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Random proportional Weibull
hazard model for large-scale
information systems
Kiyoshi Kobayashi

Proportional
Weibull hazard
model
611

Graduate School of Management, Kyoto University, Kyoto, Japan, andKiyoyuki Kaito
Frontier Research Center, Graduate School of Engineering, Osaka University,
Osaka, Japan
Abstract
Purpose – This study aims to focus on asset management of large-scale information systems
supporting infrastructures and especially seeks to address a methodology of their statistical
deterioration prediction based on their historical inspection data. Information systems are composed ofmany devices. Deterioration process i.e. wear-out failure generation process of those devices isformulated by a Weibull hazard model. Furthermore, in order to consider the heterogeneity of thehazard rate of each device, the random proportional Weibull hazard model, which expresses theheterogeneity of the hazard rate as random variables, is to be proposed.
Design/methodology/approach – Large-scale information systems comprise many components,
and different types of components might have different hazard rates. Therefore, when analyzing faults
of information systems that comprise various types of devices and components, it is important toconsider the heterogeneity of the hazard rates that exist between the different types of components. In
this study, with this in consideration, the random proportional Weibull hazard model, whose
heterogeneity of hazard rates is subject to a gamma distribution, is formulated and a methodology is
proposed which estimates the failure rate of various components comprising an information system.
Findings – Through a case study using a traffic control system for expressways, the validity of the
proposed model is empirically verified. Concretely, as for HDD, the service life at which the survival
probability is 50 percent is estimated as 158 months. However, even for the same HDD, use
environment differs according to usage. Actually, among the three different usages (PC, server, others),
failures happen earliest in the case of PCs, which have the highest heterogeneity parameter and asurvival probability of 50 percent after 135 months of usage. On the other hand, as for others, its
survival probability is 50 percent at 303 months.
Originality/value – To operationally express the heterogeneity of failure rates, the Weibull hazard
model is employed as a base, and a random proportional Weibull hazard model expressing the
proportional heterogeneity of hazard rates with a standard gamma distribution is formulated. By
estimating the parameter of the standard proportional Weibull hazard function and the parameter ofthe probability distribution that expresses the heterogeneity of the proportionality constant between
the types, the random proportional Weibull hazard model can easily express the heterogeneity of thehazard rates between types and components.
Keywords Asset management, Infrastructure management, Information systems, Decision making,
Mathematical modelling
Paper type Research paper

Introduction
In order to achieve efficiency of operation and speedy provision of information to usersof large-scale infrastructure, information systems have been developed which are

Facilities
Vol. 29 No. 13/14, 2011
pp. 611-627
q Emerald Group Publishing Limited
0263-2772
DOI 10.1108/02632771111178373

