end of each subinterval,
we will approximate this function in the same manner by (5 0.50 )?Miwhich results in
M (5) ? F (5) + 5
0
(5 ) ( )d?? Mf? ? ? ? F (5) + /210
i = 1 ( 1 ) / 2
(5 0 .5 0 ) ( )di
i
M i f ??
?
?? ?= /210
i=1 ( 1 /2
( )di
i
f ??
?? ?
+
/210
i = 1 ( 1 ) / 2
(5 0 .5 0 ) ( )di
i
M i f ??
?
?? ?= /210
i = 1 ( 1 ) / 2
1 (5 0 . 5 0 ) ( )d[]{}
?
? ?? ?i
i
+ M i f ?? (4.10)
The above Eq. (10) is similar to that of (7.10) of Elsayed [35], where his subintervals are
of length 1?t? . We first used the information M (0) ? 0 at i = 10 to calculate the last term of Eq.
(10); further, at i = 9, Eq. (10) yields ( 8 1 ) / 2
8 / 2
1 (5 4 .5 ) ( )d[] ?? ?+ M f ??= 4 .5
41 (0 .5 0 ) ( )d[]?+ M f ??
.
However, M (0.50) represents the expected number of renewals during an interval of length t??
0.50. Assuming that t? is sufficiently small relative to t such that N(t) is approximately
Bernoulli, then M( t?? 0.50) ? 1 (0.50)+?F 0 (0.50)R? . Hence, at i = 9, the value of the term
59
before last in Eq. (10) reduces approximately to 1 (0.50)[]+F ?4.5
4
( )df ??? . At i = 8, the value
of Eq. (10) is given by 4
3.51 (1 ) ( )d[]?+ M f ??
, where M (1) = 2
i = 1 1 (1 0 .5 0 )[]{ ?? + M i
/2
( 1)/2
( )d }
?
? ?i
i
f ??, where M(0.50) has been approximated. Continuing in this manner, we
backward recursively solved Eq. (4.10) to approximate M (t). The smaller ? always leads to a
better approximation of M (t).
In order to check the accuracy of this approximation method, we first used it to
approximate M(t) at ? =1 (which is the exponential failure law with M(t) = ?t), t = 10000,
minimum-life ? = 0, ? = ? = 1000 = 1/?, and t? = 50 = 0.005t, approximation yielded
M(10000) ? 9.754115099857199 compared to the exact ?t = 0.001?10000 = 10, a percent
relative error of ?2.459 with cpu-time = 65.112829 seconds. While, at the same exact
parameters, our Matlab function at t? = 40 = 0.004t, M(10000) ? 9.802640211919197 ( a %
relative error of -1.97360) with cpu-time = 567.432046. We ran the same program with same
parameters by just changing t? . We observed that for smaller values of t? we really approach
the exact value. The table below depicts the results.
60
Table 4: Time Discretizing Approximation Method Results
t? M(t) App_M(t) Relative Error
Elapsed Time
(seconds)
10 10.0000000000 9.9501662508319 -0.49834% 40238.61714
25 10.0000000000 9.8760351886669 -1.23965% 2018.707614
40 10.0000000000 9.8026402119192 -1.97360% 567.432046
50 10.0000000000 9.7541150998572 -2.45885% 82.738732
100 10.0000000000 9.5162581964040 -4.83742% 65.112829
200 10.0000000000 9.0634623461009 -9.36538% 32.320588
250 10.0000000000 8.8479686771438 -11.52031% 21.066944
500 10.0000000000 7.8693868057473 -21.30613% 16.510933
1000 10.0000000000 6.3212055882856 -36.78794% 16.347242
Figure 6 shows relationship between t? and Relative Error. As t? increases Relative Error
increases. This implies that smaller t? leads to more accurate result.
61
Figure 6: Relative Error versus t?
Next, after approximating the Weibull RF, how do we use its M(t) to obtain a fairly accurate
value of Weibull RNIF ?(t)? Because
0 ( ) ( )()( ) l i mt M t t M ttd M tt dt ? ???? ? ???
, then for
sufficiently small t? > 0 the approximate ( ) ( )() M t t M t
tt ??? ???
, which uses the right-hand
derivative, and ( ) ( )() M t M t t
tt ??? ???
, using the left-hand derivative. Because the RF is not
linear but strictly increasing, our Matlab program computes both the left-and right-hand
expressions and approximates ?t() by averaging the two, where t and t? are inputted by the
user. It is recommended that the user inputs 0 < t? ? 0.10t.
-80.00000%
-70.00000%
-60.00000%
-50.00000%
-40.00000%
-30.00000%
-20.00000%
-10.00000%
0.00000%
10.00000%
20.00000%
0 200 400 600 800 1000 1200
?t
ve
rsu
s R
el_
Err
?t
Relative Error
Linear (Relative Error)
62
As a result, all available approaches have merits and demerits in terms of complexity,
computational time and accuracy. The method here is easy to implement basically for any
lifetime distribution and it gives fairly accurate results for smaller subintervals. However, the
downside is for smaller subintervals the computational time increases. And also it is not a closed-
form approximation. Based on given parameters the program calculates both renewal and
reliability measures. Therefore, this approach cannot be used in some maintenance optimization
models if is desired to have closed-form expression.
4.6 Chapter Summary
This chapter provided the RF and RNIF for the gamma and uniform underlying failure
densities. We also devised Matlab programs that output all the renewal and reliability measures
of a 3-parameter Weibull, normal, gamma, and uniform. We have highlighted that the RNIF
?t()is different from the HZF h(t) for t > 0, except in the case of CFR.
63
CHAPTER 5
5 Expected Number of Renewals for Non-Negligible Repair
Unlike the previous chapters, we now assume that MTTR (Mean Time to Repair) is not
negligible and that TTR has a pdf denoted as r(t). This chapter gives expected number of
failures, number of cycles and availability by taking the Laplace transforms of renewal functions.
5.1 Expected Number of Failures and Cycles
Let the variates X1, X2, X3,? represent TTF i be iid with the underlying failure density
f(x) having means MTBF = ?x and variance 2x? ; further, let Y1, Y2, Y3 , ? represent the ith
Time-to-Repair (TTRi), i = 1, 2, 3, 4,? with the same pdf r(y) having means MTTR = ?y and
variance 2y? . Then, Ti = Xi + Yi represents the time between cycles (TBCs) which are also iid
whose density is given by the convolution ? ? ( ) * ( )t tg f r t? , and whose Laplace transform (LT)
is given by g(s) f (s) r (s)??. Clearly the mean and variance of the cycle-times Ti?s are ?x + ?y
and 22xy???. As described by U. N. Bhat (1984) [34] there will be two types of renewals:
(1) A transition from a Y-state (i.e., when system is under repair) to an X-state (at which
the system is operating reliably),
(2) A transition from an X-state (or operating-reliably-state) to a Y-state (where system
will go under repair).
64
Let M1(t) represent the expected number of cycles (or number of renewals of type 1), and
M2(t) represent the expected number of failures (or renewals of type 2). Then, as stated by Bhat
(1984) [34] and E. A. Elsayed (2012), the LTs (Laplace-Transforms) of the two renewal
functions, respectively, are given by
1 g (s ) f (s ) r (s )M (s ) s [1 g (s ) ] s [1 f (s ) r (s ) ]???? ? ?
(5.1a)
2 f(s) f (s )M (s ) s [ 1 g (s ) ] s [ 1 f (s ) r (s ) ]??? ? ?
(5.1b)
The corresponding LTs of RNIFs (Renewal-Intensity Functions) are given by
1 f (s ) r (s )(s ) 1 f (s ) r (s )? ?? ??
, and
2 f (s )(s ) 1 f (s ) r (s )? ? ??
(5.2)
As an example, suppose TTFi ~ Exp(?) and TTRi ~ Exp(r); then as has been documented
by numerous other authors, s
0
f ( s ) e (/e) stt dt? ? ? ? ??? ???? and ? ?rs
0
r ( s ) r e e r / r stt dt? ???? ??
. On substituting these last 2 LTs into Eq. (5.1a), we obtain
1 rM ( s ) s [ ( s ) ( r s ) r ]?? ? ? ? ? ?
