Bivariate Reliability and Availability Modeling

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					26                                                                                        IEEE TRANSACTIONS ON RELIABILITY, VOL. 50, NO. 1, MARCH 2001




         Bivariate Reliability and Availability Modeling
                                                      Sang-Chin Yang and Joel A. Nachlas


    Abstract—Equipment longevity is a resource that is consumed                   vector are specific to the equipment. Years of usage and mileage
in device operation. For many types of equipment, the resource can                are not the only quantities that might describe device longevity.
be reasonably represented in terms of two (or more) measures. We                  The terms “age” and “use” are used here, but they are generic
construct a general framework for specifying bivariate longevity
models and incorporating them in availability models. Following                   and can represent quite different measures than duration of own-
an introduction to the domain of study, a short taxonomy of model                 ership and distance traveled. In the example of an automobile
types is provided and indicates which types of models have not been               tire, age might correspond to accumulated mileage and usage
studied. For two classes of bivariate models, their construction is               might be measured as tread loss. Even more complicated mea-
shown, along with how they should be interpreted in relation to                   sures such as current flow and thermal history are appropriate
commonly used univariate concepts such as hazard functions. Sev-
eral example models illustrate these points.                                      for some integrated circuits.
   The construction of bivariate renewal models is shown and their                   Some reliability models that respond to the need to include
inclusion in basic and preventive maintenance-based availability                  both age and use have been developed. However, nearly all of
models is explained. All of these analyzes show the development of                these models are defined in a manner that permits them to be
the bivariate Laplace transforms of the probability measures being                reduced to 1-dimensional models in which the -independent
studied. Unfortunately, in nearly all cases, the inverse transforms
are not yet available.                                                            variable is time. Existing models are discussed in Section II. A
   The chief contributions of the paper are the                                   structured examination of bivariate measures of equipment use
    • definition of a structure for classifying bivariate problems,               has not been performed; thus bivariate models have not been
    • description of methods for building and analyzing bivariate                 fully developed and some potentially useful model forms have
      models,                                                                     not been defined at all.
    • general presentation of a new domain of research in equip-                     This paper provides a foundation for bivariate reliability mod-
      ment reliability.
                                                                                  eling and shows how to use that foundation to define bivariate
Potential extensions and improvements to this work are numerous
and the resulting realism in reliability and maintenance models will              renewal models that can support maintenance planning.
be important.                                                                        Section II gives a brief taxonomy of bivariate device longevity
                                                                                  model forms and indicates the forms for which models have and
  Index Terms—Bivariate distribution, corrective maintenance,
preventive maintenance, renewal model.                                            have not been constructed.
                                                                                     Section III examines the issues related to the creation of the
                                                                                  model types that have not yet been developed. Several represen-
                  ACRONYMS AND ABBREVIATIONS                                      tative example reliability models are constructed.
PM            preventive maintenance                                                 Given the definition of representative reliability models, cor-
CM            corrective maintenance.                                             responding renewal models are constructed for the case in which
                                                                                  only CM is performed and then for the case in which device
                            I. INTRODUCTION                                       management includes both CM and PM. For the models that in-
                                                                                  clude only CM, both instantaneous and noninstantaneous repair

P    EOPLE FREQUENTLY discuss equipment behavior in
     terms of age and usage. Common examples are auto-
mobiles and automobile-tires in which both model-year and
                                                                                  models are defined. The realizations of these model classes are
                                                                                  illustrated by application to a few of the reliability models in this
                                                                                  paper. In each case, only the Laplace transform of the renewal
accumulated-mileage are usually included in discussions of                        function and of the availability function is obtained. Methods
longevity. Less well recognized examples for which two mea-                       for inverting the transforms are suggested.
surement scales are quite important include factory equipment,                       The example models illustrate the key issues in the construc-
power generation machines, and aircraft. The longevity of many                    tion of bivariate models. The key motivating factor in this paper
of the devices that reliability specialists study is meaningfully                 is the fact that existing bivariate models are defined so that
described in terms of two measures.                                               they can be collapsed into single variable life-time distribution
   Device life is a resource that might best be represented, and                  models despite the fact that bivariate models can be more rep-
for which the consumption might best be measured, using a two                     resentative or descriptive of device longevity.
(or higher) dimensional vector; the quantities that comprise the
                                                                                                      II. A SIMPLE TAXONOMY
   Manuscript received May 30, 2000; revised July 26, 2000, and August 8,
2000.                                                                               There are many ways to define a bivariate reliability model,
   S.-C. Yang is with the Chung Cheng Institute of Technology, Tao-Yuan           and there are probably several alternate ways to classify the
33509, Taiwan, R.O.C. (e-mail: SCYang@cc04.ccit.edu.tw).                          model types. An informative classification scheme is based on
   J. A. Nachlas is with the Department of Industrial Engineering, Virginia In-
stitute of Technology, Blacksburg, VA 24061 USA (e-mail: Nachlas@vt.edu).         the relationship between the two variables. Specifically, this
   Publisher Item Identifier S 0018-9529(01)06799-9.                              paper distinguishes between those models for which age and use
                                                              0018–9529/01$10.00 © 2001 IEEE
YANG AND NACHLAS: BIVARIATE RELIABILITY AND AVAILABILITY MODELING                                                                        27