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composed of various monitoring sensors and processing/output devices. Assetmanagement of infrastructure is an important issue as well. Points to be noted for asset
management of information systems are as follows:.
that the information system is a large-scale system formed from an enormous
number of components;.
that the system has a hierarchical structure whereby a failure in an individual
device can possibly develop into a functional failure of the whole system; and.
that the deterioration of the information system’s function, such as throughobsolescence or physical deterioration, is an important management point.
When performing asset management of information systems, it is necessary to consider
the differences in management at the component level, system level, and function level,with the latter points requiring a more integrated consideration. Among these, theauthors develop a random proportional Weibull hazard model to carry out a fault
analysis at the component level, of the components in large-scaled information systems.
Of course, in order to perform asset management of information systems, a fault analysis
at the level of each component is not enough, and it is necessary to develop amethodology that simultaneously achieves the above three points. The fault analysis
model for the component level proposed in this study can be a basic analytical tool for
constructing integrated asset management systems of large-scale information systems.
Information systems are composed of various types of components. These components
can be divided into two groups: the accidental failure rate of equipment, which isunrelated to time, and the wear-out failure rate, which increases over time. Generally,
regarding the fault process of accidental equipment failure, the hazard rate expressed by
the moment’s failure rate density can be formulated by an exponential hazard model that
does not depend on time (or a Poisson based model of failure events). On the other hand,with wear-out equipment failure, a non-homogeneous hazard model that takes into
consideration the hazard rate’s dependency on time is necessary. With wear-out
equipment failure, there is the characteristic that the failure rate increases as time
progresses after the system is installed. Therefore, in order for decisions to be made about
renewals and replacements, information regarding the deteriorating process of wear-out
equipment failure is needed. For this reason, the authors focus on wear-out equipment
failure, using a Weibull hazard model, known as a representative non-homogeneous
hazard model, to analyze the mechanism of the failure rate increasing over time. However,large-scale information systems comprise many components, and different types ofcomponents might have different hazard rates. Therefore, when analyzing faults of
information systems that comprise various types of devices and components, it isimportant to consider the heterogeneity of the hazard rates that exist between the
different types of components. In this study, with this in consideration, the random
proportional Weibull hazard model, whose heterogeneity of hazard rates is subject to a
gamma distribution, is formulated and a methodology is proposed which estimates thefailure rate of various components comprising an information system.
Basic ideas of this study
Overview of existing studies
In traditional hazard analysis (Cox and Oakes, 1984; Lancaster, 1990), the targeted
system is assumed to be constructed of elements of the same quality, and the aim is toexpress a model of failure generation randomly attained according to a certain hazard
function. With hazard analysis, the random failure generation process is modeled, anda deterministic model called a hazard function is employed. In reality, large-scale
information systems, such as the one targeted for actual analysis in this study, havecomplex structures composed of an enormous number of many types of components.
Managing and operating a large-scale information system requires important decisions
regarding policy issues such as when to change individual components and how
components should be stored. However, the failure rate of all the components might not
be expressed by the same hazard rate. Rather, it is natural to assume that the hazard
rate differs for each type of component. As for methods expressing the heterogeneity of
the hazard rates of different types of components, it is possible to express the differenceof the component characteristics in dummy variables or to consider the probability
distribution of the hazard rate. The former method is simple and easy to understand.
However, as the number of types of components increases, the number of dummy
variables needed to express the components’ characteristics also increases, causing theefficiency of the model’s estimation results to decline considerably. In fact, inlarge-scale information systems, even components of the same type, categorized bytype and device, have different deterioration characteristics depending on how they areused and where they are located, and these systems are structures with extremely
divided deterioration characteristics. In hazard models, it is not practical to use dummy
variables for deterioration characteristics sub-divided in this way. Therefore, in thisstudy the authors choose the latter method, using a random proportional Weibull
hazard model that expresses the heterogeneity of hazard rates of different components
in probability distribution, and formulating a model for the failure process ofcomponent groups that compose information systems.
There have been many studies on the hazard analysis considering the heterogeneityof hazard rate. Especially, there have been an enormous amount of researches into themixed hazard model in which the heterogeneity of hazard rate exists for each sample(Cameron and Trivedi, 1990, 1998; Cruz, 2002, Gourieroux and Visser, 1986; McNeil
et al., 2005; Mikosch, 2000). In the mixed hazard model, it is assumed that theheterogeneity parameters that determine the hazard function is subject to a certain
probability density function. The mixed hazard model is defined with the probabilistic
convolution of the hazard function and the probability distribution of heterogeneity
parameters. The ordinary Poisson process has the condition that the average of theprobabilities of rare events is equal to the variance of them. In this situation, the mixed
Poisson process model has been researched, in order to increase the degree of freedom
in describing the variance of probabilities in the Poisson process. Kaito and Kobayashi(2008) applied it to asset management, by producing a model of the arrival process ofroad obstacles. For analyzing the failure of a device that breaks down accidentally, it iseffective to use the mixed Poisson process model that reflects the heterogeneity of
failure rate. However, monitors, input and output devices, and CPUs, etc, whichconstitute information systems, are the devices that break down due to deterioration,and so it is necessary to take into account the heterogeneity of Weibull hazard rate. In
this study, the author proposes a random proportional Weibull deterioration hazard
model, which describes the heterogeneity of Weibull hazard rate with the gamma
distribution. The gamma distribution is a general exponential probability distribution,
including an exponential distribution, and has the characteristics that can describe a

Proportional
Weibull hazard
model
613

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broad range of probability distributions. In addition, the probabilistic convolution of
the Weibull distribution and the gamma distribution is simple, and it is possible toanalytically derive the random proportional Weibull deterioration hazard model.
Accordingly, it can be considered that this model has practically excellent features.
Basic frame of modelingThis is a model for the occurrence and process of failure events in information systems.
As Figure 1 shows, the components in information systems are in three layers:(1) Type;(2) Device; and(3) Components.
The type-layer contains hard disk drive (HDD), power supply, and processing andmonitoring equipment. An information system is composed of M-unit Typecomponents and each component is represented by the suffix i (i ¼ 1, . . ., M). Typei components are used for Ni-unit Device and each Device is represented by the suffix j( j ¼ 1, . . ., Ni). In a traffic control system, for example, each Type of component isused in different Devices such as personal computers (PC), servers, and so on. Since
each Device has its own application of components, its failure probability is different
from that of others. For Device j ( j ¼ 1, . . ., Ni ), Lij-unit Type i components are usedand each component is represented by the suffix k (k ¼ 1, . . ., Lij). Components in eachType and each Device are considered to have different hazard rates. But the failure
process in components in each Device is considered to be described by using the same
hazard rates. Here, an infinitely continuous time axis starting from time point t ¼ 0 isused. If the existing information systems as a whole are renewed at t ¼ 0, thedeterioration of each component starts from t ¼ 0. When a component has a failure, it
is immediately replaced. The new one is supposed to have the same performance thatthe old one had. Now, take a look at t ¼ Twhere a certain period of time has passed.
Then, a failure history is obtained as shown in Figure 1, which cites an example of a
failure history of Device 2 (server). Device 2 is composed of L2-unit components.

Figure 1.
Large-Scaled infromation
system and history data of
failure occurrences