=
2 2 2r r rs s (s )?? ? ???? ? ? ? ?
, where r????
M1(t) = ? ?1 1M (s)?L = 1
2 2 2r r rs s ( s )? ???? ? ???????? ? ? ? ???L
=
22r r e tt ???? ? ??????
, which gives the
expected number of transitions from a repair-state to an operational-state (or expected number of
cycles). Similarly,
65
2 f (s )M (s ) s[1 f (s ) r (s ) ]? ??
= 22
2 2 2rs s (s )? ? ???? ? ? ? ?
, which upon inversion yields
? ? 22 222r e tt tM ??? ? ???? ?? ? , representing the expected number of failures during an interval of
length t. Note that the limit of both renewal functions M1(t) and M2(t) as r ? ? (i.e., MTTR ?
0) is exactly equal to ?t, as expected. Further, a comparison of M2(t) with M1(t) reveals that
M2(t) > M1(t) for all t > 0, which is intuitively meaningful because the expected number of
failures must exceed the expected number of cycles for all t > 0. As an example, if ? =
0.0005/hour and repair-rate = 0.05, then M1(t =1000 hours) = 0.485246544456426, while
M2(1000) = 0.495147534555436, and hence the availability will be shown below that
A(at t = 1000 hours) = 1+0.485246544456426?0.495147534555436 = 0.990099009900990.
We now obtain a general expression for the RF s M1 and M2 by inverting equations (5.1).
Eq. (5.1a) shows that 1 g (s)M (s)[ 1 g (s) ] s?? 11g (s)M (s) M (s) g (s)s??
? ? ? ? 11 t
0
M ( x ) g ( x ) dG x,M ttt ??? ?where G(t) is the cdf of f(x)*r(x) = g(x). Eq. (5.1b) now
shows that 2 f(s)M (s )[ 1 g (s ) ] s?? 22f(s)M (s ) M (s ) g (s )s??
? ? ? ?
02
t
2M ( x )F g ( x ) d xtM t t ??? ?
; thus, in general the expected number of cycles is given by
66
? ? ? ?
01
t
1M ( x )G g ( x ) d xtM t t ??? ?
(5.3a)
While the expected number of failures
? ? ? ?
02
t
2M ( x )F g ( x ) d xtM t t ??? ?
(5.3b)
For example, suppose TBFs ~ N(?x = MTBF, 2x? ) and TTR is also N(?y, 2y? ); then TBCs
~ N(?x+?y, 22xy???). Then, M1(t) =
1 ()n
tnn?
?
??? ?? , where ? =?x+?y, 22xy? ??? ? , and M1(t)
gives the expected number of renewals of the first type, i.e., the expected number of cycles.
However, because the system is under repair a fraction of the times, then
? ? 12 ()xnxtn nMt ?? ?? ?? ??. In order to obtain a good approximation for M2(t) and the resulting
A(t), we may argue that the expected duration of time the system is under repair is given by M1(t)
?MTTR; letting t2 = t ? M1(t) ?MTTR, then Eq. (5.3b) shows that the expected number of
failures is approximately given by ? ? 2
2 2( ) ( )
?
?
??? ? ? ????? ?xx
x xn
tn nt tM . Clearly, M2(t) ? M1(t)
for all t ? 0.
5.2 Availability
Because we are assuming that a system can be either in an operational-state, or under
repair, then the reliability function must be replaced by the instantaneous (or point) availability
67
function at time t, denoted A(t), which represents the Pr that a repairable unit or system is
functioning reliably at time t. Thus, if there is no repair, the availability function is simply A(t) =
R(t), the reliability function . However, if the component (or system) is repairable, then there are
two mutually exclusive possibilities:
(1) The system is reliable at t, in which case A1(t) = R(t),
(2) The system fails at time x, 0 < x < t, gets renewed (or restored to almost as-good-as-
new) in the interval (x, x+?x) with Pr element ?(x) dx, and then is reliable from time x to time t
Trivedi (1982) [104].
This second Pr is given by ? ?
02
( x ) d x ( x)? ??t RA tt ? . Because the above two cases are
mutually exclusive, then
? ? ? ? ? ? ? ?
1 02 ( x ) ( x ) d x ? ? ? ? ??
t RA t A t A t tRt ? (5.4).
Taking Laplace transform of the above Eq. (5.4) [and observing that the integral is the
convolution of R(t) with ?(t)] yields
A(s) = R(s) + R(s) (s)? = R(s) [1+ (s)? ]=
R(s) [1 + f (s) r (s)
1 f (s) r (s)???
] = R(s)
1 f (s) r (s)??
, (5.5)
where r(t) is the density of repair-time. For the case when the TTF (of a component or a system)
has a constant failure rate ? and time to repair is also exponential at the rate r,
68
s
0
R ( s ) e e ( )/stt dt? ? ? ? ?? ??? ? ; hence, the Laplace-transform of availability from Eq. (5.5) is
given byA(s) = 1 / ( s )
1 [ / ( s )][ r / ( r s )]??? ? ? ? ?
= rs
s[s ( r)]?? ??
= r/
ss???? ??
?
? ? ? ?1trA ( s ) eAt ? ? ??? ? ???L where ? = ? + r, which is provided by many other authors in
Reliability Engineering such as C. E. Ebeling (2010) [4], E. A. Elsayed (2012) [6], etc. For
example, given that ? = ?failure-rate = 0.0005 and r = ?repair-rate = 0.05 per hour, then ? = ? + r =
0.0505 and the Pr that the unit is available (i.e., not under repair) at t = 1000 hours is given by
? ? 0 . 0 5 0 5 ( 1 0 0 0 )0 . 0 5 0 . 0 0 0 5 e10 0 . 0 5 0 5 0 .0 0 0 . 9 9 0 005 990099005 1A ????, while R(1000 hours W/O Repair) =
0.5e? = 0.60653066 < A(1000) = 0.9901. Note that in the exponential case, we can also obtain
the availability function A(t) directly from Eq. (5.4) as follows:
? ? ? ? ( x )11
00
( x ) ( x ) d x e e ( x ) d x? ? ? ? ??? ? ? ???tt ttRAR t ?tt ?, where
xx11 22d r r r r r( x ) d ( x ) / d x x e edx?M ? ? ? ????? ? ? ? ?? ? ? ? ? ???? ? ?????. Upon substitution of this
RNIF into the expression for A(t), we obtain ? ? ( x ) x
0
rre e ( 1 e ) d x e? ? ? ? ? ? ? ? ???? ? ? ? ?? ? ??tt t tAt ,
as before. As pointed out by E. A. Elsayed (2012, pp. 466-467) [6], we also observe that
69
s
0
R (s) ( )te R t dt? ?? ? = s
0
e [(1 ( )]t F t dt? ? ?? = s
0
1 e ( )s t F t dt? ?? ? =1 F(s)s? . Hildebrand (1962) [98]
proves that F(s) f(s) / s? so that 1 f(s)R(s)= s? ; on substitution into Eq. (5.5) we obtain
1 f (s )A (s ) s [1 f (s ) r (s )]?? ?? = 1 f ( s )s [1 f ( s ) r ( s ) ] s [1 f ( s ) r ( s ) ]?? ? ? ? =
1 f ( s ) r ( s ) f ( s )s s [1 f ( s ) r ( s ) ] s [1 f ( s ) r ( s ) ]???? ? ? ?. Inverting these 3 LTs, we obtain
? ? ? ? ? ?12 1 A t M t M t? ? ? (5.6)
for all underlying failure distribution f(t) and TTR-distribution r(t). Eq. (5.6) is the same as that
of E.A. Elsayed (2012) [6] on his page 467.
5.3 Markov Analysis
Note that we can also use Markov analysis, as has been done by many authors in
stochastic processes, in the case of constant failure and repair rates to obtain the availability of a
simple on & off- system as depicted in the following Figure:
0 1
?
r
Figure 7: On & off system
70
where state ?0? represents a system in the reliable-state and ?1? represents the same system
under repair. The above figure clearly shows that 0 0 1d P ( ) / P ( ) r P ( ) ,t d t t t? ?? ? where P0(t)
represents the unconditional Pr of finding the systems in the operational state ?0? at time t, and
similarly for P1(t). Because P1(t) = 1?P0(t) for all t, we obtain 0 0 0d P ( ) / P ( ) r (1 P )t d t t? ?? ? ?
and hence 00d P ( ) / ( r ) P ( ) rt d t t? ? ? ?, or 00d P ( ) / P ( ) rt d t t? ? ?. This is a simple differential
equation with the integrating factoret? . Solving and applying the boundary condition
0P ( 0) 1t ?? results in 0 rP ( ) e tt ???????, which is identical to the A(t) obtained above in 2
other different methods.