are functionally related and those in which they are -correlated       are used to obtain a distribution in time. Thus, nearly all of the
rather than functionally dependent.                                    bivariate models that have been formulated have been collapsed
   The models in which the two variables are functionally re-          into univariate models and have not been used to study bivariate
lated are further separated on the basis of whether the functions      longevity.
are deterministic or stochastic. The models based on -correla-            It appears that the only effort to date to study a true bivariate
tion of the two variables are further classified by whether or not     reliability model is that in [10], [11]. Their focus is the determi-
the -correlation coefficient,        . From a reliability perspec-     nation of necessary warranty reserves for products such as au-
tive, the case in which age and use are -independent is unlikely       tomobiles. They develop models using both deterministic and
to be practically meaningful.                                          stochastic functions and obtain useful results. In fact, they de-
   Our study is only for bivariate models of longevity of a single     fine the method used here to construct the models based on sto-
device. There is extensive literature devoted to the formula-          chastic functions. In our development below, we include their
tion and analysis of bivariate (and multivariate) models of the        model and show how their approach can be extended to define
  -correlated or -dependent aging of two (or more) components.         other models.
Starting with the work in [1], [2] which developed multivariate           This section provides our view of how bivariate reliability
models for the reliability of series systems comprised of non-         models should be classified, and indicates the types of models
-independent components, numerous papers have addressed bi-            that already exist. The models that have been defined are very
variate reliability. However, in these models, each variable cor-      useful for the study of equipment reliability, but there are prob-
responds to the age of one of the components. In the construction      lems for which they are not well suited. For example, the def-
of the models, [1], [2] provide some useful definitions but they       inition of PM plans on the basis of both use and age requires
do not treat bivariate (or multivariate) reliability in the sense of   a bivariate failure model. Therefore, the remaining sections ex-
this paper.                                                            plore the definition of bivariate failure models, and consider the
   Returning to the idea of the bivariate longevity of a single        cases of both stochastic functional relationships and simple cor-
device, note that models in which the age and use variables are        relation between the age and use variables.
  -correlated have not been developed at all. All of the previously
developed bivariate reliability models treat the two variables as                    III. EXAMPLE DISTRIBUTION MODELS
functionally related. Models of wear processes [3], [4] and of
cumulative damage [5] portray equipment reliability in terms             Notation:
of deterministically-defined deterioration occurring at random                        time to failure (the subscript is omitted when-
points in time. Important defining features of all of these models                    ever it is clear there is only one interval)
are:                                                                                    th usage to failure
                                                                                      function relating usage and time
   1) The focus is on a single failure-mechanism or phenom-                           parameters of
       enon.                                                                          pdf on the parameter
   2) The use-variable determines failure by crossing a use-                          age, use dependent hazard functions, (used here
       threshold value.                                                               as ,        ; thus , are parameters of the hazard,
   3) They are ultimately reduced to univariate time models on                        life Cdf
       the basis of the functional dependence. The focus of the                       coefficient of -correlation between age and use
       models is to determine how much time will elapse before                        bivariate life Cdf with pdf
       the use threshold is surpassed.                                                Sf for the bivariate life distribution
   An interesting and not completely obvious observation is that                      marginal Cdf for
while the wear process and cumulative-damage models treat use                           -fold convolution of
as a function of time, the proportional hazards models [6] treat                      truncated life Cdf based on
age as a deterministic function of use covariates. Models that                        moment generating function for
are based upon deterministic functions relating use and age are                       marginal pdf on use at failure
readily reduced to univariate forms. They have been studied ex-                       conditional pdf on age given usage
tensively and do not lead to bivariate longevity models. Thus,                        bivariate hazard function
while they are often useful, they are not pertinent to the discus-                    conditional hazard function on age given usage
sion here and are not examined further.                                               means of the age and usage distributions
   The analytic emphasis with models based on stochastic func-                        standard deviations of the age and usage distribu-
tions has also been their reduction to a single dimension—time.                       tions
The stochastic-wear models and cumulative-damage models                               time until the th renewal
that treat damage magnitude as a r.v. are all defined in a manner                     total usage at the th renewal
that permits a focus on reliability in time. The same is true of                      bivariate renewal function based on
the shot-noise models [7]. Even the diffusion process models                          and having renewal pdf
[8] which really are expressed comprehensively in terms of                            number of renewals by time and usage
both variables are analyzed in terms of first-passage time to                         Laplace transform for
a failure state. Finally, the time dependent stress-strength [9]                      Cdf on CM interval [with pdf                 and pa-
interference models have the same characteristic in that they                         rameters              ]
28                                                                                            IEEE TRANSACTIONS ON RELIABILITY, VOL. 50, NO. 1, MARCH 2001



