Implicit in the set-up of section 4.7.5 is the assumption that the operating environment, be it harsher or milder than the test-bed environment, is static. Thuswas taken to be an unknown constant. The more realistic scenario is the one wherein the operating environment changes over time so that its effect on componenti i=12, is to changeitoti, wheret is some unknown function oft; it is called themodulating function. When such is the case, the operating environment is said to bedynamic, and the topic of survivability under dynamic environments is interesting enough for an entire chapter to be devoted to it (Chapter 7). The purpose of this section, however, is to introduce an absolutely continuous bivariate distribution with exponential marginals that arises in the context of dynamic environments, and to compare the properties of this distribution with other bivariate distributions discussed here.
Sincetis unknown, we start by introducing a probability model for describing our uncer- tainty about it. There are several approaches for doing so, a natural one being to suppose thatt is a polynomial function of time with unknown coefficients. For example,t=a0+$n
i=1aiti, where theai’s are unknown. This kind of strategy is the essence of the approach used by Cox (1972) in his celebrated paper on ‘proportional hazards’. Another approach is to assume that t t≥0 is a meaningful stochastic process. That is, for every value of t∈0 t is a random variable with a specified distribution, and for every collection of k time points, 0< t1< t2<· · ·< tk k≥1, the joint distribution of thekrandom variablest1 tkis also specified. In essence, a stochastic process is simply a collection of random variables whose marginal and joint distributions can be fully specified. Thesample pathof a stochastic process is the collection of values taken by the Poisson process. The homogenous and the non-homogenous Poisson counting processes of section 4.7.4 are examples of stochastic processes, and so are the other point processes discussed in Chapter 8.
The advantage of describingt by a stochastic process over assuming thatt is some deterministic function of time is that the latter describes the effect of a systematically changing
PROBABILITY MODELS FOR INTERDEPENDENT LIFELENGTHS 111 environment, whereas the former best encapsulates the features of a haphazard one. In Singpur- walla and Youngren (1993), two stochastic process models for t are considered. The first entails the assumption that!t
0udYu t≥0"
is a ‘gamma process’;uis the intrinsic fail- ure rate function of a component, and dYu=udu, wheneveruexists. The gamma process will be formally defined in section 4.9.1. An overview of these processes and their generalizations is in Singpurwalla (1997) and in van der Weide (1997). With!t
0udYu t≥0"
i=12, described by a gamma process, Singpurwalla and Youngren (1993) produce a generalization of the BVE whose essence boils down to the feature that in a non-fatal shock model, each shock induces its own probability of failure on its affiliated component. Recall that in Marshall and Olkin’s (1967) set-up, the probability of failure from shock to shock (within a shock-generating process) is a constant. Whereas the above generalization of the BVE is of interest, the focus here is on the second process fort t≥0, namely, a ‘shot-noise process’.
The Shot-noise Process
Shot-noise processes have been used in the physical and the biological sciences to describe phenomenon having residual effects. Examples are surges of electrical power in a control system, or the after effects of a heart attack. Such residual effects are captured via an attention function, ht, which typically is a non-decreasing function oft(details are in Cox and Isham, 1980, p. 135).
In the context considered here, suppose that the operating environment consists of shocks, or a series of events, whose effect is to induce stresses of unknown magnitudesXk>0 k=01 , on a component or components; Xk is the stress induced by the k-th shock. The shocks are assumed to occur according to a non-homogenous Poisson process with a specified intensity functionmt t≥0. Suppose that when a stress of magnitudeX is induced at some epoch of timet, then its contribution to the modulating functiontat timet+uist+u=Xhu.
Specifically, if 0≡T0< T1< are the epochs of time at which stresses of magnitude X0 X1 , respectively, are induced then
t=
k=0
Xkht−Tk
hu=0, foru <0 (Figure 4.16). We are assuming here that time is measured from the instant the first shock occurs; thus0=X0. The processt t≥0is then ashot-noise process.
Time T(0)=0 T(1)
X0
X1
X2
T(2) η(•)
h(t)
Figure 4.16 A shot-noise process for·.
Whenhtis a constant, the effect of the induced stresses is cumulative; whenhtdecreases int, the unit reveals some form of healing or recovery.
Ift t≥0is described by a shot-noise process, then the processtt t≥0is also a shot-noise process;tis the intrinsic model failure rate of a component.
A Single-parameter Bivariate Distribution with Exponential Marginals
In what follows, we assume that the failure rate of componenti is a constanti i=12, and that for all values ofk Xkis independent ofTk. Furthermore, theXis are mutually independent and have a common distribution functionG. Let G∗ denote the Laplace transform of G, and letMt=t
0muduandHt=t
0hudube the cumulative intensity and attention functions, respectively. Then, in the case of a solo component experiencing the kind of environment described above, and having a constant failure rate , Lemoine and Wenocur (1986) give arguments which can be used to show thatT, the time to failure of the component is such that
PT≥t=G∗Htexp
−Mt+ t 0
G∗Hu mt−udu
as before, in writing out the above, the conditioning parameters have been suppressed. The proof is based on some well-known properties of a non-homogenous Poisson process. The details can be found in Singpurwalla and Youngren (1993), who also show that in the bivariate case, with 0≤t1≤t2,
PT1≥t1 T2≥t2=G∗1Ht1+2Ht2· exp
t1 0
G∗1Ht1−u1+2Ht2−u1mu1du1
· (4.47)
exp t2
t1
G∗2Ht2−u2mu2du2−Mt2
here again, the conditioning parameters have been suppressed.