It should be noted that, although the solution
0 rP ( ) e tt ???????
is valid exactly iff both
failure and repair rates are constants, it can be used to obtain a rough approximate solution for
the A(t) when only the MTBF and MTTR are available. For example, suppose that a system?s
TTF ~ U(0, 2000 hours), while TTR is also U(10, 30 hours); then, the MFR (Mean Failure Rate)
? 1/2000 = 0.0005, and the MRR = 1/20 = 0.05. Thus, a rough approximate solution for A(t) in
this case of uniform TTF and TTR is also given by
0 0 . 9 9 0 0 9rP ( ) e 901tt ???? ? ???
, while this
last availability value is exact only for the exponential cases of TTF and TTR.
71
5.4 Renewal and Availability Functions when TTF is Gamma and TTR is Exponential
It is well known that the LT of an underlying gamma failure density with shape ? and
scale ? = 1/? is given by f (s) ? / (? + s)? ??; note that only when ? is a positive integer this last
closed-form is valid. When ? is not an exact positive integer, there is no closed-form solution
for the LT of a gamma density because the integration-by-parts never terminates. Thus, in the
case of shape being an exact positive integer, i.e., Erlang underlying failure-density, we have:
A(s) = 1 f (s)s[1 f (s) r (s)]???=
?1
(? + s)
?rs[ 1 ]
rs(? + s)
?
???
?
?
?
?
= (? + s) (s r) ? (s r)
s [ (? + s) (r s) ? r]? ? ???
??
??
. At ? = 2 this last
LT reduces to A(s) = 2
22s ( 2 r ) s 2 ?rs [ s ( 2 r ) s 2 ? r]? ? ? ?? ? ? ? ? ?
= 12
2
3
1r
cccs s s r????, where 1r & 2r are the
roots of the polynomial 22s ( 2 r ) s 2 ? r 0=? ? ? ? ? ?. Thus, 21 ( ) ( r / 2 ) rr r / 2? ? ? ? ???,
22r r / 2 ( ) ( r / 2 ) r?? ? ? ? ?? , 1 2r /c (2r? ??, 1
2 2( 2 r )c ( 2 r ) r 4 rr?? ? ? ?? ? ? ? ?
, and
23 2( 2 r )c ( 2 r ) r 4 rr? ? ? ?? ? ? ? ? . Inverting back to the t-space we obtain
? ? 12rr232 r / 2 r c ec( ) e ttAt ?? ? ? ?. This last availability function clearly shows that as t ? ?,
??At ? 2r/ 2( r )?? = /2(rr )/?? = MTBF/(MTBF+MTTR), and further, A(0) ? 1, as expected.
For example, if a system has an underlying gamma failure distribution with shape ? = 2, scale
72
? = 1/? = 1000 hours and TTR has a constant repair-rate r = 0.05, then the availability at 500
hours is given by A(500) = 0.993855509027565; while the same system with minimal repair has
an ? ? ? ? ( ) e5 0 0 5 0 0 0 . 9 0 9 7 9 5 9 8 9 5 6 8 9 5 0( 1 + )xt
t
x e dA t R xt? ? ? ? ?? ? ? ? ?? ?? ?. That is, repair will
improve availability by 9.24%. The same system has an A(1000 hours) = 0.991466622031406,
and R(1000 hours, no repair) = 0.735758882342885; now repair will improve availability by
34.75%. Thus, the steady-state (or long-term) availability of such a system as discussed by many
other authors is A = 0.05/(0.05+0.0005) = 0.99009901.
At ? = 2 the LT of expected number of cycles reduces to
2
1 2 2 2r ?M ( s ) s [ s ( 2 r ) s 2 ? r ]? ? ? ? ? ? ?
= 5 6 74
2 12c c ccs s s r s r? ? ???
, where 1r & 2r are the same
roots,
24 r(2 r)2)c (r? ?????
, 5 (c r / 2r)?? ?? , 4
6 522c c rc r 4 r?? ??
, and 5
7 412c r cc r 4 r?? ??
. Upon inversion,
we obtain ? ? 12rr671 4 5 cc e c ec ttM t t? ???. For the same parameters as above, we obtain
M1(t = 10,000 hours) = 4.695618077086354 expected cycles. Similarly, it can be shown that the
LT of the expected number of failures is given by 2
2 2 2 2? ( r s )M ( s ) s [ s ( 2 r ) s 2 ? r ]+? ? ? ? ? ? ?
=
8 9 1 0 112 12c c c cs s r s rs? ? ???, where 228 2( r ))c ( 2r????? , 9 (c r / 2r)?? ?? , 810 192( 2 r r )rc cc4r??? ????, and
281 921 ( 2 r r ) cc cr 4 r? ? ?? ?? ?? . Upon inversion to the t-space we obtain
73
? ? 12rr1028 119 c e c c c ettM t t ?? ? ?. The value of expected number of failures during a mission of
length 10,000 hours is M2(t =10000) = 4.705519067168098, which exceeds M1(10000) =
4.695618077086354, as expected. Further, M1(10000) ?M2(10000) + 1 = 0.990099009918256,
which is identical to the value availability function obtained from
???At 12r t r t( ) c 22 r / ee2r c3???? at t = 10000.
Unfortunately, when TTF is Erlang at ? = 3, 4, 5 & 6 and a specified constant repair rate
r, the corresponding denominators ? ? s [1 f (s)Ds r (s) ]??? has at least 2 complex roots, which
are generally complex conjugate pairs. Yet, after partial-fractioning, the LT?s can be inverted to
yield real-valued M1(t) and M2(t), as demonstrated below.
At ? = 3,
1 f (s ) r (s )M (s ) s[1 f (s ) r (s ) ]?? ??
=
3
3
3
3
?r
rs(? + s)
?rs[ 1 ]
rs(? + s)
? ?
???
= 3
33?rs [(? + s) r s) r? ]( ??
=
3
2 3 2 2 3 2?rs [ + ( 3 ? + r ) s ( 3 ? r 3 ) s ? 3 r ]s+ ? ? ? ? ?
= 351 2 4
2 1 2 3ccc c cs s s r s r s r? ? ? ?? ? ?
, where the root 1r
will be real, while 2r and 3r will be complex conjugates, i.e., both 23rr? and 23rr? will be real
numbers. In order to maintain equality in the above PFRAC (Partial Fraction), it can be shown
that 3
2 1 2 3rc rr r???
, 1 2 1 3 2
2 31 2 31c r r r r + r rrc rr? ??
; further, letting the constants
? ?1 2 1 2 3 1 1 2 1 3 2 3 c ( r r r r + r r )r r r ca ? ? ? ? ?, 21 1 2 32 ( r r +c )c ra ?? ?, then 3c , 4c , and 5c
74
are the unique solution given by C = ? ?34 15c c c Ab?? ??, where C is the 3?1 solution vector, b
is a 3?1 vector b= 1
2
1
a
a
c
????
?????
??
and the 3?3 matrix A = 2 3 1 3 1 2
2 3 1 3 1 2
r r r r r r
r r r r r r
1 1 1
????
? ? ?
??
. A Matlab program
was devised to obtain the expected number of cycles M1(t) as outlined above. The program also
uses similar procedure as above to compute M2(t) and the resulting A(t). The Matlab program
has the capability to compute the 3 renewal measures M1(t), M2(t), and A(t) for ? = 2, 3, 4, 5, 6
and 7.
75
CHAPTER 6
6 The Approximate Expected Number of Renewals for Non-Negligible Repair
As in chapter 5, we assume that MTTR (Mean Time to Repair) is not negligible and that
TTR (Time to Restore, or repair) has a pdf denoted as r(t) but this chapter gives the approximate
number of cycles, number of failures and the resulting availability for particular distributions.