                Cdf on PM interval [with pdf          and param-                      we encourage the study of functions other than the ones treated
                eters            ]                                                    here.
                convolution of               with         [                              The use of the pdf          to construct the marginal proba-
                and has pdf           ]                                               bility distribution on usage is accomplished using well known
                convolution of               with         [                           methods. In general [11], we can construct the marginal pdf on
                and has pdf           ]                                               use as:
                mixture of            and
                bivariate renewal function based on                                                                                                    (1)
                bivariate renewal function based on
                bivariate renewal function based on                                   For example, with                     , solving for    yields:
                PM policy time
                PM policy usage
                bivariate point availability                                          so:
                standard      -normal Cdf corresponding to
                            .                                                                                                                          (2)
   For all functions to which they apply, Laplace transforms and
convolutions are indicated in the same manner as shown in the                         Once the marginal distribution on usage is obtained, we con-
Notation list.                                                                        struct the joint failure pdf using the well known conditioning
   We have suggested in our taxonomy of model types that there                        relation:
are two model classes that are not yet well developed, are inter-
esting, and might be practically applicable. These are the models                                                                                      (3)
based on a stochastic functional relationship between the two                         and the conditional pdf       is obtained using the well known
variables and the models that represent the variables as -corre-                      relationship between a pdf and its hazard function:
lated. We define example models in both of these classes in this
section.1                                                                                                                                              (4)

A. Stochastic Functions                                                               We use this form specifically so that we can focus upon the
   The definition of failure models on the basis of stochastic                        hazard function in the definition of the failure model. We assume
functions relating age and use, begins with the specification                         that the bivariate device failure hazard function can be stated as:
of how the stochastic feature of the longevity variables is por-                                                                                       (5)
trayed. This paper assumes:
    1) The time and use to failure are related by               .                     so that the definitions of     ,     ,      determine the hazard,
    2) The stochastic nature of the relationship in #1 can be rep-                    and ultimately, the bivariate life distribution. This paper treats
         resented by treating one or more of the parameters of                        the simplest conceivable form of the hazard function. More in-
         as r.v.                                                                      tricate, and perhaps realistic, forms should be addressed in ex-
   The interpretation of         is: Across a population of devices                   tensions to this work. Thus, we assume that          and       are
the accumulated usage by age is              . This is equivalent to                  simple linear functions:
saying that the mileage traveled by 2-year old cars is              .
Of course, we impose a probability measure on              to model
its dispersion. To illustrate this construction, consider 4 exam-                        Under this modeling format, the bivariate life distribution cor-
ples:                                                                                 responding to form #i is obtained by constructing:
      i)                                                                                                                                               (6)
     ii)
   iii)                                                                               then apply (3) and (4) to obtain:
    iv)                                                .2
In each case, we introduce randomness into the function by
treating as a r.v. having pdf         . This imposes random varia-
tion on the extent-of-use experienced by any age. Consequently,                                                                                        (7)
both age and usage at failure are r.v. We readily concede that
there can be many other functional forms that could be de-                            The same analytic approach yields:
fined and that might represent observed behavior. The analytic
methods described here might apply to those other forms, and                          for form   ii,