Among them, component A has no failure from t ¼ 0. The time of use of component A
is T and the lifetime of component A is considered to be longer than T. On the other
hand, component B had a failure twice at T1 and T2. The first lifetime z ¼ T 1 and thesecond z ¼ T 2 2 T 1 .
Here, it is supposed that each type of component has the failure characteristics of
the wear-out failure component. With the wear-out failure component, as shown in
Figure 2, the generation rates of failures (hazard rates) grow as the elapsed time moves
farther from the nearest time of renewal. This type of lifespan distribution for thewear-out failure component is often used to express the components’ time-dependent
deterioration. It is assumed that this is subject to the Weibull distribution.
Furthermore, the hazard rates of different types of components can be expressed intime functions, as are shown in Figure 2. Functions like this that express the hazard
rate’s temporal change are called hazard functions. The hazard functions of eachcomponent are in an expansion or contraction by a factor of a baseline hazard function.
A model that expresses proportionally expanded or contracted functions is called a
proportional hazard model. If the failure process of each type of component thatcomposes each device can be mutually expressed with a proportional hazard model, theheterogeneity in hazard rates can be expressed by the probability distribution of
proportionality constants of hazard functions. Information systems comprise many
devices, but in most cases, the number of the same type of components in each device is
not that large. By estimating the parameter of the standard proportional Weibull
hazard function and the parameter of the probability distribution that expresses theheterogeneity of the proportionality constant between the types, the random
proportional Weibull hazard model can easily express the heterogeneity of the hazard
rates between types and components. On the other hand, if the heterogeneity of the
Weibull hazard rate cannot be expressed by the proportional hazard model, it becomes

Proportional
Weibull hazard
model
615

Figure 2.
Heterogeneity of hazard
functionsF
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necessary to estimate the Weibull hazard model for the different types and devices.
However, if the number of components of the same type that compose each device issmall, it becomes difficult to estimate the Weibull hazard model. If the above points are
considered, the random proportional Weibull hazard model proposed in this study isvery effective for expressing the failure process of information systems that have
sub-divided component structures.

616
Random proportional Weibull hazard model
Formulation of random proportional Weibull hazard model
Large-scale information systems are, as shown in Figure 1, composed of M types ofcomponents, and the component at number i (i ¼ 1, . . . , M) is used in a total of Ni
devices. Furthermore, the total number of type i components used in device j is Lij.
Among the type i components, the component at number k (k ¼ 1, . . . , Lij) out of thecomponents that compose device j ( j ¼ 1, . . . , Ni) is picked up. The elapsed time since
this component has been renewed will be expressed as tkij . The arrival rate of eachcomponent’s failure occurrence is based on the random proportional Weibull hazard
model.

lij ðt kij Þ ¼ 1ij gi mðtkij Þm21

ð1ÞHowever, gi is the parameter expressing the arrival density of type i, and m is theacceleration parameter. Equation (1) is the general Weibull hazard function (Aoki et al.,2007; Tsuda et al., 2006) with the parameter 1ij (hereinafter, heterogeneity parameter)
expressing the heterogeneity (Maher and Summersgill, 1996) of the hazard rates of typei and device j. The heterogeneity parameter expresses the heterogeneity of the hazard
rates between the components of different devices and types. Especially, in the casethat 1ij ¼ 1, the random proportional Weibull hazard function (1) matches the general
Weibull hazard function. This kind of hazard function is called a baseline hazard
function (see Figure 2). However, for the same component used in the same device, theheterogeneity parameter has a common value. The heterogeneity parameter takes adeterministic value in reality, but is a parameter that is impossible to observe. Also, theprobability density function f ij ðtkij Þ and survival probability F ij ðt kij Þ of the lifespan oftype i component k in device j are as follows.
nof ij ðt kij Þ ¼ 1ij gi mðt kij Þm21 exp 2gi 1ij ðtkij Þmð2aÞnoF ij ðtkij Þ ¼ exp 2gi 1ij ðtkij Þm

ð2bÞ

The value of the heterogeneity parameter is a probabilistic variable, which is subject toa certain probability distribution. The random proportional Weibull hazard model (1)has the same deterioration acceleration parameter m for all components, but the hazard
arrival density 1ij gi m expresses a proportionately different deterioration characteristic
for each type and device. In this study, a Weibull hazard model in which the hazard
arrival density has a proportional distribution for each type and device, is defined as arandom proportional Weibull hazard model.
Here, the heterogeneity parameter 1ij is subject to the gamma distribution.
Furthermore, use the case in which there are different averages of the heterogeneity

parameter for each type. Gamma distribution as a special form contains exponential
distribution, and can express the exponential probability density function family
defined by 0; 10. Also, it has the merit that it is easy to handle analytically. Here, theparameter gi expresses the average hazard arrival density of the type i component, andthat the heterogeneity parameter 1ij is a probability error term that is subject to the
gamma
distribution of average 1, variance f 21 . The gamma distribution is defined by0; 10, and regarding the arbitrary explanatory variable and probability error term,
the right side of equation (1) is guaranteed to have a positive value. Generally, theprobability density function gð1ij : a; bÞ of the gamma distribution Gða; bÞ can be
defined as follows.


1
1ij
1ija21 exp 2
ð3Þ
gð1ij : a; bÞ ¼ ab GðaÞ
b
The average of gamma distribution Gða; bÞ is m ¼ ab, and the variance is s 2 ¼ ab 2 .
Also, GðÞ is the gamma function. Furthermore, the probability density functiong ð1ij : fÞof the gamma distribution of average 1, variance f 21 can be expressed asfollows.
g ð1ij : fÞ ¼