Availability was explained in the previous chapter. The inverse Laplace transform of
Equation 5.5 results in the point availability A(t). If the underlying distributions are not
exponential, problems arise in inverting the Laplace transform [105]. Therefore numerical
solutions and approximations become the only alternatives for obtaining A(t) [35]. There are
numerous approximation techniques in the literature such as Sarkar & Chaudhuri (1999) [105]
uses Fourier transform technique to determine the availability of a maintained system under
continuous monitoring and with perfect repair policy. They also obtain closed-form expressions
when the system has gamma life distribution and exponential repair time. Ananda and Gamage
(2004) [106] consider statistical inference for the steady state availability of a system when
repair distribution is two-parameter lognormal and failure distributions are Weibull, gamma and
lognormal. There are also other papers in the literature that work on confidence limits for steady
state availability of a system like [107], [108] etc.
In this chapter in order to approximate availability and renewal functions two different
approximation techniques are discussed. First for some cases like Weibull TTF and uniform
TTR we managed to obtain the convolution of failure density f(t) and repair density r(t). Then
76
we used these convolution densities to approximate M1(t), M2(t) and A(t) by using time
discretizing approximation method that was discussed in Chapter 4.
However, obtaining the convolutions of f(t) with r(t) for the general classes of failure and
repair distributions is not always tractable, such is the case of both TTF and TTR being Weibull.
In these cases we used moment based approximation which only requires knowing the first four
row moments of failure and repair distributions. ?There are a number of cases where the
moments of a distribution easily obtained, but theoretical distributions are not available in closed
form? [109]. And also, efficient estimators for the various moments of the underlying
distribution could be calculated from the observed sample data [92]. Kambo et. al. (2012) [92],
uses first three moments of failure distribution in order to approximate the renewal function for
negligible repair and they conclude that the method produces exact results of the renewal
function for certain important distributions like mixture of two exponential and Coxian-2.
In this chapter, we propose an approximation for the evaluation of expected number of
cycles, number of failures and availability based on first four row moments of failure and repair
distributions where convolution of f(t) and r(t) is intractable. We conclude that the method
produces very accurate results for especially large values of time t.
6.1 Weibull TBF and Uniform TTR
Let the variates X1, X2, X3,? represent TTF i be iid with the underlying failure density
f(x) having means MTBF = ?x and variance 2x? ; further, let Y1, Y2, Y3 , ? repres ent the ith
Time-to-Repair (TTRi), i = 1, 2, 3, 4,? with the same pdf r(y) having means MTTR = ?y and
77
variance 2y? . Then, Ti = Xi + Yi represents the time between cycles (TBCs) which are also iid
whose density is given by the convolution ? ? ( ) * ( )t tg f r t? , and whose Laplace transform (LT)
is given by g(s) f (s) r (s)??. Clearly the mean and variance of the cycle-times Ti?s are ?x + ?y
and 22xy???.
Suppose the TBFs of a component (or a system) has the Weibull distribution with minimum
life ? = t0 ? 0, characteristic life ? > ? = t0, and shape (or slope) ? > 0, i.e., TTF ~ W(?, ?, ?).
Letting ? = 1/(? ? t0), the density of X = TBFs (Time Between Failures) is given by
000[ ( x t ) ]1( x ) [ ( x t ) ] e , t x 0. We are
considering only the simpler case of the TTFF (Time to first Failure) and TTFi, i = 2, 3, 4, ?
having the same Weibull distributions, and also succeeding repairs have the same identical U(a,
b) distributions. Then, the Time-Between-Cycles is given by TBCs = TBF + TTR; we used a
geometrical mathematical statistics method to obtain the exact convolution of f(x) with r(y),
denoted g(t). The corresponding pdf of TBCs, g(t), is given below.
? ? 0
00
[ ( t ) ] 00
[ ( t ) ] [ ( t ) ] 0
1 e c t t( ) * ( )
e e c , t <
{ } /
{ } /
ta
t b t a
, a +gt t b +f t r t
b + t
? ? ? ?
? ? ? ? ? ? ? ?
? ??? ???
?? ? ? ?
?
?
??
(6.1a)
78
The above density has no closed-form (or explicit) antiderivative, except when ? ? 1, but
Matlab can integrate g(t) within any desired limits (t1, t2), a+t0 ? t1 < t2 < ?.
6.2 Uniform TBF and Weibull TTR
Conversely, suppose that the TBFs of a component or system has the U(a, b) density
function and its TTR has the W(?, ?, ?) density. Thus, the repair-rate function is given by
?r[r(t??]?, where ? represents the minimum repair-time. Only when ? =1, the repair-rate is
constant and is denoted by r, and at ? =1 the TTR has the exponential distribution. Because most
of TTR distributions in Reliability Engineering are positively-skewed, it is recommended that the
value of shape ? not to exceed 3. Then, the failure density is f(x) =1/c, a ? x < b, c = b?a, and
repair density is given by
? ? 1 [ r ( y ) ]y r [ r ( y ) ] e , y 0 and scale ? =1/?. The density of X = TBF (or uptime) is given by
1 ?(
()?() ? ) e 0xf x = x , x 0.
We are considering only the simpler case of the TTFF (Time to first Failure) and TTFi, i = 2, 3,
4, ? having the same gamma distributions, and also succeeding repairs have the same identical
U(a, b) distributions. Then, the Time-Between-Cycles is given by TBCs = TBF + TTR, and it
can be proven that TBCs has the following density, which is the convolution of f(t) with r(t), and
is denoted by g(t).
? ? ( ) * ( )
, <[ ( ) , ] /[ ( ) , ] [ ( ) , ] /{} c , a tgt bf t r t c b ttat a t b ???? ? ???? ??? ? ? ? ???? ? ? ?
(6.2a)
where () 1x
0
1 x e d x()[ ( ), ] ? ????? ?tata ? ??? ? ? is the cdf of the standard gamma density, at
?(t??). The above density has no closed-form antiderivative but Matlab can integrate g(t) for
any interval within a ? t < ?.
80
6.4 Uniform TTF and Gamma TTR
Conversely, suppose that the TTF of a component or system has the U(a, b) density
function and its TTR has the gamma density. Then, the failure density is f(x) =1/c, a ? x < b, c =
b?a, and repair density is given by
? ? 1rrx ( r
() ) e 0xr = x , x 0
2
( 2 ) 2 2() 2( 2 ) /( 2 ) / ata t cb abft a b ttc b
? ? ? ????
?
??
?????
? ? ? ? ? ?? ?
23
2 23
23
( 3 )
( 3 ) / 2
3 3 3 3 / 2 /
( 3 ) /
32
( ) 2 2
2( 2 ) 3
a t a b
f t a b t
a t c
a t c a t c ab
ab
c
t tbbc
?
?
? ? ? ?
??
? ? ? ? ??
?
? ? ? ?
? ? ?
?
?
?
? ? ? ?
? ? ? ? ? ? ? ?
? ? ? ? ? ? ? ?
3 4
32 2 3 4
( 4 )
32 2 3 4
34
4 / 6
( 3 4 1 2 4 1 2 4 4 ) / 6
( 3 4 2 4 4 6 0 4 4 4 ) / 6
(4
43
3 2 2
()
2
) / ( 6 )
23
34
a t c
a t c a t c a
a t a b
a b t a b
ft
a b t a b
t c c
a t c a t c a t c
bb
c
bt tc a
?? ???
? ? ? ? ? ?
?
?
? ? ? ? ?
??
?
? ? ? ? ?
?
? ??
? ? ? ? ? ?
?
?
? ?
? ? ? ?
? ? ? ? ? ? ? ? ? ?
? ? ? ? ? ? ? ? ? ?
? ? ? ? ? ? ? ? ? ?
4 5
4 3 22 3 4 5
4 3 22 3 4 5
4 3 22 3 4
(
5
5)
5 / 2 4
( 4 5 2 0 5 3 0 5 2 0 5 5 )
54
4 3 2/ 2 4
( 6 5 6 0 5 2 1 0 5 3 0 0 5 1 5 5 ) / 2 4
( 4 5 6 0 5 3 3 0 5 7 8 0 5
() 3 2 2 3
236 5 5 ) / 2 4
a t c
a t c a t c a t c a t c c
a t c a t c
a t a b
a b t a b
ft a b a ba t c a t c c
a t c a t c a t c a c c bt a
?