  1There are very many conceivable approaches to the construction of these
models and, in order to initiate the study of bivariate models, we use the simplest
possible approaches.
  2This form is the logistic model analyzed in [11].
                                                                                                                                                       (8)
YANG AND NACHLAS: BIVARIATE RELIABILITY AND AVAILABILITY MODELING                                                                     29



for form   iii,                                                      and the marginal pdf are the constituent exponentials regardless
                                                                     of the value of .
                                                                        The second candidate model is the obvious bivariate
                                                                      -Normal. The pdf for this model is well known; thus it is
                                                                     not restated here. As is also well known, the marginal pdf are
                                                              (9)     -Normal.
for form   iv,                                                          The final model for this Section III-B is from [13] in a queuing
                                                                     context but is also consistent with reliability interpretations




                                                                                                                                   (13)

                                                                               modified Bessel function of order . The marginal pdf
                                                                     for this model are not obvious.
                                                                        In summary, we have defined examples of two classes of
                                                             (10)    models that might be used to portray the dispersion in equip-
                                                                     ment longevity, as defined using two variables. We next examine
                                                                     the general probability concepts commonly associated with re-
For form #iv (10), the definition of the use function limits the     liability analysis and use some of the suggested example forms
variable to [0, 1] so the functions can require rescaling for        to illustrate these concepts. Sections V and VI construct relia-
some applications. Also, in forms #i and #ii, the forms of           bility and maintenance models using the probability concepts
allow a nonzero minimum value for usage.                             and some of the example model forms.
   All 4 forms are well defined and require only the specification
of the pdf        to be complete bivariate life distributions. On                IV. BIVARIATE PROBABILITY ANALYSIS
the other hand, for each form, it is unlikely that a closed-form        Consider a device for which longevity is defined in terms of
expression can be obtained for the marginal distribution on age      2 variables, say age and usage. Assuming the Cdf on longevity
at failure. These examples illustrate the construction of a model    has been constructed and is bivariate, there are some subtle and
in which the usage variable is a stochastic function of the age      sometimes difficult questions and concepts that arise in applying
variable.                                                            the bivariate model to reliability.
                                                                        First, we interpret the Cdf             as the probability that
B. -Correlation Models                                               failure occurs by time and usage :
   In many applications, the two life variables appear to be
 -correlated rather than functionally dependent. The definition                                                                    (14)
of models that can represent -correlations in the life variables
appears initially to be somewhat simpler than the construction       This probability corresponds to the proportion of the population
in Section III-A. We simply choose a bivariate distribution.         of devices that have longevity vector values at failure that do
However, it is important that the distribution can accurately        not exceed          in either vector component. We emphasize
represent equipment behavior and, in particular, that it have        this definition because, for a bivariate distribution, probability
marginal distributions that are consistent with experience. We       is generally computed over rectangles such as                     ,
have selected 3 example models that appear to hold promise for                       . Consequently, for any specific longevity vector,
representing bivariate failure processes in which the two vari-            , the range of age and use values implies that there are 4
ables are -correlated. Once again our choices are not the only       rectangles in the          plane for which probabilities can be
conceivable ones. Observed equipment behavior can suggest            meaningfully calculated. Fig. 1 shows that in addition to the
the use of a different distribution, and the ideas developed here    rectangle used in (14), there are the rectangles:
should apply to those cases as well as the ones treated here.
   The first candidate model is a generalization of the bivariate
exponential model [12]. The reliability function is:                 Relative to the cumulative probability              , the probabil-
                                                                     ities:
                                                             (11)
                                                                                                                                   (15)
so the corresponding pdf is:
                                                                                                                                   (16)

                                                                     are Sf. They correspond to the proportions of the population that
                                                             (12)    do not have longevity vectors less than         , either because
30                                                                             IEEE TRANSACTIONS ON RELIABILITY, VOL. 50, NO. 1, MARCH 2001



                                                                       A useful special case of (18) applies to the reliability function,
                                                                       which can be represented by:



                                                                                                                                       (19)

                                                                       Equation (19) can also be used to compute cumulative proba-
                                                                       bilities where the reliability function is easier than the Cdf to
                                                                       analyze.
                                                                          Very often, the first question that follows the definition of a
                                                                       probability model for device failure is that of the identity and be-
                                                                       havior of the associated hazard function. For a bivariate failure
                                                                       distribution, a return to first principles yields:



Fig. 1. Bivariate probabilities.


their failure ages exceed or their failure usages exceed . We
do not have informative names for the probabilities represented
by (15) and (16) but have considered names such as marginal
survival probabilities.
   A further, subtle point is that the reliability at     does not
include the probabilities represented by (15) and (16). The reli-
ability at longevity vector value           corresponds to the pro-
portion of the population for which failure age exceeds and
failure usage exceeds . Therefore, the reliability function cor-
responding to                is:                                                                                                       (20)

