Freund’s bivariate distribution is absolutely continuous and possesses the bivariate lack of memory property; however, it does not allow for the simultaneous failure of both components.
Also, it does not generalize easily to the multivariate case. To account for scenarios that involve the simultaneous failure of two (or more) components, Marshall and Olkin (1967) introduced a bivariate distribution with exponential marginals, which have the bivariate lack of memory property. They termed this distribution the ‘BVE’ and its multivariate version the ‘MVE’. As
we shall soon see, the BVE has many attractive features; its main disadvantage is that it is not absolutely continuous, a consequence of which is that statistical methods based on densities cannot be easily invoked. To appreciate the structure of the BVE, it is necessary for us to overview the notion of a Poisson counting process.
The Poisson Counting Process – An Overview
Apoint processis, roughly speaking, a countable random collection of points on the real line.
LetNt be the number of points in0 t. ThenNt, as a function oft≥0, counts the events of the point process, and is hence called acounting process. To see why point processes are of interest to us in reliability, consider a scenario wherein certain events (like shocks) occur over timet≥0, according to the postulates of a special and an important kind of a point process, namely a Poisson process. The postulates of a Poisson process are:
(i) The probability of an event occuring in an interval of timet t+h=h+oh, where is a specified constant.
(ii) The probability of two or more events occurring int t+h=oh;
ohis a function ofh, saygh, forh >0 with the property that lim
h↓0gh/h=0.
As before, letNt denote the (unknown) number of events that have occurred in the time interval0 t, with the proviso thatN0=0. Then the sequence of random variablesNt t≥0 is called ahomogeneous Poisson counting process with intensity. The following properties of this process are well-known (for example, Ross, 1996).
(a) For time point s andt with s < t Nt−Ns, the number of events that occur in the intervals thas aPoisson distributionwith parametert−s; i.e.
PNt−Ns=k=e−t−st−sk k! consequently, takings=0,
PNt=k=exp−ttk
k!
withENt=t.
(b) If T1 T2 , denote the times between the arrivals of the events, i.e. the inter-arrival times, then given, theTis are independent and identically exponentially distributed. Thus PTi≥t=e−t i=12
(c) The processNt t≥0 has stationaryindependent increments, i.e. for 0≡t0< t1<
t2<· · ·< th< th+1<· · ·, where ti=t0+i, for >0, the random variables Nt1− Nt0 Nt2−Nt1 Nth+1−Nthare independent and identically distributed.
The sequence of random variablesNt t≥0is called anon-homogeneous Poisson process with a mean value function t, if the process has independent increments – see (c) above – and if for allsandtwiths < t,
PNt−Ns=k t=e− t− s t− sk
k!
hereENt= tand d t/dtdef=tis called theintensity function of the nonhomoge- nous Poisson process.
PROBABILITY MODELS FOR INTERDEPENDENT LIFELENGTHS 95 Structure of the Bivariate Exponential Distribution – BVE
Consider a system comprising of two components, connected in series or in parallel. The system operates in an environment in which three types of shocks occur, each occurring per the postulates of a homogenous Poisson process. LetNit t≥0 i=123, denote the associated counting processes, with intensities1 2and12, respectively. Suppose that the above three sequences of random variables are contemporaneously independent. The shocks associated withi have an effect on componenti alone,i=12, and those associated with12 have an effect on both components. Whenever a component receives a shock that is associated with it, the component fails. The set-up given here is very general and one can conceive of many scenarios for which it is reasonable. A simple example is an electro-mechanical system wherein certain shocks are pertinent to the mechanical part alone; certain pertinent to the electrical part alone, and certain, like an earthquake, pertinent to both. Probability models based on scenarios wherein shocks lead to failure are called ‘shock models’. LetTidenote the time to failure of componenti, and Uj j=123the time to occurrence of the first shock in the processNjt t≥0. Since the three Poisson processes mentioned above are independent, i.e., for anyt, the random variables N1t N2tandN3tare independent, the correspondingUjs are also independent. Furthermore, by property (b) of homogenous Poisson processes, eachUjhas an exponential distribution.