? ? ? ? ? ? ? ? ?
? ? ? ? ? ? ? ?
? ? ? ? ?
? ? ?
??
? ? ? ?
??
? ? ? ?
?
45( 5 ) / ( 2 4 )
4
45
t a b
a b tb t c b
?
?
?
?
?
?
?
? ? ? ?
?
?
? ? ?? ?
109
? ? ? ?
? ? ? ? ? ? ? ? ? ? ? ?
? ? ? ? ? ? ? ? ? ? ? ?
? ? ? ? ? ?
5 6
5 4 3 22 3 4 5 6
5 4 3 223
( 6 )
4 5 6
5 4 32
6 1 2 0
[ 5 6 3 0 6 6 0 6 6 0 6 3 0 6 6 1 2 0
[ 5 6 6 0 6 2 7 0 6 5
65
] 5 4 2
]47 0 6 5 8 5 6 2 3 7 2 3 360
[ 5 6 9 0 6 6 3 0 6( 2)
/
/
/
a t c
a t c a t c a t c a t c a t c c
a t a b
a b t a b
a b ta t c a t c a t c a t c a t c c
a t c a
ab
ft t c a t
? ? ?
? ? ? ?
? ? ?
??
? ? ? ? ? ? ? ? ? ?
? ? ? ? ? ? ? ? ? ? ?
? ? ? ? ?? ?
?
? ? ? ? ? ?
? ? ? ? ? ? ? ? ? ?
? ?
23 4 5 6
5 4 3 22 3 4
56
56
1 3 0 6 3 4 6 5 6 2 1 9 3 6 0
[ 5 6 1 2 0 6 1 1 4 0 6 5 3 4 0 6
] 3 3
1 2 2 7 0 6
1 0 9 7 4 1 2 0
24
24
( 6 ) ( 1 2 0 )
5
]
56
/
/
/
c a t c a t c c
a t c a t
a b t a b
a b t a b
ab
c a t c a
t
t c a t
cc
b t c b
? ? ? ?
?
?
?
?
?
?
??
? ? ? ? ?
?
?
?
? ? ? ??
?
?
? ? ?
? ? ? ? ? ? ? ? ? ?
?
?
??
? ? ? ?
? ? ? ? ? ? ? ?
? ? ? ?
? ? ? ? ? ? ? ?
? ? ? ?
6 7
6 5 4 323
24 5 6
6 5 4 323
24
7
( 7 )
5 76
76
6 5 2
)
7 72 0
6 7 42 7 10 5 7 14 0 7
10 5 7 42 7 7
15 7 21 0 7 11 55 7 32 20 7
49 35 7 39 90 7 13 3
()
7
52
,
] ( 72 0 ,
] / ( 72
/
0,
[
)
[
/
?
? ? ? ? ? ? ? ? ?
? ? ? ?
? ? ? ? ? ? ? ?
? ? ?
??
?
?
? ? ? ?
?
?
a t c
a t c a t c a t c a t
c a t c a t c
a t c a t c
a t a b
a b t a b
a t c a t
c a t c a t
a
t
c
c
c
f
? ? ? ? ? ? ? ?
? ? ? ?
? ? ? ? ? ? ? ?
? ? ? ?
7
6 5 4 323
24 5 6
6 5 4 323
2 56 74
10 7 21 0 7 17 85 7 78 40 7
18 79 5 7 23 52 0 7 12 08 9
15 7 42 0 7 48 30 7 29 12 0 7
96 81 0 7 16 80 00 7 11 91 8
43
4 3 3 4
34
2
] / ( 36 0 ) ,
] / ( 7
[
20 ) ,
[
? ? ?
??
? ? ? ? ? ? ? ? ?
? ? ? ?
? ? ? ? ? ? ? ?
??
?
?
??
?
a t c a t c a t c a t
c a t c a t c
a t c a t c a t c a t
c a t
b
c a t c
t a b
a b t a b
a
c
c
? ? ? ? ? ? ? ?
? ? ? ?
6 5 4 32
7
3
24 5 6
67
6 7 21 0 7 30 45 7 23 38 0 7
10 00 65 7 22 57 50 7 20 89 43
( 7 )
25
( 72 0 )
2 5 6
67
] / ( 72 0 )
[
/
,
,
? ? ? ? ? ? ? ? ?
? ? ? ?
? ? ?
?
? ? ? ?
? ? ?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
a t c a t c a t c a t
c a t c a t
b t a b
c
b
a b t a b
btt bc a
c
110
? ? ? ?
? ? ? ? ? ? ? ?
? ? ? ? ? ? ? ?
? ? ? ? ? ? ? ?
? ? ? ?
7 8
7 6 5 423
324 5 6 7 8
7 6 5 423
3245
(8
6
)
8 / 504 0
[ 7 8 56 8 168 8 280 8
280 8 168 8 56 8 8 ] 504 0
[ 21 8 336 8 218 4 8 756 0 8
1540
87
7 6 2
(
0 8 186 48 8 1
)
2488
/
a t c
a t c a t c a t c a t
c a t c a t c a t c c
a t c a t c a t c a t
c a t c a t
a t a b
a b t a b
ft
c
??
? ? ? ? ? ? ? ?
? ? ? ? ? ?
? ? ? ? ? ? ? ? ?
? ? ?
? ? ?
? ? ? ?
?
?
? ? ? ?
? ? ? ? ? ?? ? ? ?
? ?? ? ? ? ? ?? ? ? ? ? ?
? ? ? ? ? ? ? ?? ?
? ? ? ?? ?
78
7 6 5 423
324 5 6 7 8
7 6 5 423
3245
8 357 6 504 0
[ 8 / 144 8 / 6 5 8 / 3 9 8
256 8 / 9 53 8 488 8 / 9 247 7 / 105
6 2 5 3
]
53
]
[ 8 / 144 2 8 / 9 3 8 199 8 / 9
96 8 73
44
7 8 / 3
/
/
a b t a b
ab
a t c c
a t c a t c a t c a t
c a t c a t c a
t a b
t c c
a t c a t c a t c a t
c a t c a t
? ? ? ?
? ? ? ?
??
? ? ? ? ? ? ? ?
? ? ? ? ? ?
? ? ? ? ? ? ? ? ?
? ? ? ? ? ? ? ? ?
? ? ? ? ? ?? ? ? ?? ?
? ?? ? ? ?? ? ? ?? ? ? ? ? ?
? ? ? ? ? ? ? ?
6 7 8
7 6 5 423
324 5 6 7 8
7 6 5 423
344 8 642 49 / 315 ]
[ 8 / 240 8 / 6 17 8 / 6 53 8 / 2
264 7 8 / 18 967
4 4 3 5
35
8 / 2 156 83 8 / 18 139 459 / 210 ]
[ 7 8 336 8 6
2
8
6
88 8 781 20 8
5
/
/
c a t c c
a t c a t c a t c a t
c a t c a t c a t c c
a t c a t c
a b t a b
a
a t c a t
b t a b
? ? ?
? ? ? ? ? ? ? ?
? ? ? ? ? ?
? ? ? ? ?
? ? ? ?
??
? ? ?
?
?
?
? ? ? ? ? ? ? ?
324 5 6 7 8
78
289 20 8 213 544 8 8 475 333 6 8 449 119 2 ] 504 0
( 8 ) / ( 5
2
040 )
67
78
/ a b t a b
ab
c a t c a t c a t c c
b t c tb
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
? ? ? ?
? ? ??
?
?
?
? ? ?
?
? ? ?
??
111
? ? ? ?
? ? ? ? ? ? ? ?? ?
? ? ? ? ? ? ? ? ? ?
? ? ? ? ? ? ? ?
? ? ? ?
8
9
4 3 2
2 3 4
4 3 2
2 3 4 9
8 7 6 5
23
4
4
(
5
9)
9 / 403 20
[ 2 9 12 9 18 9 12 9 3 *
( 4 9 12 9 18 9 12 9 3 ) ] 403 20
[ 28 9 504 9 780 9 315 624 9
396 90 9 640 08
98
8 7 2
()
9
/
a t c
a t c a t c a t c a t c
a t c a t c a t c a t c c
a t c a t c a t c a t
ca
a t a b
a b t a b
f
t
t
t c a
?