                                                               (17)
                                                                       which is a very appealing result. The next question is whether
                                                                       or not the hazard function is increasing. Reference [4] defines
Because it does not include the probabilities represented by (15)      MIFR (multivariate increasing failure rate), and the application
and (16), we call this the reliability rather than the Sf. Also, the   of that definition to the bivariate life distributions is that a dis-
cumulative failure probability and the reliability no longer sum       tribution is MIFR iff:
to 1.
   The apparent paradox in the definitions of                   and
              arises from distinctions in point of observation.                                                                        (21)
When considering the distribution, all positive valued longevity
vectors can potentially occur and, across a population of
devices, all do occur. Relative to the distribution, the cumula-       is nonincreasing in       , and the same statement applies to the
tive probability at         does not include devices for which         marginal distributions, viz:
either          or          . On the other hand, all copies of a
device population that have achieved a longevity of             will
have longevity vectors at failure that lie within the rectangle                                        and                             (22)
                  ,                     so at        the rectangles
corresponding to the marginal survival probabilities are not
accessible.                                                            are nonincreasing in and , respectively. Applying these condi-
   The computation of bivariate probabilities is reasonably clear.     tions is far from direct. Most models require numerical analysis
For any rectangle say                 ,                in the plane,   to characterize hazard function behavior and that behavior can
the probability of observing a failure at a point included in the      be rather complicated. For example, for some choices of its pa-
rectangle is:                                                          rameters, the bivariate exponential distribution of (13) displays
                                                                       a hazard function that is increasing in usage and decreasing in
                                                                       time.
                                                                          There is an important and subtle difference between the “con-
                                                                       struction of univariate reliability models on the basis of an as-
                                                               (18)    sumed hazard form” and the “corresponding model definition
YANG AND NACHLAS: BIVARIATE RELIABILITY AND AVAILABILITY MODELING                                                                       31



for a bivariate longevity distribution.” As is well known, an as-
sumed univariate hazard function,        , directly implies the life
Cdf by:

                                                               (23)

which is the solution of the differential equation:

                                                               (24)

The corresponding bivariate (partial) differential equation is:



                                                               (25)

The solution for (25) has not yet been found so one can not build
a bivariate reliability model from the hazard function as is done      Fig. 2. Bivariate renewal process.
in the univariate case.
   A further question is how one computes the mean and other           renewals at any coordinate point in the plane, say    , corre-
descriptive measures for a bivariate longevity distribution. The       sponds to the largest value of for which renewal occurs on
answer is that, as with univariate distributions, one begins by        or before time and usage . Therefore, the number of renewals
constructing the moment generating function (or Laplace Trans-         up to        is:
form) and then obtains moments as successive derivatives of the                                                                      (28)
moment generating function. The moment generating function
for the bivariate failure distribution is:                               Under this definition,              is a bivariate renewal
                                                                       counting process for which the distribution is obtained using
                                                               (26)    the usual “time-frequency duality” relation. Reference [13]
                                                                       confirms that this leads to:
and its construction is not always simple.                                                                                           (29)
   Also of critical importance to bivariate failure modeling is the
                                                                       which implies that:
question of how convolutions are constructed and how bivariate
renewal functions are defined and interpreted. Fortunately, the                                                                      (30)
convolution theorem extends directly to the bivariate case. On         which is intuitively appealing. In addition, using (29) to obtain
the other hand, the definition and interpretation of the associated    the renewal function yields:
counting process and the bivariate renewal function is less ob-
vious and can depend upon the application. This is a topic that                                                                      (31)
is treated in Section V, and for which considerable further study
is needed.                                                             which corresponds to the univariate form. As for the univariate
                                                                       function, the recursive statement of (31) is the “key integral re-
              V. FAILURE AND RENEWAL MODELS                            newal equation”:
   Consider the renewal model that is obtained when -identical
copies of a device are operated to failure and then replaced
instantaneously with new ones. We call this: Failure model with
instantaneous replacement. It is a model that has been studied
extensively for univariate life distributions. For a bivariate                                                                       (32)
longevity distribution, the sequence of device-lives forms a           and this function is the basis for analysis of the renewal process.
bivariate renewal process.                                             As a final point, assuming                is absolutely continuous
   Reference [13] defines a bivariate renewal process using the        implies that the renewal pdf exists and is:
vector            where it is assumed that the         are i.i.d.
with the common joint distribution              and:


                                                               (27)