f f f21
1ij exp 2f1ij
GðfÞ

ð4Þ

Estimating method of the model
The random proportional Weibull hazard model has unknown parameters such asarrival density parameters gi (i ¼ 1, . . . , M) for each type, an acceleration parameter m,different heterogeneity parameters 1ij (i ¼ 1, . . . , M; j ¼ 1, . . . , Ni) for each type anddevice and the variance parameter f of the heterogeneity parameter. Normally with
Weibull hazard models, the parameters gi and m can be estimated from failure history
data. However, with the random proportional Weibull hazard model, it is necessary toestimate the variance parameter f of the heterogeneity parameter, and theheterogeneity parameters 1ij (i ¼ 1, . . . , M; j ¼ 1, . . . , Ni) for each type and device, as
well. For reader’s convenience, the proposed estimation flow is shown in Figure 3 withthe following detail description.
Now the database of the failure history of the targeted system is available. All theinformation regarding the time that each component failed (was exchanged) since thetime of installation, of the targeted system, is stored in the database. Then, the failure
history of the component is expressed as J ¼ ðj1 ; :::; jM Þ, and ji ¼ ðji1 ; :::; jiNi Þ is thefailure history of type i ncomponent.
 Also, jij is the ofailure history for the type i componentof device j, and is jij ¼

Lij
d1ij ; t1ij ; :::; dLij
ij ; t ij

ði ¼ 1; :::; M ; j ¼ 1; :::; N i Þ. Also, dkij isthe dummy variable which takes the value 1 if the type i component k (k ¼ 1, . . . , Lij) of
device j fails, and the value 0 if it does not. t kij is the in-service time (or lifespan) of the typei component k of device j. Therefore, when dkij ¼ 0, t kij is the length of time since
installation to the present. On the other hand, when dkij ¼ 1, t kij is the lifespan. Let us
discuss it in detail with the servers A and B (device 2) of the power source (type 2) shownin Figure 1. Server A has been used continuously from the installation of the system tothe present T without failure or replacement. Accordingly, ðdA22 ¼ 0; tA22 ¼ TÞ is recorded

Proportional
Weibull hazard
model
617

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618

Figure 3.
Estimation flowin the database as the information on the history of failure regarding Server A. In thisstudy, the information on whether there is any failure and usage time is considered as abasic information unit. On the other hand, Sever B broke down twice at T1 and T2 andwas replaced since the installation of the system. Accordingly, as the information on the
B2
B1
B2
history of failure regarding server B, ðdB1
22 ¼ 1; t 22 ¼ T 1 Þ, ðd22 ¼ 1; t 22 ¼ T 2 2 T 1 Þ,
B3
B3
and ðd22 ¼ 0; t 22 ¼ T 2 T 2 Þ are recorded. Obviously, even if several devices areobserved for the same period of time, the basic information unit varies according to thehistory of failure. When it is possible to gain several basic information units from asingle device like server B, we should discriminate each basic information unit by adding
a superscript to B like dB1
22 , but let us omit this superscript notation.
Here it is assumed
that the heterogeneity parameter 1ij is given. The conditional
likelihood lij jij : gi ; m; 1ij with observed data jij concerning faults of type icomponents of device j can be expressed as follows:lij ðjij : gi ; mi ; 1ij Þ ¼

Lij n
Y

F ij ðt kij : gi ; mi ; 1ij Þ

oð12dkij Þ nodkij
f ij ðt kij : gi ; mi ; 1ij Þ :

ð5Þ

k¼1

However, with the above equation, the probability density function of the lifespandistribution fij ðtkij : gi ; mi ; 1ij Þ and the survival function F ij ðt kij : gi ; mi ; 1ij Þ are explicitly
expressed as the parameter gi ; mi ; 1ij function. Here, when the heterogeneity parameter1ij is subject to the standard gamma distribution g ð1ij : fÞ, the likelihood function of
the observed data jij can be expressed as follows:
Lij ðjij : ui Þ ¼

Z 1Y
Lij n
oð12dkij Þ nodkij
f ij ðt kij : gi ; mi ; 1ij Þ g ð1ij : fÞd1ij
F ij ðt kij : gi ; mi ; 1ij Þ
0

¼

k¼1

Lij n
odkij Z 1ff Ys þf21gi mðtkij Þm21
1ijij
exp 2ðf þ gi tij Þ1ij d1ij :GðfÞ k¼1
0

ð6ÞP
PLij k m21
kHowever, u ¼ ðgi ; mi ; fÞ. Also, sij ¼ Lij. With the above
k¼1 dij ; tij ¼
k¼1 ðt ij Þequation, the heterogeneity parameter 1ij of all type i components of device j takes thesame value. Notice that to express this, the authors define the likelihood function
Lij ðjij : ui Þ as the expected value regarding the probability variable 1ij of theconditional likelihood lij ðjij : gi ; mi ; 1ij Þ. Hence, by the transformation of the variable
xij ¼ 1ij ðf þ gi tij Þ, the following is obtained:619

sij þf21
Lij nodkij Z 1  x
ff Y
dxij
ij
gi mðt kij Þm21
expð2xij Þ
Lij ðjij : ui Þ ¼GðfÞ k¼1
f þ gi tij
f þ gi tij
0
¼

Lij n
odkij
Y
ff
Gðsij þ fÞ
k m21
g
mðt
Þ
:
i
ij
GðfÞ ðf þ gi tij Þsij þf21 k¼1

ð7Þ

Therefore, the logarithmic likelihood function with the observed data J ¼ ðj1 ; :::; jM Þ
can be expressed asln LðJ; uÞ ¼