? ? ? ? ? ? ? ? ?
? ? ? ? ? ? ? ?
? ? ? ?
? ? ?
? ? ? ?
?
? ? ? ?
? ? ? ? ?
? ? ? ?
? ? ? ? ? ?
? ? ? ? ? ?
? ? ? ? ? ?
? ?
32
6
7 8 9
8 7 6
2
5 4 3
3 4 5
2
6 7 8 9
8
642 60 9
367 92 9 920 7 ] 403 20
[ 56 9 151 2 9 173 88 9
111 384 9 43
7 2 6 3
6 3 5659 0 9 107 906 4 9
165 034 8 9 143 287 2 9 541 917 ] 403 20
[ 70
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116
Appendix B: 3rd and 4th Moment of Sums of iid rvs
The Skewness of the Sum of n independent and Identically Distributed (iid) Variates
Suppose X1, X2, ?, Xn are independently and identically distributed random variables
each with means ? and variances ? ? 22 2Xi V X ? ?? ? ? ?, where each Xi is identically
distributed like X. Let the nth partial sum n
ii1nS X???
; then, clearly ? ?=nE S n? and
? ? n2S 2== nV S n?? . It is well known that the ? ? ? ? ? ?33 3 3 / () /,nia S X x n a X n? ? ??
where ? ? 33 3 ,aX ?? ? and 33 , [( ) ]EX? ? ? ? the 3th central moment of X. The proof
follows.
3 3 3
113 ( ) [ ( ) ] [ ( ) ] [ ]? ()
nnn n i i
iiS E S n E X n E X? ? ???? ? ? ? ? ???
? ? ? ? ? ?321 1 16n n ni i i ji i jEX ? X ? X ?? ? ?????? ? ? ?????? ? ?=
? ? ? ? ? ? ? ?
3 3 3 3
11[ ] [ ] [
nn
i i i i i i iiiEX ? E X ? n E X ? n ? X??? ? ? ? ? ? ???
The corresponding skewness of Sn is given by;
? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ?3 / 23 / 2 233 3 2 3 3 3() nn i i? S n ? X ? n ? X n ? ? X ? n ? X nS ??
The Kurtosis of the Sum of n iid Variants
Suppose X1, X2, ?, Xn are independently and identically distributed random variables
with means ? and variances ? ? 22 2Xi V X ? ?? ? ? ?, where each Xi is identically distributed
like X. It is well known that the kurtosis of each xi is equal to ? ?44 3,X???? where
? ? 444, /X? ? ? ? and 44 , [( ) ]EX? ? ? ? the 4th central moment of X.
117
Now consider the partial sum
ii1S
n
n X???
; our objective is to compute the 4th central
moment of Sn from the known central moment of each identical Xi, i =1, 2,?, n. Clearly, the
mean of Sn is given by ? ? nE S n??, the variance is given by ? ? ? ? 2 niV S n V X n? ? ?, and
thus ? ? ? ? 44
ii4 ii 1 1E X E ( )S
nn
n nX????
??? ? ? ???? ? ? ?
? ? ? ???? ? ? ?? ? ? ?
?
?
???
14 2 2
i i ji 1 i 1 j42 1E ( ) ( ) ( C )
n n nX X X? ? ??
? ? ?
??? ? ? ? ? ???
??? ? ?
? ?4i ijn2 6 C V( ) V( )XnX X?? ???
Note that in the binomial expansion of 4n
ii1(X )????????? ?
, the expectation of odd products
such as 312[( )( ) ]E X X???? vanish due to mutual independence of Xi and Xj for all i ? j.
Hence,
? ? ? ? 444 3 ()1nS n X n n? ? ? ? ? ?
Thus, ? ? 4
4
44
22( S ) ( X ) 3 ( 1 )S V ( S ) () ( X ) / + 3 ( 1 ) /nn n 4n n nn ? n n n??? ? ? ???? ? ?
The corresponding kurtosis of Sn is given by,
? ? ? ? ? ?4 4 4 / .( X ) / + 3 ( 1 ) / 3 ( X ) 3 /[]nn44 Sn=SXn? n n ? n? ? ??? ? ? ? ?
118
Appendix C: Moments of the Most Common Base-line Distributions in Reliability
Table 8: Parameters and Density Functions of Most Common Baseline Distributions in
Reliability
Lifetime
Distribution
Failure Density f(t) Threshold
Or: Minimum-life
Location
Or: Shape
Scale
Exponential ()e t ??? ?? ? 1/?
Three
Parameter
Weibull
()1( ) ( ) ttf t e ??? ????? ? ? ? ??? ??? ??
?
?
???
Gamma
? ? 1 ( )[ ( ) ]() tteft ? ? ?? ???? ? ? ???
?
?
? =1/?
Lognormal
2l n ( )[]1() (): , , 2 tt tf e ????? ??? ? ???? ?
?
?
?
Normal ? ? ? ? ? ?2 2/21
2 t?f t e? ? ????
? ?
Logistic ( ) /
( ) / 2e[1 e ]g; 1),(
t
tt
??
??? ??
??
?????
? ?
Loglogistic [ ln ( t ) ] /
[ ln ( t ) ) ] / 2e[1 e ]1() ( t )ft
? ? ? ? ? ?
? ? ? ? ? ?????? ?
?
?
?
119
The Normal N(?, ?2); 1?? = ?; 2?? = ?2 + ?2; 3?? = 3?2?+ ?3; 4?? = 3?4 + 6?2?2 + ?4 ; 5?? = 15?4?+
5?2?3 + ?5 .
Two-Parameter Exponential pdf: 1?? = MTTF = ? + 1/?; 2?? = 2/?2 + 2?/? + ?2
The skewness ?3 = 2, while the kurtosis ?4 = 6. 3?? = 6/?3 + 6?/?2 + 3?2/? +?3
4?? = 9/?
4 + 4
31???? ?6 221???? + 3 41?? .
Three-Parameter Weibull pdf: Shape =?; 1?? = MTTF = ? + (? ? ?)??[(1/?) + 1];
Note that when ? = 1, then (? ? ?) = 1/?; the 2nd raw moment is given by
2?? = (? ? ?)
2??[(2/?) + 1] + 2?(? ? ?)??[(1/?) + 1] + ?2 ;
2? = V= (? ? ?)
2?[?[(2/?) + 1] ?
?2[(1/?) + 1)] = ?2
3
3 2 3/ 2
3 2 1 1? ( 1 + ) - 3 ? ( 1 + ) ? ( 1 + ) + 2 ? ( 1 + )
21[ ? ( 1 + ) - ? ( 1 + ) ]
? ? ? ?
??
?? , and the kurtosis is
4
4 3 1 2 1 124( 1 ) 4 ( 1 ) ( 1 ) 6 ( 1 ) ( 1 ) 3 ( 1 )
321 22
[ ( 1 ) ( 1 ) ]
? ? ? ? ? ? ? ? ? ? ? ? ? ? ?
??
? ? ? ? ?
? ? ? ? ? ? ?
??
= ?4-3
The 3rd raw moment is
3?? = (? ? ?)
3??[(3/?) + 1] + 3?(? ? ?)2??[(2/?) + 1] +3?2(? ? ?)??[(1/?) + 1] + ?3
4?? = ?
4?4 +4
31???? ?6 221???? + 3 41??
120
Lognormal pdf: 2l n ( )[]1()
(): , , 2
tt
tf e
?????
??? ?
????
?
, where ? =Threshold, or Min-Life,
? is a location parameter, and ? is scale. The MTTF = 1?? = ? +exp(?+?2/2) and
? ? 222 2 2eT eV ? ? ? ?????= 22ex ( ) ( )p 2 e x p 2? ? ?? ? ? ? = 222e (e 1)? ? ?? ? , and
222 2 / 2 22 e 2 e? ? ? ??????? ? ? ? , the skewness is ?3 = ?3 /?3 = 22
2
3
1.5
e 3e 2
(e 1)
??
?
??
?
> 0.
?4 = ?4 /?4 = 2 2 2
2
63
2
e 4 e 6 e 3
( e 1)
? ? ?