Of course, by convention,                         . The renewal
vector is illustrated in Fig. 2, which shows that the number of                                                                      (33)
32                                                                              IEEE TRANSACTIONS ON RELIABILITY, VOL. 50, NO. 1, MARCH 2001



The Laplace (Laplace–Stieltjes) transform is the usual method           Unfortunately, the inverse transforms for (38) and (40) cannot be
of analysis for the renewal models. For the bivariate case,             constructed in closed form. For (39), working with the transform
the Laplace transform of the pdf associated with the Cdf,               of the associated renewal pdf permits its inversion [13] to:
            is:



                                                                (34)                                                                     (41)
and
                                                                        It is rare that the transform is invertible in closed form. For most
                                                                (35)
                                                                        of the models developed in this paper, closed-form transform-
                                                                        inversions are not available. We are exploring the use of the
Using (34) and (35) in the analysis of the key renewal equation
                                                                        Hartley transform [14] for numerically inverting the transform
leads to:
                                                                        expressions constructed in this paper.
                                                                            Next, consider the cases in which repair is not instantaneous.
                                                                        Assume that:
                                                                            1) repair effort extends over a bivariate interval that is
                                                                (36)            random.
                                                                            2) The Cdf on the magnitude of the repair effort is
                                                                                and is of the same family as the failure distribution.
                                                                            There is no formal justification for assumption 2. Our reason
                                                                (37)    for using it is because it sometimes makes the analysis easier. In
                                                                        addition, for cases in which it is appropriate, the repair distribu-
                                                                        tion can easily be collapsed into a univariate form.
which correspond to the univariate forms.
                                                                            To construct renewal models for the failure with noninstan-
   Applying these results to specific bivariate distributions is
                                                                        taneous repair, we obtain the convolution on the operating and
never simple. For the stochastic functions (7)–(10), the alge-
                                                                        repair intervals and then the renewal function based on the con-
braic complexity of the models implies that the Laplace trans-
                                                                        volution. Because availability is of interest for these cases, the
forms cannot be obtained in closed form but must be computed
                                                                        renewal function is used to obtain it. That is, considering a
numerically. Even the numerical construction of the transforms
                                                                        longevity cycle to be the sum of an operating interval and a re-
is taxing. The procedure is to select a grid-size over the
                                                                        pair interval, the cycle has distribution               which is ob-
plane and to perform the integration of (34) numerically. The re-
                                                                        tained as the inverse transform of:
sults can then be stored in an array for further use or fit (approxi-
mately) using a second-order regression model. We have imple-
mented this process for the models in (7)–(10) without special                                                                           (42)
difficulties but, because they are voluminous, the numerical re-
sults are not given here.                                               Then using the same reasoning as for univariate models, a device
   For the bivariate distributions selected to represent -corre-        is available at coordinate point         if it experiences no failures
lation between the variables, expressions for the Laplace trans-        prior to that point or else it is renewed at some earlier coordinate
forms can be obtained. For the bivariate exponential distribution       point and experiences no further failures before            :
in (11) and (12), the renewal function has:



                                                                (38)                                                                     (43)
Similarly, the bivariate exponential distribution of (13) has:
                                                                                   is obtained using (36).
                                                                (39)      Also comparable to the univariate case is that the availability
                                                                        function is ultimately obtained as the inverse transform of:
and the Laplace Transform of the renewal function for the bi-
variate -Normal distribution is:

                                                                                                                                         (44)


                                                                        As indicated is this section, inversion of the transform is quite
                                                                (40)    a challenge. Taking the bivariate exponential pdf of (12) as the
YANG AND NACHLAS: BIVARIATE RELIABILITY AND AVAILABILITY MODELING                                                                    33



failure pdf and assuming the repair-time pdf comes from the         Each of the availability function expressions obtained is far to
same family, means that the repair time pdf is:                     intricate to allow for closed form inversion. Even numerical
                                                                    evaluation of the usual contour inversion integral is unmanage-
                                                                    able. We continue to work on the inversion problem and believe
                                                                    that the use of the Hartley transform holds considerable promise.
                                                          (45)
                                                                                            VI. PM MODELS
                                                                       Consider a PM policy that corresponds in 2 dimensions to the
                                                                    age replacement policy for univariate models. The policy defi-
                                                                    nition is: the device is replaced when it fails and when it attains
                                                                    either age or usage . The vector               is the “replacement
                                                                    policy longevity.”
                                                                       To define a model of device behavior under PM we follow
                                                                    [15] and define 2 types of renewal intervals corresponding to
                                                                    device replacement. The intervals are:
                                                          (46)
                                                                        • the failure intervals in which the device fails before at-
                                                                          taining the policy longevity,
Thus,
                                                                        • the PM intervals in which the device survives to the policy
                                                          (47)            longevity and is replaced preventively.
                                                                    The aggregate renewal process for the device is the mixture of
The corresponding construction for Hunter’s bivariate exponen-      these two interval-types; the availability function is constructed
tial distribution of (13) yields:                                   using this mixture renewal function.
                                                                       To implement this process for the bivariate case, we assume
                                                                    again that the duration of the PM service is random and has
                                                                    Cdf             that is of the same family as the life distribution
                                                                    and the repair time distribution. Then, whenever the device fails
                                                          (48)      prior to policy longevity, replacement is performed according to
                                                                              , and the resulting duration of the renewal interval has
Then, using (19) simplifies (47) slightly:                          distribution             which is the convolution of           and
                                                                    the truncated distribution              :