M X
Ni
X

ln Lij ðjij : ui Þ

i¼1 j¼1¼ N f ln f 2

M X
Ni
X

ðsij þ fÞ lnðf þ gi tij Þ þ

i¼1 j¼1

þ

Lij
M X
Ni X
X

M X
Ni sX
ij 21
X

lnðf þ kÞ

i¼1 j¼1 k¼0

no
dkij ln gi þ ln m þ ðm 2 1Þ ln t kij :ð8Þi¼1 j¼1 k¼1

However, each element of u ¼ ðu1 ; :::; uMþ2 Þ is expressed as ðu1 ; :::; uM Þ ¼ ðg1 ; :::; gM Þ,P
uM þ1 ¼ m, uM þ2 ¼ f. Also, N ¼ M
N , and with the third item on the right side of
Psij 21 i¼1 iequation (8), when sij ¼ 0,
k¼0 lnðf þ kÞ ¼ 0 is defined. Also, when sij ¼ 1,
Psij 21
lnð
f
þ
kÞ
¼
ln
f
.
k¼0
The maximum likelihood estimator of the parameter value u, which maximizes thelogarithmic likelihood function (8), is given as u^ ¼ ðu^1 ; :::; u^M þ2 Þwhich simultaneously
satisfies› ln LðJ; uÞ¼ 0; ði ¼ 1; :::; M þ 2Þ:ð9Þ›ui^ u^Þ of the asymptotic covariance matrix of the
Furthermore, the estimated value Sðparameter can be expressed as follows:2^ 21^Sðu^Þ ¼ › ln LðJ; uÞ :›u›u‘

Proportional
Weibull hazard
model

ð10Þ

However, the inverse matrix of the right side of the above equation is the inverse
matrix of the Fisher information matrix after ðM þ 2Þ £ ðM þ 2Þ with elements

F
29,13/14620›2 ln LðJ; u^Þ=›ui ›uj . The parameter’s maximum likelihood estimator can be obtainedby solving the non-linear simultaneous equation (9) of M þ 2 dimension. In this study,
the maximum likelihood estimator is estimated by the Newton-Raphson method. If the
maximum likelihood estimator u^ is obtained, the covariance matrix estimated value^ u^Þ is employed to estimate the t-test statistics.
Sð
Next, with the maximum likelihood estimator u^ of the parameter vector as postulate,the maximum likelihood estimator of the heterogeneity parameter 1ij (i ¼ 1, . . . , M;
j ¼ 1, . . . , Ni) is obtained. Here, the partial likelihood function is defined as follows:^ Lij nodkij
f^ f Ys þf^21^^ kij Þm21g^i mðt
1ijij
exp 2ðf^ þ g^i t^ij Þ1ij :ð11Þ
L+ij ðjij ; 1ij : u^i Þ ¼^GðfÞ k¼1
P
k m
However, t^ij ¼ Lij
k¼1 ðt ij Þ . At this time, the maximum likelihood estimator of theheterogeneity parameter 1ij can be obtained as 1^ij that satisfies›L+ij ðjij ; 1ij : u^i Þ
¼ 0:
›1ij

ð12Þ

The maximum likelihood estimator of the heterogeneity parameter obtained in this
^ i ; f^Þ as postulate. To
way is the estimated value obtained with the parameter u^ ¼ ðg^i ; mexpress this explicitly, the solution of equation (12 is expressed as 1^ij ðu^i Þ. Finally, fromequation (11), (12), the following equation can be obtained:sij þ f^ 2 1
1^ij ðu^i Þ ¼:
f^ þ g^t^ij

ð13Þ

Empirical study
Overview of empirical study
Targeting the traffic control system managed by a highway company, the random
proportional Weibull hazard model is estimated. The traffic control system is a systemthat has been sequentially renewed from the old system since 1990, and has been in
operation continuously. It comprises nine central station systems (hereinafter,
stations), manages the conditions of expressways, and provides appropriate real-time
information to users. The operational conditions of the traffic control system are also
under real-time surveillance, and in the case of faults, the failure-generated component
will be specified and the time and content of the failure will be recorded.
In this study, out of all the components that compose the traffic control system, theauthors targeted the component group that, in case of a fault, has the possibility ofdeveloping into a serious functional failure of the whole system, and estimated therandom proportional Weibull hazard model. After investigating the failure history
database of the traffic control system and interviewing the system manager, theauthors selected a component group to target for model estimation, ultimately deciding
on three types, HDDs (Hard disk drive), power supply, and processing devices. In thecurrent traffic control system, there are 177 HDDs, 306 power supply, and 180
processing devices. From the same research and interview, HDDs and processing
devices are used as three different devices: PCs (used as monitors and terminals),

servers (used as servers and processing equipment), and other devices (other uses notincluded in the former two). Furthermore, these components are used in nine different
stations, each type being categorized into 27 categories.
Estimation results
In this study, three types of components: HDD, power supply and processing device areused. Therefore, three arrival density parameters gi (i ¼ 1,2,3) are introduced. Also,
assuming the hazard rates of the PCs, servers and other devices in the nine stations areheterogeneous, heterogeneity parameters 1ij (i ¼ 1,2,3; j ¼ 1, . . . , Ni) are defined foreach device. Therefore, the unknown parameters to be estimated are arrival density
parameter gi (i ¼ 1,2,3), acceleration parameter m, heterogeneity dispersion parameterf, 1ij . The random proportional Weibull hazard model estimated by the method
proposed is shown in Table I. The value in parentheses shows the t-value, and thet-value of either parameter; as a result, the null hypothesis that they have no
explanatory power for each explanatory variable model, is rejected at significance level
0.95. It is possible to show more clearly the difference in hazard rates for each type ofcomponent. As shown in Table I, the maximum likelihood estimator of the acceleration^ ¼ 2:174. From equation (2b), it can be seen that the survival probability
parameter is mof each type of component gradually decreases as the in-service time increases.
^ ¼ 1:00, it would be an accidental failure component whose failure rate
Generally, if mdoes not depend on time, but all the components used in this study have characteristicsof the wear-out failure component.