?
? ? ?
?
, and the kurtosis is
2 2 2 2
2
6 3 24
4 24 e 4 e 3 e/ 3 1 2 e 6( e 1 )
? ? ? ?
??
? ? ? ???
?? ? ?
, and the 3rd origin-moment is given by
3?? = 2 2 23 4 . 5 2 2 2 / 2 3e 3 e 3 e? ? ? ? ? ?? ? ?? ? ?? ? ?
4?? = 243 1 2 1 14 4 4 6 3? ? ? ? ?? ? ? ? ? ?? ??a
Gamma pdf: ? ? 1 ( )[ ( ) ]
() tteft ????? ? ? ? ??? ? ?
, ? = Threshold, ? = Shape, and ? =1/? = scale.
1?? = ? +?/?; V(T) = ?/?
2 =?2 22
2 / ( / )? ? ??? ? ? ? ? ? ; ?3 =2/? , ?4 =3+6/?
and the kurtosis is ?4 = 6/?,
3 3 2 2 3 3 2 2 3 23 2 / 3 / / 3 / 3 / 3 /? ? ? ? ? ? ? ? ? ??? ? ? ? ? ? ? ? ? ? ? ? ? ?, and 4?? =
243 1 2 1 14 4 4 6 3? ? ? ? ?? ? ? ? ? ?? ??a .
121
Logistic pdf: ( ) /
( ) / 2e[1 e ]g; 1),(
t
tt
??
??? ??
??
?????
; ? ? 1E T ???? ? and ? ? 2 2( ) / 3VT ?? ???
? ? 1ET? ?? = location = ?, ? = Scale; 2221( ) / 3 ( )???? ? ? ? ?, ? ? < t < ?; therefore, for RE-
analyses ? must exceed 10(??/ 3 ); the cdf is
G(t; ?, ?) =
( ) / ( ) / 11[1 e ] [1 e ]t t?? ???? ? ? ?? ??
? ? < t < ?.
The skewness is zero due to symmetry about ? and the kurtosis is 4
224 7 / 1 5( / 3 3 1 .2 0 0 0 0)? ? ? ???
,
i.e., the Logistic distribution has thicker tails than the corresponding normal with mean ? and
variance 2 2 2 /3?? ?? , i.e., the N(?, 2 2 2 /3???? ). For example, if the location of the mean
is ? = 200 hours and scale ? = 4.5 hours, then the cdf G(t = 185; 200, 4.5) =
0.034445195666211, while ?[(185-200)/ 8.162097139054] = Pr(Z? ?1.837762984739307) =
0.033048669103432; so, it seems the left-tail of the Logistic is a bit heavier than that of the
corresponding normal. At t = 180, the respective cdfs are 0.011607316445305 (Logistic), and
0.007135857827108 for the corresponding normal. So, as we move further on the tail, the
Logistic seems to become even heavier than the normal. The 3rd raw moment is given by:
233 ()? ? ??? ? ? ?, and 4 2 2 44 7 ( ) / 1 5 2 ( )? ? ? ??? ? ? ? ? ?.
Loglogistic pdf: A rv T has a Loglogistic pdf with threshold ? iff X = ln(T??) has logistic pdf .
T = eX + ?, where X is logistic. The cdf of Loglogistic is obtained as follows:
FT(t) = Pr(T ? t) = Pr(eX + ? ? t) = Pr(eX ? t??) = Pr[X ? ln(t ? ?)] = FX[ln(t ? ?)]; thus,
122
f(t) = dFT(t)/dt = dFX[ln(t ? ?)]/dt = dFX[ln(t ? ?)]/dx]?(dx/dt) = fX[at ln(t ? ?)]?(t??)?1 =
= -[ ln ( - ) - ]/
-[ ln ( - ) - )] / 2 -1e[ 1 + e ]1 ? ? ( )-
t ? ? ?
t ? ? ? t ??
= [ ln ( ) ] /
[ ln ( ) ) ] / 2e[1 e ]1()
t
tt
? ? ?
? ? ???
? ? ?
? ? ????
, t > ?, ? > 0.
Thus, cdf is F(t) =
[ln( ) )]/1[1 e ]t ? ? ?? ? ??
= [ln ( ) )]/ 1[1 e ]t ? ? ?? ? ? ?? , ? ? t < ?, where ? is the
Location-parameter and ? is the scale of the Loglogistic pdf. It seems that the 1st two moments
do not generally exist from Johnson & et al. on p. 152, where the rth raw moment is given by
E(Tr) = r/e ( r / ) c o s e c ( r / )?? ??? ??, where on a comparison of their Eq. (23. 89) atop their p. 152
with the above cdf of Loglogistic we must have 1/? = ?, and ?? = ?/?; then csc(r / )?? =
1/sin(r / )?? = 1/sin(r )? ; for the 1st raw moment r = 1, and 1/sin( )? = 1/0 , which does not
exist. Similarly none of the raw moments for r = 2, 3, 4, 5, ? , n exist.
Note that Johnson, Kotz & Balakrishnan define on their p. 151 the log-logistic variate,
X, as Y = ? + ?ln(X), where Y is the standard logistic, and X is Log-logistic according to their
definition. The pdf of their standard logistic Y is given by: ? ? y
y2Y e(1 ef )y
?
???
,
? ? y
y2Y e(1 ef )y ??
? ? < y < ? (Eq. 23.8 J. K.,& B [110])
In our above notation, we have X = ? +? ln(T), where now X is logistic and T is Loglogistic.
Then, T = e(X??)/?; this last expression clearly shows that the 3 authors (J. K. and B) [110] are
using ? as location and ? as scale, and no minimum-life. Proceeding as before, we have:
123
FT(t) = Pr(T ? t) = Pr[e(X??)/? ? t ] = Pr[(X??)/? ? ln(t) = Pr[X ? ?+ ?ln(t)] = FX[?ln(t) +? ]; thus,
f(t) = dFT(t)/dt = dFX[?ln(t)+ ? ]/dt = dFX[?ln(t) + ? ]/dx]?(dx/dt) = fX[at ?ln(t) + ?]?( ?/ T) =
[ ln(t) ]
[ ln(t) ] 2et{1 e }
? ? ??
? ? ????
; However, [ ln(t) ]e?? ?? = ln(t)ee?? ??? = ln(t )ee?? ??? = et?? ??? ;
Thus, f(t) =
2e[1 e ]ttt
?
?
?
??
?
?
?
??
=
2
2
2(e[1 e) ]ttttt
?
?
??
???
?
?
?
???
=
2e[e]ttt
?
?
?
??
?
???
=
2e[e 1]ttt
?
?
?
????
f(t) =
2
1e
(e 1)
t
t
?
?
?
??
?
?
t ? 0 and scale ? > 0. (Eq. 23.88, p.151 of J. K. &B) [110]
The cdf of T is given by F(t) = 1
1et ?? ???
= 1(1 e )t ??? ? ?? .
The raw moments of T are given atop p. 152 of [110] as follows:
E(Tr) = r/ rre csc( )?? ??? ???? , where rcsc( )?? = rcosec( )?? = r1/sin( )?? . (Eq. 23.90 p.152 of
[110])
From the above Eq. of [110] we obtain ? ? /e / s n )T i(E ?? ??? ???? ; this last clearly shows that
the mean exists iff sin( ) 0?? ? . Further, as ? ? ? , sin()?? ? 0 and then E(T) does not exist.
Further, for some 0 < ? < 1, sin()?? < 0 leading to a negative MTTF, which is not admissible.
124
We now compare the cdf of Loglogistic from Minitab, F(t) =
[ln( ) )]/1[1 e ]t ? ? ?? ? ??
[ ln ( ) ) ] / 1[1 e ]t ? ? ?? ? ? ?? , ? ? t < ?, with the form provided by [110] as F(t) = 1
1et ?? ???
=
ln( )
1
1 e et ? ?? ??
=
ln( )
1
1et ? ?? ??
=
ln( )11e t??? ? ??
=
[ ln( ) ]11e t??? ? ??