                                                                                                                                  (54)


                                                                    The distribution is truncated to show that         cannot exceed
                                                                            . An important implication of the limits on the longevity
                                                                    vector is that the Laplace transform of the distribution is also
                                                          (49)      truncated. The truncated transform of the pdf corresponding to
                                                                                 is          .
                                                                       If the device survives to any of the coordinate points corre-
For the bivariate -Normal models, again use (19):
                                                                    sponding to the policy longevity, the renewal interval has pdf
                                                                              :


                                                          (50)




                                                          (51)


                                                          (52)



                                                                                                                              (55)
                                                                    Combining these definitions, the distribution on the length of
                                                          (53)      each renewal interval is a mixture of the distributions on the
34                                                                            IEEE TRANSACTIONS ON RELIABILITY, VOL. 50, NO. 1, MARCH 2001



lengths of the intervals that include failure and of those that       in each of the convolution terms in (57) and the truncated bi-
result from device survival:                                          variate transform along with the corresponding truncated trans-
                                                                      forms for the marginal distribution:

                                                              (56)
                                                                                                                               (61)
                                                                      as suggested by (19).
                                                                         The realizations of these expressions are intricate. For
                                                                      example, the transform for the availability function when the
                                         ;                            life distribution and both service distributions are bivariate
                                                                        -Normal is obtained by inserting (62) into (61):




                                                                                                                                              (62)

                                                                      and using:




                                                                                                                                              (63)


                                                                      to obtain the renewal pdf.
                                                                         Comparably intricate expressions are obtained for the other
                                                                      distributions treated here. In all cases, there is no feasible al-
                                                                      gebraic inversion of the transform for the availability function.
                                                                      Instead, numerical methods are necessary. We are exploring this
                                                                      issue.

                                                              (57)                       VII. CONTINUING RESEARCH
                                                                         The chief purpose of this paper is to initiate ongoing efforts
The renewal pdf based on             ,            is still obtained   to develop bivariate and multivariate reliability and maintenance
as in (36) and is incorporated in the availability function as:       planning models. We are convinced that there are many devices
                                                                      for which these types of models are meaningful. Our descrip-
                                                                      tion here points to the need to define additional distributions on
                                      otherwise,                      longevity and to analyze them further as well as the ones in this
                                                                      paper. Methods for numerical inversion of availability function
                                                                      transforms are needed, and incorporation of the resulting func-
                                                                      tions in maintenance optimization models are only a portion of
                                                                      the rich domain of research identified here. An important di-
                                                                      mension of the continuing study of bivariate reliability models
                                                              (58)    should be their application to real equipment along with exam-
                                                                      ining the resulting advantages for reliability specialists.
Using this formulation, the Laplace transform of the availability
function is:                                                                                        REFERENCES
                                                                        [1] A. W. Marshall and I. Olkin, “A multivariate exponential distribution,”
                                                              (59)          J. American Statistical Assoc., vol. 62, pp. 30–44, 1967.
                                                                        [2]      , “A generalized bivariate exponential distribution,” J. Applied
                                                                            Probability, vol. 4, pp. 291–302, 1967.
The key to implementing (59) is to use the truncated Laplace            [3] A. Mercer, “Some simple wear-dependent renewal processes,” J. Royal
                                                                            Statistical Society, ser. B, vol. 23, pp. 368–376, 1961.
transform:                                                              [4] R. E. Barlow and F. Proschan, Statistical Theory of Reliability and Life
                                                                            Testing: Holt, Reinhart, Winston, 1975.
                                                              (60)      [5] Z. W. Birnbaum and S. C. Saunders, “A new family of life distributions,”
                                                                            J. Applied Probability, vol. 6, pp. 319–327, 1969.
YANG AND NACHLAS: BIVARIATE RELIABILITY AND AVAILABILITY MODELING                                                                                               35