Proportional
Weibull hazard
model
621

Analysis results
The authors categorized the three types of components (HDD, power supply, andprocessing device) according to which of the nine stations of the traffic control systemthey are installed in, and further subdivided them according to the three types of usage:(1) PCs (used as monitors, terminals);(2) servers (used as servers, processing equipment); and(3) other devices (used in other ways other than described in the former two).
As shown in Table II, this creates 27 categories for each type. From equation (13), themaximum estimator of heterogeneity parameters for each category can be identically
Parameter

g

Model

g1
g2
g3
m

f
Logarithmic likelihood

1.251E-5(25.104E6)
1.631E-6(22.311E7)
5.293E-6(29.182E6)
2.174(49.031)
1.193(2.182)2402.441

Table I.
Estimation results

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622

Table II.
Heterogeneity parameters

HDD

Power supply

Processing device

Station 1

PC
ServerOthers

0.154
0.148
–

(1.536)(1.744)
–

0.006
–
–

(2.401)
–
–

0.154
0.148
0.008

(1.398)(1.484)(3.694)

Station 2

PC
ServerOthers

0.123
2.208
0.161

(2.973)(0.541)(1.321)0.120––(2.586)
–
–

0.770
0.125
0.008

(8.94E-02)(1.897)(3.313)

Station 3

PC
ServerOthers

0.146
0.669
0.133

(1.799)(3.357)(2.337)

0.007––(5.549)
–
–

0.146
0.008
0.860

(1.507)(3.715)(7.62E-02)

Station 4

PC
ServerOthers

1.437
0.768
0.006

(5.38E-02)(2.136)(11.207)

0.004
–
–

(3.174)
–
–

0.688
0.674
0.833

(0.190)(0.213)(7.82E-02)

Station 5

PC
ServerOthers

0.753
0.600
–

(0.479)(1.500)
–

0.008
–
–

(1.983)
–
–

0.113
0.628
–

(2.170)(0.303)–

Station 6

PC
ServerOthers

0.134
0.114
–

(2.310)(3.365)
–

0.008
–
–

(1.954)
–
–

0.132
0.802
0.142

(1.752)(6.24E-02)(1.579)Station 7

PC
ServerOthers

0.147
1.304

(1.779)(0.246)0.007––(2.090)
–
–

0.147
0.136
0.009

(1.500)(1.674)(3.070)Station 8

PC
ServerOthers

5.400
1.833
–

(12.044)(0.181)
–

1.360
–
–

(3.519)
–
–

0.481
1.508
0.581

(0.830)(0.325)(0.424)

Station 9PC
ServerOthers

0.844
0.138
–

(0.178)(2.140)
–

0.632
–
–

(3.44E-02)
–
–

0.416
0.948
0.123

(1.260)(3.43E-03)(1.937)

obtained. However, some categories (i, j) have no applicable components, and thenumber of heterogeneity parameters to be estimated are these: 21 for HDD, nine forpower supply, and 26 for processing device. Table II shows each of the estimatedheterogeneity parameters 1^ij ðu^i Þ and the number of samples of each component of each
type. The same table shows that the maximum estimators of heterogeneity parameters
are distributed variously depending on the device, and it can be seen that in order toexpress the deterioration characteristic of components in an information system it isnecessary to consider the heterogeneity of hazard rates. Also, because the component
structure is extremely subdivided, it can be surmised that the estimation efficiency will
fall if the different component characteristics with dummy variables are expressed. By
using the maximum estimator of heterogeneity parameters, we can obtain adeterministic Weibull hazard model that expresses the deterioration characteristic ofeach device. However, as shown in Table II, there are some device categories with avery small number of samples. Therefore, there is the possibility of a problem
occurring with the reliability of the Weibull hazard models obtained for each device
category. In fact, of the heterogeneity parameters shown in Table II, there are some forwhich the null hypothesis that it has no explanatory power for implementing theheterogeneity parameter could not be rejected, at a significance level of 95 percent.
Therefore, by grouping the device categories, aggregative Weibull hazard models that
express the average deterioration characteristic for each group can be obtained. Let’s
define the heterogeneity parameters for each type and device 1ij , as aggregative
average heterogeneity parameters for each type E 1i (i ¼ 1, . . . , M). The aggregative
partial likelihood functions for all devices are defined as

Proportional
Weibull hazard
model
623

L++i ðji ; E1i : u^i Þ¼^ Ni Y
Lij n
odkij

f^ f Ys þf^21^^ kij Þm21g^i mðt
E1ijijexp 2ðf^ þ g^i t^ij ÞE1ij :
Gðf^Þ

ð19Þ

j¼1 k¼1

The maximum estimator of the heterogeneity parameter E1i satisfies›L++i ðjij ; E1i : u^i Þ
¼0
›E1i