As a special case of (4.47), suppose that mu=m, so that the shock generating process is a homogenous Poisson and thatG is an exponential distribution with a scale parameterb.
Furthermore, suppose that hu=1, so that the effect of the imposed stresses is cumulative.
Then, for 0≤t1≤t2
P T1≥t1 T2≥t2=
b b+1t1+2t2
b+2t2−t1 b+1t1+2t2
−mb/1+2
·
b b+2t2−t1
−mb/2
exp−mt2 (4.48) and its marginal
PTi≥t=
b+it b
mb i−1
exp−mt (4.49)
i=12, is a Pareto distribution of the third kind (Johnson and Kotz, 1970, p. 234). Withm=i/b the survival function (4.49) becomes an exponential distribution. It can be easily verified that wheni/b > <m, the model failure rate ofPTi≥tdecreases (increases) intfromi/btom (Figure 4.17).
PROBABILITY MODELS FOR INTERDEPENDENT LIFELENGTHS 113
0 Failure rate
Values of t m
0
b= m λi
b< m λi
m+b λi
m– b λi
b< m λi
Figure 4.17 Model failure rate function of the marginal.
In (4.48), if we set1=2=, and/b=m, then we see that for 0≤t1≤t2
PT1≥t1 T2≥t2=
%
1−mt1+mt2
1+mt1+mt2
exp−mt2 similarly, its symmetric version holds for 0≤t2≤t1.
Thus fort1 t2>0 PT1≥t1 T2≥t2
=
%
1−mmint1 t2+mmaxt1 t2
1+mt1+t2 exp−mmaxt1 t2 (4.50)
=
%
1− t−s
1+t+s exp−maxs t (4.51)
ifmt1=tandmt2=s.
This is a single parameter bivariate distribution with exponential marginals. The distribution generalizes easily to the multivariate case, and because it has only one parameter, namelym, it is highly scalable. Whereas the feature of scalability makes the distribution attractive, the question of realism needs to be resolved. Of particular concern here is equating the Poisson parameterm to the ratio of the component’s intrinsic model failure rateand the scale parameter bof the exponential distribution for the inflicted stress.
Strength of Dependence in the Single-parameter Bivariate Distribution
It can be shown, details omitted, that the joint survival functionPT1≥t1 T2≥t2of (4.50) has a probability density att1 t2for allt1 t2>0, and that it is absolutely continuous. Furthermore,T1 andT2are positively quadrant dependent, and the distribution does not experience the bivariate lack of memory property. With PTi> t i=12, known, it is evident that the conditional survival function
PT1≥t1T2≥t2=
%
1−mt1+mt2
1+mt1+mt2 0< t1< t2<
=
%
1+mt1−mt2
1+mt1+mt2exp−mt1−t2 t1≥t2>0
To investigate the regression ofT2 onT1, we need to findPT1≥t1 T2=t2. After some routine, but laborious manipulations, this can be shown to be:
⎧⎪
⎪⎪
⎪⎨
⎪⎪
⎪⎪
⎩
1+mt1
1+mt1−t21+mt1+t2
%1+mt1−t2
1+mt1+t2e−mt1−t2 fort1≥t2 and
1− mt1
1+mt2−t11+mt1+t2
%1+mt2−t1
1+mt1+t2 for 0< t1< t2
IntegratingPT1≥t1T2=t2with respect tot1over0would give usET1T2=t2. This is analytically difficult to do. However, its numerical evaluation shows thatET1T2=t2 tends to be a convex function oft2becoming linear ast2gets large (Figure 4.18).
A comparison of Figures 4.15 and 4.18 indicates that the regression curves of the bivariate Pareto, the model of Downton, and the model of this section are generally similar. The regression ofT1 on T2 suggests that T1 andT2 are positively correlated. Indeed Kotz and Singpurwalla (1999) show that the correlation betweenT1 andT2is 0.4825, irrespective of the value of m.
Thusmis purely a measure of location and scale.
Systems with Interdependent Lifelengths
IfTsdenotes the time to failure of a series system of two components having lifelengthsT1and T2described by the joint survival function of (4.50), then by settingt1=t2=t, we see that the reliability of the system is
PTs≥t= 1
1+2mt
1
2exp−mt t≥0 (4.52)
Contrast this expression with exp−2mt, the reliability of the system assuming indepen- dent exponentially distributed lifelengths with a scale parameter m. Clearly, the assumption of independence underestimates the reliability of a series system functioning in a shot-noise environment.
Similarly, ifTpdenotes the lifelength of a parallel redundant system, then its reliability is PTp≥t=
2−
1 1+2mt
1 2
exp−mt t≥0 (4.53)
Regression
0 Values of t2
E(T1
|
T2=t2)Figure 4.18 The Regression ofT1onT2for the single parameter bivariate exponential.
PROBABILITY MODELS FOR INTERDEPENDENT LIFELENGTHS 115 Comparing the above with 2−exp−mtexp−mt, the reliability of the system assuming independent exponentially distributed lifelengths, suggests that independence overestimates sys- tem reliability of parallel redundant systems.
An intriguing aspect of (4.52) and (4.53) is the feature that ast→ , the reliability of both the series system and the parallel system are approximated by exp−mt. This suggests that the contribution of redundancy to the reliability of the system asymptotically diminishes, so that both the series system and the parallel system behave like a single unit having an exponentially distributed lifelength.