. Because, [110] do not have
minimum life, then comparing ln( )t???? against [ln(t?0) ??]/? shows that ? =1/?, and ?/? =
??. Hence, the raw moments of Minitab?s Loglogistic at zero-min-life are E(Tr) =
r/e ( r / ) c sc ( r / )?? ??? ??= re (r ) cosec(r )? ????= re (r ) / sin(r )? ????; this again shows that when
scale ? =1, 2, 3, 4, 5, ?. E(T) and E(T 2) do not exist. When ? = 0.50, E(T) exists but E(T2) does
not.
We now derive the first four moments of Loglogistic as follows.
E(T) = [ l n ( ) ] /
[ l n ( ) ) ] / 2
e
[1 e ]()
t
t
t d tt ? ? ?
? ? ?? ??
? ? ?
? ? ?
?
?? ??
; letting z= [ln( ) )] /t ? ? ??? results in
dz = 1[( ) / ]t dt???? E(T) = z
z2
e
[1 e ]
dzt ?
?
?
?? ??
; however, ?z + ? = ln( )t ??
zet ??? ??? ; thus E(T) = zz
z2
( e e
[1 e ]
) d z??? ??
?
?
??
?
??
= z (1 )
z2
e
[1 e ]
dze ??? ??
?
?
?? ?
? ? .
We now let u = ze? in the above last integral; then, du = ze dz?? , and du ( ze? ) = dz
1zue?? and 1dz u du???
125
E(T) = (1 )
2
0 zu
[1 u ]
d u ( e )e ??? ?
?? ?
?? ? = (1 )
20
u
[1 u ]
d u (1 / u )e ??? ???
?? ?
=
20
u
[1 u ]
due ??? ???
?? ?
.
Hildebrand (1962, p. 91) [98] proves that c
20
x
(1 x )
dx (1 c) (1 c) ,??
? ? ? ? ? ? ??
for all c within the open
interval (?1, +1), i.e., c must lie within ?1 < c < +1, or else the integral does not exist.
Therefore, from Hildebrand?s formula, we obtain
1?? = E(T) = e (1 ) (1 )?? ? ?? ? ? ? ? ? = e B e ta [ (1 ) , (1 ) ]?? ? ? ? ? ? =
e [(1 ), (1 )]B?? ? ?? ? ? , where 0 < ? < 1, where B represents the Beta-function.
The above Eq. clearly shows that the 1?? = E(T) of a Loglogistic exists iff 0 < ? < 1.
The Beta-function is defined as Beta( , )ab = ( , )Bab = 1 11
0
x (1 x ) dxab???? =
( ) ( ) / ( )a b a b? ?? ? ?, a & b > 0. Next we derive the 2nd raw moment of the Loglogistic
density.
E(T2) = [ l n ( ) ] /
[ l n ( ) ) ] / 2
2 e
[1 e ]()
t
t
t d tt ? ? ?
? ? ?? ?
? ? ?
? ? ?
?
? ??
?? ; The same z-transformation yields
2?? = E(T
2) = z
z22
e
[1 e ]
dzt ?
?
?
?? ?
?? = z z2z2 e[1 e ]dz( e )??? ? ?? ?
?? ?
??? =
zz
z 2 z 22 z 2 z 2
ee
[ 1 e ] [ 1 e ]
d z d z2 e e? ? ? ??? ??
??
????
? ? ? ???
? ? ? ???
126
2?? = z ( 1 ) z ( 1 2 )
z 2 z 222
ee
[ 1 e ] [ 1 e ]
d z d z2 e e?????? ? ? ? ?
??
??
? ? ? ???
????
Again, letting u = ze? results in u?1 = ez , ze dzdu ??? , or z1d z e d u u d u?? ? ? ?, we obtain
z(1 )
z2
e
[1 e ]
dz???
?
?
?? ??
= (1 )
2
0 1
(1 u )
u ( u du )?? ?
? ?
?? =
20 (1 u)
u du???
??
= (1 ) (1 ) (1 ,1 )B? ? ? ?? ? ? ? ? ? ? ?.
Similarly, z(1 2 )
z2
e
[1 e ]
dz???
?
?
?? ??
= (1 2 ,1 2 )B ????, 0 < ? < 0.50. Hence,
2?? = E(T2) = 222 e ( 1 , 1 ) e ( 1 2 , 1 2 )BB??? ? ? ? ? ?? ? ? ? ? ?, 0 < ? < 0.50
The variance of the Loglogistic is given by
2 2 2V ( T) 2 e ( 1 , 1 ) e ( 1 2 , 1 2 ) { e [ ( 1 ) , ( 1 ) ] }B B B? ? ?? ? ? ? ? ? ? ?? ? ? ? ? ? ? ? ? ? ?
= 2 2 2e ( 1 2 , 1 2 ) e [ ( 1 ) , ( 1 ) ]BB??? ? ? ?? ? ? ? ?
?2= 22e [ (1 2 , 1 2 ) (1 , 1 ) ]BB? ? ? ? ?? ? ? ? ?, 0 ? ? < 0.50; thus the variance does not exist
outside the range 0 ? ? < 0.50; it does not exist at ? = 0.50, and is identically equal to zero at ? =
0, which is not permissible. On a comparison with J. K. B. Eq.(23.90, p. 152), we may deduce
that V(T) = 2 2 2 2e [ 2 c s c ( 2 ) c s c ( ) ]? ? ? ? ?? ? ? ? ? ?, where csc( ) 1 / sin( )??? ? ?.
To determine the skewness, we proceed as follows:
3?? = E(T
3) = z
z23
e
[1 e ]
dzt ?
?
?
?? ?
?? ; because we know that minimum life does not impact the
variance, then for simplicity we obtain the 3rd raw moment for the case of ? = 0; thus zet ???? ,
127
and hence E(T3) = z
z23 z 3
e
[1 e ]
dze ?? ?
?
? ?
?? ?
?? = z (1 3 )z23 e (1 e dze )?? ????
?? ??
= 3e (1 3 ,1 3 )B? ????, 0 < ? <
1/3. The 3rd central moment is given by
? ?23 33e ( 1 3 , 1 3 ) 3 e ( 1 2 , 1 2 ) e ( 1 , 1 ) 2 e ( 1 , 1 )B B B B? ? ? ?? ? ? ? ? ? ? ?? ? ? ? ? ? ? ? ? ? ??? =
3 3 3 3e ( 1 3 , 1 3 ) e ( 1 2 , 1 2 ) ( 1 , 1 ) 2 e ( 1 , 1 )3B B B B? ? ?? ? ? ? ? ? ? ?? ? ? ? ? ? ? ? ? ? ?Hence, the
skewness is
3
2 1 . 5 03 ( 1 3 , 1 3 ) ( 1 2 , 1 2 ) ( 1 , 1 ) 2 ( 1 , 1 ) ,[ ( 1 2 ,3 011 2 ) ( 1 31 , ) ] /B B B BBB? ? ? ? ? ? ? ??? ???? ? ? ? ? ? ? ? ? ? ?? ? ? ?? ? ? ??
The 3rd raw moment is given by
3?? = E(T
2) = 3 2 2 33 e ( 1 , 1 ) 3 e ( 1 2 , 1 2 ) e ( 1 3 , 1 3 )B B B? ? ?? ? ? ? ? ? ? ? ?? ? ? ? ? ? ? ? ?
Similarly, the 4th standardized moment is
?4 =
2
24
2( 1 4 , 1 4 ) 4 ( 1 3 , 1 3 ) ( 1 , 1 ) 6 ( 1 2 , 1 2 ) ( 1 , 1 ) 3 ( 1 , 1 )( 1 2 , 1 2 ) ( 1 , 1 )[]B B B B B BBB? ? ? ? ? ? ? ? ? ? ? ?? ? ? ?? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ?? ? ? ? ?
,
Letting (1 ,1 ) B e t a (1 , )C 1B ? ? ? ?? ? ? ? ? ?, the above reduces to
?4 =
2
24
2( 1 4 , 1 4 ) 4 C ( 1 3 , 1 3 ) 6 C ( 1 2 , 1 2 ) 3 C( 1 2 , 1 2 ) C[]B B BB? ? ? ? ? ???? ? ? ? ? ? ? ? ?? ? ?
; thus, the kurtosis
is ?4 = ?4?3, only for 0 < ? < 1/4 = 0.25.