ð20Þ

and can be expressed as
PNi^
j¼1 sij þ f 2 1:
E 1^i ðu^i Þ ¼ PNi^
j¼1 f þ g^t^ij

ð21Þ

Similarly, the maximum estimators E 1^i ðu^i Þ of heterogeneity parameters E1il
(i,l ¼ 1,2,3) aggregated for each type and usage (PC, server, others) isP^
j[vl sij þ f 2 1^:
ð22Þ
E 1^il ðui Þ ¼ P^
j[vl f þ g^t^ij
However, l is the usage of the components, and l ¼ 1 means PC, l ¼ 2 means servers,
and l ¼ 3 means others. Also, vl is the set of devices of the usage l. The maximum
estimators of the average heterogeneity parameters aggregated with the above idea areshown in Table III. If the average heterogeneity parameters aggregated by type arecompared, E 1^1 ðu^Þ . E 1^3 ðu^Þ . E 1^2 ðu^Þ is true. Furthermore, the same relation applies
to the arrival density parameter, so the hazard rate of HDD is highest, and that ofpower supply is smallest. By obtaining the average heterogeneity parameter E 1^i ðu^Þ,which is the heterogeneity parameters of each type of component aggregated, theWeibull hazard model that expresses the average deterioration characteristic of eachtype of component can be obtained. By using the average heterogeneity parameterE 1^i ðu^Þ (i ¼ 1,2,3) aggregated from each type, the average survival function of each typeHDD 
E 1^i u^ E 1^i1 u^
E 1^i2 u^
E 1^i3 u^

0.923
1.302
0.900
0.095

Power supply(9.746)(0.022)(6.403)(14.552)

0.205
–
–
–

(8.737)–––

Processing device
0.431
0.366
0.527
0.410

(20.086)(8.344)(3.973)(8.243)

Table III.
Heterogeneity parametersF
29,13/14

624

Figure 4.
Survival probabilities ofeach typeFigure 5.
Survival probabilities
of HDD

is obtained, and these are shown in Figure 4. Of all the samples, the shown survival
probabilities are the relative ratio of the samples that survived in the targeted period.
The lifespan of components is generally evaluated as service life. The same figure canbe illustrated to show the service life according to the survival probability. Therefore,
by using the survival probability as a management indicator, the service life can beevaluated by an arbitrary management indicator. However, the management indicator
should be prepared with the importance of components in mind. From Figure 4, it canbe seen, for example, that the use period (service life) at which the survival probability
is 50 percent is 158 months for HDD, 804 months for power supply, and 332 months forprocessing device. Furthermore, the survival probability of power supply used for 120
months is 98.9 percent, and it is 95 percent at 240 months. For processing device, it is92.7 percent at 120 months and 71.1 percent at 240 months. It can be seen that the
failure rate increases as use time lengthens for power supply and processing device aswell as for HDD. However, among the three types, HDD has the most rapid incline,
while power supply has the gentlest. Furthermore, the survival functions of HDD andprocessing device obtained by aggregating heterogeneity parameters by usage areshown in Figures 5-6. Figure 5 shows the survival function of HDD. Even for the same
HDD, use environment differs according to usage. As seen from the figure, among theProportional
Weibull hazard
model
625

Figure 6.
Survival probabilities ofprocessing device

three different usages (PC, server, others), failures happen earliest in the case of PCs,
which have the highest heterogeneity parameter and a survival probability of 50
percent after 135 months of usage. On the other hand, devices of other uses have thesmallest heterogeneity parameter, and the survival probability is 50 percent at 455
months. In Figure 6, of processing parts, the hazard rate when they are used as servers
is the highest, with a survival probability of 50 percent at 303 months. The hazard rateis smallest for PCs, with a survival rate of 50 percent at 356 months.
Conclusions
In this study, a deterioration failure estimation model for components in information
systems is proposed, aiming to improve asset management at the component level oflarge-scale information systems, which support infrastructure. The authors focus on
the point that information systems are composed of many types of components, andpoint out the necessity for a fault analysis model that can express the heterogeneity ofhazard rates of different types. To operationally express the heterogeneity of failure
rates, the Weibull hazard model is employed as a base, and a random proportional
Weibull hazard model expressing the proportional heterogeneity of hazard rates with astandard gamma distribution is formulated. Furthermore, through a case study using a
traffic control system for expressways, the validity of the proposed hazard model is
empirically verified. As mentioned above, by using the random proportional hazard
model considering the heterogeneity in the deterioration process, it is possible toimprove the decision-making process regarding deterioration prediction for facilities
and equipment, including infrastructure, and asset management based on deterioration
prediction. The following are the findings of this study:.
The random proportional Weibull hazard model was formulated, in order to take
into account the heterogeneity of hazard rate of each device for large-scale
information system composed of a variety of component. The authors proposed a
two-step estimation method using observational data and indicated thatheterogeneity parameters can be calculated identically.
.
The random proportional Weibull deterioration hazard model was applied to a
traffic control information system for expressways, and examined theappropriateness of the proposed method. Actually, three types of devices:

F
29,13/14.

626

HDD, power supply, and processing device were classified into 27 categories(nine sections, three usages), and a hazard model estimation was conducted whileconsidering each heterogeneity.
The results of estimation focused on the difference among device types indicatethat expected lifespan is 158 months for HDD, 804 months for the power source,and 332 months for the processing part. It was also found that expected lifespanvaries from 135 to 455 months for HDD and from 303 to 356 months for theprocessing part according to the difference in purpose.