Thus for the ‘fatal shock model’ described above
Ti=minUi U3 i=12 and so
PT1≥t1 T2≥t21 2 12=PU1≥t1 U2≥t2 U3≥maxt1 t2
=exp−1t1−2t2−12maxt1 t2 (4.37) A generalization of the fatal shock model is the ‘non-fatal shock model’, wherein each shock is fatal to its associated component with a certain probability. Specifically, letpibe the probability that a shock generated by the processNit t≥0destroys componenti i=12, and letp10
be the probability that a shock generated by the processN3t t≥0destroys component 1 but keeps component 2 intact. Similarly,p01 p00andp11, wherep00+p10+p01+p11=1. Then, using property (b) of the homogenous Poisson process, it can be verified that for 0≤t1≤t2, and p=p1 p2 p00 p10 p01 p11,
PT1≥t1 T2≥t21 2 12 p=
exp−t11p1+12p10−t22p2+121−p00−p10 the details can be found in Marshall and Olkin (1967). Similarly, for 0≤t2≤t1,
PT1≥t1 T2≥t21 2 12 p=
exp−t11p1+121−p00− p01−t22p2+12p01 If we set 1=1p1+12p10 2=2p2+12p01and 12=12p11, we obtain
PT1≥t1 T2≥t2 1 2 12=exp− 1t1− 2t2− 12maxt1 t2 (4.38) To gain insight about the arguments that lead to the above equation, consider the case of a single component with lifelengthT, which experiences shocks per the postulates of a homogenous Poisson process with intensity . Suppose that each shock is fatal to the component with
probabilityp. Then the component survives to timetif all the shocks it experiences in0 tare non-fatal. Thus
PT≥t=
j=0
e−ttj
j! 1−pj=e−pt
If every shock is a fatal shock, then p=1, and PT ≥t=e−t, an exponential chance distribution with parameter.
Similarly, whenp1=p2=p11=1, the non-fatal shock model of (4.38) reduces to the fatal shock model of (4.37). In deriving (4.38) we have assumed that there are no after-effects of each shock that is not fatal to its component(s). WhenT1andT2have a survival function of the form given by (4.37) or (4.38), they are said to have abivariate exponential distribution, abbreviated the BVE. The distribution easily generalizes to then-variate case forn >2. However, its number of parameters grows exponentially. For example withn=3, the number of parameters is seven:
1 2 3 12 13 23 and123. Thus unless one is prepared to assume that some of the s, particularly those having multiple indices such as123, are zero, the MVE as a model for failure is unwieldy. Engineers refer to such models as being non-scalable, and models with severals set to zero are said to have a loss of granularity. Thus in using the MVE as a model for failure one may have to trade off between scalability and granularity; this is perhaps the MVE’s biggest disadvantage. However, the model has many attractive features, some of which serve to illustrate the finer aspects of modeling interdependency. These are described below with the bivariate case as a point of discussion. In the interest of giving a broad coverage, many of the results given below are without proof; the details can be found in Marshall and Olkin (1967) or in Barlow and Proschan (1975, pp. 127–138).
Moments, Marginals and Memory of the BVE
Consider the joint survival function of (4.37); here again, in all that follows, we suppress the conditioning parameters1 2, and12, to write the BVE as
F t1 t2=PT1≥t1 T2≥t2=exp−1t1−1t1−12maxt1 t2 (4.39) Claim 4.1 below is a natural consequence of the construction of the BVE.
Claim 4.1. IfT1and T2have the BVE, then there exist independent exponentially distributed random variablesU1 U2andU3such thatT1=minU1 U3andT2=minU2 U3.