  [6] L. M. Leemis, Reliability: Prentice Hall, 1995.                             Sang-Chin Yang is an Assistant Professor of Engineering in the Logistics Edu-
  [7] A. J. Lemoine and M. L. Wenocur, “On failure modeling,” Naval Re-           cation Center of the Chung Cheng Institute of Technology (CCIT), the National
      search Logistics Quarterly, vol. 32, pp. 497–508, 1985.                     Defense University of Taiwan. He received his Ph.D. (1999) in industrial and
  [8]       , “A note on shot-noise and reliability modeling,” Operations Re-     systems engineering from Virginia Tech. He also holds a M.S. in systems engi-
      search, vol. 43, pp. 320–323, 1986.                                         neering from Virginia Tech, and a B.S. in civil engineering from CCIT. Prior to
  [9] K. C. Kapur and L. R. Lamberson, Reliability in Engineering Design:         undertaking his graduate studies, he served as director of the Logistics Educa-
      John Wiley & Sons, 1977.                                                    tion Center at CCIT. His research interests include reliability analysis, mainte-
 [10] S. P. Wilson, “Failure modeling with multiple scales,” Doctoral disser-     nance planning, systems engineering, and logistic support analysis.
      tation, George Washington Univ., 1993.
 [11] J. Eliashberg, N. D. Singpurwalla, and S. P. Wilson, “Calculating the
      reserve for a time and usage indexed warranty,” Management Science,
      vol. 43, pp. 966–975, 1997.                                                 Joel A. Nachlas is an Associate Professor of Industrial and Systems Engi-
 [12] G. E. Baggs and H. N. Nagagaja, “Reliability properties of order statis-    neering at Virginia Tech where he has served on the faculty since 1974. He
      tics from bivariate exponential distributions,” Communications in Sta-      also teaches regularly at the Ecole Superiore d’Ingenieures de Nice-Sophia
      tistics, vol. 12, pp. 611–631, 1996.                                        Antipolis. He received his B.E.S. (1970) from the Johns Hopkins University,
 [13] J. Hunter, “Renewal theory in two dimensions: Basic results,” Advances      M.S. (1972) and Ph.D. (1976) both from the University of Pittsburgh. His
      in Applied Probability, vol. 6, pp. 376–391, 1974.                          research interests are in applying probability and statistics to problems in
 [14] C. Hwang and M. Lu, “Numerical inversion of 2-D Laplace transforms          reliability and quality-control. His work in microelectronics reliability has
      by fast Hartley transform computations,” J. Franklin Institute, vol. 336,   been performed in collaboration with, and under the support of, the IBM
      pp. 955–972, 1999.                                                          Corp, INTELSAT, and Bull Corp. Part of that work earned him the 1990 P.K.
 [15] W. P. Murdock, Jr., “Component availability for an age replacement          McElroy Award from the AR&MS. He is a member of INFORMS, IIE, and
      preventive maintenance policy,” Doctoral dissertation, Virginia Tech.,      SRE and is a Fellow of ASQ. He serves on the management committee of the
      1995.                                                                       Annual Reliability and Maintainability Symposium and is an Associate Editor
                                                                                  of the IEEE TRANSACTIONS RELIABILITY. He is head coach of the Virginia
                                                                                  Tech men’s lacrosse team.

				
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Description: Equipment longevity is a resource that is consumed in device operation. For many types of equipment, the resource can be reasonably represented in terms of two (or more) measures.We construct a general framework for specifying bivariate longevity models and incorporating them in availability models. Following an introduction to the domain of study, a short taxonomy of model types is provided and indicates which types of models have not been studied. For two classes of bivariate models, their construction is shown, along with how they should be interpreted in relation to commonly used univariate concepts such as hazard functions. Several example models illustrate these points. The construction of bivariate renewal models is shown and their inclusion in basic and preventive maintenance-based availability models is explained. All of these analyzes show the development of the bivariate Laplace transforms of the probability measures being studied. Unfortunately, in nearly all cases, the inverse transforms are not yet available. The chief contributions of the paper are the • definition of a structure for classifying bivariate problems, • description of methods for building and analyzing bivariate models, • general presentation of a new domain of research in equipment reliability. Potential extensions and improvements to this work are numerous and the resulting realism in reliability and maintenance models will be important.