Applying the random proportional Weibull hazard model to asset management showsthat there are issues requiring future study. First, it is necessary to develop an asset
management methodology for components, using the proposed hazard model.
Especially, there are cases in which it is difficult to acquire necessary equipment if an
information component goes out of production during the operation of an information
system. It is expensive to use a different component as a substitute, so that in order toavoid the problem of out of stock components, it is necessary to store replacement
components. Or, to carry out preventative maintenance on information equipment, it isnecessary to decide logically on the renewal time of components. The hazard model
proposed in this study can be used for performing asset management at the component
level. Secondly, fault analysis is necessary at the function level of information systems.
With asset management at the function level, it is necessary to focus on the seriousnessof the effect a failure in one component or component group can have on the functional
level of the whole system, and it is necessary to consider the maintenance strategy ofeach component and component group. The authors are in the process of developing amethodology to analyze the system’s dynamic failure characteristics, which expresses
the failure process of each component using the hazard model proposed in this study,
and expresses the impact of each failure on the function of the whole system using a
fault-tree. Thirdly, it is necessary to work on asset management at the system level of
information systems. For this, it is necessary to consider simultaneously the technicalobsolescence of the information system, the delay in processing time, and the dynamic
failure process, and to develop a real-time option model to determine the best timing forrenewing the information system. Fourth, the proposed method is applied to asset
management of other types of infrastructure, for instance pavement, since the effect of
the heterogeneity of individual sections of pavement would be larger than that of
information systems.
References
Aoki, K., Yamamoto, K. and Kobayashi, K. (2007), “Optimal inspection and replacement policy
using stochastic method for deterioration prediction”, paper presented at the 11th World
Conference on Transportation Research, Berkeley, CA.
Cameron, A.C. and Trivedi, P.K. (1990), “Regression-based tests for overdispersion in the Poissonmodel”, Journal of Econometrics, Vol. 46, pp. 347-34.
Cameron, A.C. and Trivedi, P.K. (1998), Regression Analysis of Count Data, Cambridge
University Press, Cambridge.
Cox, D.R. and Oakes, D. (1984), Analysis of Survival Data, Chapman & Hall, London.
Cruz, M.G. (2002), Modelling, Measuring and Hedging Operational Risk, Wiley, New York, NY.

Gourieroux, C. and Visser, M. (1986), “A count data model with unobserved heterogeneity”,
Journal of Econometrics, Vol. 79, pp. 247-68.
Kaito, K. and Kobayashi, K. (2008), “Obstacle emergence risk and road patrol frequency”,
Proceeding of the International Seminar on Asset Management for Developing World,
pp. 42-55.
Lancaster, T. (1990), The Econometric Analysis of Transition Data, Cambridge University Press,
Cambridge.
Maher, M.J. and Summersgill, I. (1996), “A comparative methdology for the fitting predictive
accident models”, Accident Analysis & Prevention, Vol. 28, pp. 281-96.
McNeil, J.A., Frey, R. and Embrechts, P. (2005), Quantitative Risk Management, Princeton
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Mikosch, T. (2000), Non-life Insurance Mathematics, Springer, New York, NY.
Tsuda, Y., Kaito, K., Aoki, K. and Kobayashi, K. (2006), “Estimating Markovian transition
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Lawless, J.F. (1987), “Negative binomial and mixed Poisson regression”, Canadian Journal of
Statistics, Vol. 15, pp. 209-25.
Piegorsch, W.W. (1990), “Maximum likelihood estimation for the negative binomial dispersion
parameter”, Biometrika, Vol. 46, pp. 863-7.
About the authors
Kiyoshi Kobayashi is Dean, Professor, whose achievements and affiliations are: 1978, MA, Kyoto
University; 1978-1986, Research Associate, Kyoto University; 1987-1990, Associate Professor,
Department of Social Systems Engineering, Tottori University; 1990-1996, Professor, Department
of Social Systems Engineering, Tottori University; 1996-present, Professor, Graduate School ofEngineering, Kyoto University; 2006-2009, Vice Dean, Professor, Graduate School ofManagement, Kyoto University; 2007-present, Professor, Research Institute of Management
Studies, Kyoto University; 2010-present, Dean, Professor, Graduate School of Management,
Kyoto University. Kiyoshi Kobayashi is the corresponding author and can be contacted at:
kkoba@psa.mbox.media.kyoto-u.ac.jp
Kiyoyuki Kaito is Associate Professor whose achievements and affiliations are: 2000, PhD,
Department of Civil Engineering, The University of Tokyo; 2000-2001, Visiting Researcher,
Department of Civil Engineering, The University of Tokyo; 2001, Visiting Scholar, Department
of Civil Engineering and Engineering Mechanics, Columbia University; 2001-2002, Postdoctoral
Fellow, Department of Civil Engineering, The University of Tokyo; 2002-2004, R&D Manager,
Bridge Maintenance Consultant Corporation; 2004-2006, Senior Researcher, Bridge MaintenanceConsultant Corporation; 2006-2007, Visiting Researcher, Department of Urban Management,
Kyoto University; 2007-present, Associate Professor, Frontier Research Center, Graduate Schoolof Engineering, Osaka University.

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Proportional
Weibull hazard
model
627