By settingt1(ort2) equal to zero, we see that for anyti>0 PTi≥ti=exp−i+12ti i=12. Thus we have
Claim 4.2. The marginal distributions of the BVE are exponential. Consequently, ETi= i+12−1, andVTi=i+12−2 i=12.
For anyt≥0, (4.39) leads to the result that for alls1 s2≥0
PT1≥t+s1 T2≥t+s2T1≥s1 T2≥s2=PT1≥t T2≥t (4.40) Thus we have
PROBABILITY MODELS FOR INTERDEPENDENT LIFELENGTHS 97
Claim 4.3. The BVE enjoys the bivariate lack of memory property.
Indeed, it can be shown (cf. Barlow and Proschan, 1975, p. 130) that, besides a bivariate distribution that is based on independent exponential marginals, the BVE is the only bivariate distribution having exponential marginals which satisfies the bivariate lack of memory property.
A consequence of this feature, plus (4.40) is
Claim 4.4. The probability of survival of a two-component series system is independent of the ages of each component if, and only if, the joint lifelengths have the BVE.
The Nature of Dependence in the BVE
To investigate the dependence of T2 on T1, we consider the quantityPT2≥t2 T1=t1def= F t2t1. This quantity, evaluated via an evaluation of limt1↓0PT2≥t2t1≤T1< t1+t1, leads to the result that
F t2t1=
exp−2t2 t2≤t1
1
1+12exp−12t2−t1−2t2 t2> t1 (4.41) The regression ofT2onT1, obtained by integratingF t2t1over 0 to, is
ET2T1=t1= 1
2− 12e−2t1 21+122+12
where=1+2+12. Thus the regression ofT2onT1is an exponentially increasing function oft1that is bounded by 1/2(Figure 4.15). This behavior parallels that ofET2T1=t1in the case of Gumbel’s bivariate exponential distribution, Version II, for >0; here the conditional mean is bounded by1+/2.
Positive dependence in the case of the BVE can be asserted via the covariance. Specifically, it can be seen (cf. Barlow and Proschan, 1975, p. 135) that
CovT1 T2= 12
1+122+12
so thatT1 T2, Pearson’s coefficient of correlation is 12/. When 12=0 T1 T2=0, and (4.37) together with Claim 4.2 imply thatT1 andT2 are independent. Thus for the BVE, T1 T2=0 implies that T1 andT2 are independent. Another distribution which shares this property, namely that T1 T2=0 implies that T1 and T2 are independent, is the bivariate Gaussian.
The BVE of (4.39) generates the survival copula
Cu v=uvminu− v−
where=12/1+12and=12/2+12(Nelson, 1999, p. 47). SinceCu v≥uv, the lifelengthsT1andT2are positively quadrant dependent.
WhereasF t2t1is well defined att2=t1– see equation (4.41) – it does take a downward jump of size12/1+12exp−2t1at that point. The size of the jump depends on both1 and2, and is greater than zero, even if1=2; the jump vanishes when12=0. I am able to show, details omitted, that the size of the jump is (approximately) equal toPT2=t1T1=t1, (Figure 4.10).
Values of T2 0
1
λ1+λ12 λ12e–λ2t1 F (t2
|
t1)t1
Figure 4.10 Jump in the conditional survival function of the BVE.
0 t1 T2 t1 T2 t1 T2
F (t2
|
t1)dt2
−d
Jump
(a) λ1>λ2
0 (b) λ1<λ2
0
Cusp
(c) λ1=λ2
Figure 4.11 The conditional density of the BVE.
Corresponding to the jump of the conditional survival function of Figure 4.10 is a jump in its conditional probability density−dF t2t1/dt2, which exists for allt2=t1; fort1=t2, the density does not exist. It is easy to verify that the size of the jump in the density, att2=t1, is 12/1+121−2e−2t1. The jump is upward when1< 2, and the density has a cusp when1=2; see Figures 4.11bandc, respectively.
Even though the probability density ofF t2t1is not defined at the pointt2=t1, its failure rate, by virtue of (4.8), is defined everywhere. Specifically, if
rt2t1def
= −dF t2t1/F t2t1/dt2 then it is easy to verify that
rt2t1=
⎧⎪
⎪⎨
⎪⎪
⎩
2 t2< t1 2+12 t2> t1 and
12
1+12 t2=t1
and the last equation is a consequence of the jump inF t2 t1. Figure 4.12 shows a plot of rt2t1. It is instructive to compare Figures 4.9 and 4.12; recall that the former pertains to the
PROBABILITY MODELS FOR INTERDEPENDENT LIFELENGTHS 99
Values of T2 0
Point mass of λ12/(λ1+λ12) λ2+λ12
λ2
Failure rate of F (t2 |t1)
t1
Figure 4.12 Failure rate function of the second to fail component in the BVE.
0
(a) Survival function (b) Failure rate function
1
Failure rate P(T2 ≥ t2)
|
T1 ≥ t1)t1 T2 0 t1 T2
λ2+λ12
λ2
Figure 4.13 The conditional survival functionT2≥t2T1≥t1of the BVE and its failure rate function.
failure rate of the second component to fail in Freund’s model. Both the failure rate functions experience a jump at t2=t1, but the BVE has a point mass of size 12/1+12at t2=t1. The point mass could be smaller than2, between2and2+12, or greater than2+12. In Figure 4.12, we show only the last case.
Our discussion thus far, pertaining to the nature of dependence of T2 on T1, has been based on a consideration of PT2≥t2 T1=t1. An analogous discussion based uponT1≥ t1 as a conditioning event can also be conducted. The major difference between the cases PT2≥t2 T1=t1 andPT2≥t2 T1≥t1is that whereas the former experiences a jump at t2=t1 (Figure 4.10), the latter experiences a cusp at t2=t1. That is PT2≥t2 T1≥t1 is continuous, but not differentiable at t2=t1 (Figure 4.13 (a)). A consequence is that the failure rate of PT2≥t2 T1≥t1 is not defined at t2=t1 (Figure 4.13 (b)). Thus, when investigating the dependence of T2 onT1, the nature of the conditioning event is to be borne in mind.
A comparison of Figures 4.9, 4.12 and 4.13(b) is instructive.
Survival Function of the BVE and its Decomposition
The BVE has other noteworthy features, many of which are of mathematical interest alone. Some of these are given below; they help us gain a deeper appreciation of this remarkably interesting most referenced but least used joint distribution.
We start by noting that fort1> t2>0, limt2↑t1
Ft1 t2 t1
=lim
t2↓t1
Ft1 t2 t1
fort2> t1>0. Thus we have
Claim 4.5. The BVE has a singularity along the linet1=t2; the singularity disappears when 12=0.
We next note that
P T1=T2= 0
e−1t e−2t12e−12tdt
= 12
1+2+12
The integral term is a consequence of the fact that the lifelengths of the two components can only be equal at the time of occurrence ofU3, but provided thatU3 precedes bothU1andU2. Thus we have
Claim 4.6. The BVE has a probability mass along the linet1=t2, unless of course12=0.
After some tedious but routine calculations, it is evident that 2F t1 t2/t1t2 exists for botht1> t2>0 andt2< t1<0; however, it does not exist fort1=t2. Sincet1=t2 has a two- dimensional Lebesgue measure zero, we claim that2F t1 t2/t1t2exists almost everywhere.
However, it can be seen – again after some tedious calculations – that
t1
t2
2F t1 t2
t1t2 =F t1 t2
thus2F t1 t2/t1t2cannot be regarded as a density. We therefore have
Claim 4.7. The BVE is not absolutely continuous; it does not have a probability density with respect to the two-dimensional Lebesgue measure.
A consequence of Claim 4.7 is that one is unable to use methods based on densities because writing out the likelihood poses difficulties (Bemis, Bain and Higgins, 1972; and section 5.4.7).
Since the BVE is not absolutely continuous, its Lebesgue decomposition (section 4.2) will entail discontinuities and/or singularities. However, as shown in Barlow and Proschan (1975, p. 133), the BVE has an absolutely continuous part and only a singular part; it has no discrete part. This is summarized via
Claim 4.8. For=1+2+12, the Lebesgue decomposition ofF t1 t2yields F t1 t2=1+2
Fat1 t2+12
Fst1 t2 where the absolutely continuous part is
Fat1 t2= 1+2
e−1t1−2t2−12maxt1t2− 12
1+2
e−maxt1t2
PROBABILITY MODELS FOR INTERDEPENDENT LIFELENGTHS 101 and the singular part is
Fst1 t2=e−maxt1t2
Fat1 t2is the absolutely continuous part because2Fat1 t2/t1t2exists almost every- where, and because
t1
t22Fat1 t2/t1t2=Fat1 t2. SimilarlyFst1 t2is the singular part because2Fst1 t2/t1t2=0 for 0< t1< t2< and for 0< t2< t1<; it does not exist whent1=t2. A verification of these statements entails several laborious steps.
Letfat1 t2=2Fat1 t2/t1t2be the probability density function ofFat1 t2. Then, it is evident that
fat1 t2=
⎧⎪
⎪⎪
⎨
⎪⎪
⎪⎩
1+122
1+2 e−1+12t1−2t2 t1> t2>0 2+121
1+2 e−1t1−2+12t2 t2> t1>0 it is undefined fort1=t2.
The marginals ofFat1 t2 Fat1andFat2, obtained by setting in Fat1 t2 t2=0 and t1=0, respectively, turn out to be mixtures of exponential distributions. Specifically,
Fat10=Fat1=
1+2 e−1+12t1− 12
1+2 e−t1 t1>0 and Fa0 t2=Fat2=
1+2
e−2+12t2− 12
1+2
e−t2 t2>0 Interestingly, if infat1 t2we set
=1+ 112
1+2
=2+ 212
1+2
=1+12 =2+12
then the resulting expression takes the form of (4.31) with parameters and. This, we recall, is the probability density function of Freund’s bivariate exponential distribution. We thus have
Claim 4.9. Freund’s bivariate exponential distribution can also be derived via a shock model.
The survival function of Freund’s distribution isFat1 t2, the absolutely continuous part of the BVE, and its marginals are mixtures of exponential distributions.
In Fat1the weights assigned to the exponential components have opposite signs; thus its failure rate is increasing.
Block and Basu (1974) deriveFat1 t2via the bivariate lack of memory property, and call Fat1 t2an ‘absolutely continuous BVE’, abbreviated ACBVE. Because of Claim 4.9, Block and Basu (1974) state that the ACBVE is a variant of Freund’s distribution, which we recall has the bivariate lack of memory property; thus the ACBVE Fat1 t2also possesses the lack of memory property.
The marginals ofFst1 t2, the singular component of the BVE, areFsti=e−ti ti>0 i= 12. Analogous to Claim 4.8, which is a decomposition ofF t1 t2, is a decomposition ofF t1 andF t2, the marginals of the BVE. This decomposition turns out to be a mixture ofFatiand Fsti i=12, the marginals of the absolutely continuous and the singular components of the BVE. Specifically
Claim 4.10. The marginals of the BVE are mixtures of the marginalsFatiandFsti, respec- tively. That is, fori=12,
F ti=1+2
Fati+12
e−ti ti>0 Systems Having Interdependent Lifelengths Described by a BVE
Let Ts and Tp be the times to failure of a series and a parallel system, respectively, for a two-component system whose lifelengthsT1andT2have the BVE of (4.39). Clearly,
PTs≥t=PT1≥t T2≥t=exp−t
where=1+2+12. Thus the survival function of the series system is exponential, and its failure rate is constant,.
The case of the parallel system is more interesting. For one, it brings into play the problem of identifiability of the parameters1 2and12. Specifically, should the first failure experienced by the system be an individual failure (i.e. the failure of component 1 or component 2, but not both), then the cause of the second failure cannot be identified; it could be due to a component- specific shock or a common shock. With the matter of identifiability, it is difficult to use the observed time of the second failure, sayt2, to estimate2and12. Identifiability also manifests itself in the form of identical survival functions for the fatal and the non-fatal shock models;
((4.37) and (4.38)).
It is easy to verify (section 4.7.4) that since P
Tp≥t
=1−PTp≤tdef
= Fpt
=exp−1+12 t+exp−2+12 t−exp−t
the probability density ofFptexists everywhere and thatFptis absolutely continuous. The failure rate ofFpt rpttherefore exists, and can be shown to be
rpt=12+1
e−1+12t1−e−2t Fpt +2
e−2+12t1−e−1t Fpt
Figure 4.14 is a plot of this failure rate function for1=1 2=2, and12=1 5.
Figure 4.14 reveals some interesting features. The first is thatrp0=1 5, suggesting that the two-component system can fail instantly, as soon as it is commissioned into operation.
The second is thatrptinitially increases and then decreases (albeit slightly), asymptoting to a constant 2.5. This feature of rpt is reminiscent of the failure rate function of a gamma distribution, which starting from zero increases and then asymptotes to a constant (cf. Figure 3.5.2 of Barlow and Proschan 1975, p. 75). The analogy is not surprising, because when1=2= and12=0 Tphas a gamma distribution with scaleand shape 2. The decrease inrptprior to its asymptoting to 2.5 is a consequence of the fact that1=2; the constant 2.5 can be identified with12+min1 2.
The above behavior ofrptmotivates us to consider its decomposition. The components of this decomposition are shown by the dotted graphs of Figure 4.14; their interpretation – given below – is instructive.
We start by noting that for the two-component parallel redundant systemPT1< tTp> tis the conditional probability that component 1 has failed bytgiven that the system has not. Since
PROBABILITY MODELS FOR INTERDEPENDENT LIFELENGTHS 103
0 1.5 2.5
0.5 1
0 2 3
1 2 3 4 5 6 7 8 9 10
rP(t)
λ1P(T2<t
|
TP>t)λ2P(T1 < t
|
TP>t)Time t Failure rate
λ12
Figure 4.14 The Failure rate function for a two-component parallel system whose lifelengths are a BVE.
P
T1< tTp> t
=PT1< t Tp> t
Fpt and since P
T1< t Tp> t
=PN1t >0 N2t=N3t=0 P
T1< tTp> t
=1−exp−1texp−2+12t Fpt
Similarly, we can findPT2< t Tp> t,mutatis mutandis. Thus, we may write the following as a decomposition ofrpt:
rpt=12+1PT2< tTp> t+2PT1< tTp> t
Sincerptdt≈Pt≤Tp< t+dtTp≥t, and sinceidtis approximately the probability that thei-th shock occurs in an interval dt, the above decomposition has a probabilistic interpretation in terms ofiandrpt.
Generalizations and Extensions of the BVE
Besides a generalization to then-variate case, a consequence of which is the problem of scala- bility, the structure and the manner of construction of the BVE suggests some natural ways to build upon it. Some of these are overviewed below.
One possibility is to allow the shocks in a fatal shock model to have after-effects. Thus, for example, we may assume that each component can withstand exactly k shocks, be they of a component-specific type or a common type;kis assumed to be known. When such is the case, the bivariate survival function F t1 t2will have gamma distributions for their marginals (cf.
Barlow and Proschan, 1975, p. 138). Such survival functions do not posses the bivariate lack of memory property. The above theme can be extended by supposing that componentican withstand