Our discussion so far has been limited to the development of failure models (univariate and multivariate) that are indexed by a single scale, namely, time. However, there are scenarios which require that the occurrence of failure be registered in terms of two (or more) scales;
time and some other metric which may or may not be related to time. For example, when an automobile fails, we often note its chronological age as well as its mileage. This type of information is required for making claims against an automobile’s warranty (cf. Singpurwalla and Wilson 1993). In the biological and medical contexts we would be interested in knowing the cumulative radiation from diagnostic radiotherapy as a function of age, in order to assess if the hazards of therapy outweigh its benefits. Often the two scales bear a strong relationship to each other, and this is what makes the topic of multiple scales interesting. The relationship could be a deterministic one, though most likely it tends to be stochastic. For example, the mileage accumulated by an automobile increases with its age, but its value depends on usage, and usage varies from one individual to the other.
Singpurwalla and Wilson (1998) propose a strategy for constructing failure models indexed by two scales, time and a time-dependent quantity such as usage. A time-dependent quantity is considered because most other measures of failure turn out to be functions of time; the approach generalizes to multiple scales. In what follows I describe the strategy proposed by Singpurwalla and Wilson (1998) since the emphasis there is on probabilistic modeling in a reliability setting. However, it is helpful to note that there have been other proposals for the treatment of multiple scales, namely, those by Nelson (1995), Oakes (1995), Kovdonsky and Gertsbach (1997), and Jewell and Kalbfleisch (1996). The work of Jewell and Kalbfleisch comes close in spirit to that described here; the works of Nelson and of Oakes boil down to combining scales so that the actual analysis is conducted on a single scale.
In section 4.9.1, I outline an overall strategy for developing failure distributions with multiple scales. Included here are candidate models for relating usage with chronological age, and an approach for capturing the effect of usage, which now becomes acovariateof the time to failure.
A covariate of the time to failure is to be interpreted as a variable that influences the time to failure. For example, the operating environment, discussed in section 4.7.5, whose effect on the failure rate of the item is encapsulated by the parameter, can be viewed as being a covariate.
Our strategy for relating usage with age is based on Cox’s (1972) celebrated proportional hazard model. The relationships between usage and time are prescribed by some well-known stochastic processes, such as the Poisson and its variants. In section 4.9.2, I summarize results on a specific model that is a consequence of the techniques of section 4.9.1.
4.9.1 Model Development
By way of notation, letT denote the time to failure of an item andMtits cumulative usage (or dose) at timet≥0. ThusMTdef=U is the cumulative usage at failure. Both T andU are
FAILURE DISTRIBUTIONS WITH MULTIPLE SCALES 121 random variables; furthermore, by construction they are dependent. Our aim is to specify the joint probability density function ofT andU, attandu, respectively, assuming that it exists;
bothtanduare non negative. LetfTUt udenote this joint density. Then fTUt u=fTtfU Tut
by the multiplication rule, wherefU Tutis the density atuof the conditional distribution ofU givenT; we suppose that this conditional density also exists. SinceU=MT, the above decomposition can also be written as
fTUt u=fTtfMTTut
=fTtfMtTut since we are conditioning atT=t
=fMTufT Mttu by symmetry of the multiplication rule.
Observe that infTMttu tappears two times: as an argument of the variableTMt, and as an indexing parameter of the random variableMt. Indeed as a function oft fTMttu is not a probability density function. Our next step is to prescribe meaningful forms forfMtu andfT Mttu. This is done next.
Candidate Usage Processes
Since usage varies from unit to unit, a natural model for Mtis a non-decreasing stochastic processMt t≥0, with M0≡0. For usage characterized by simple counts, such as the number of times a unit is turned on (and off), a suitable model forMt t≥0would be a Poisson process with mean value function t; (section 4.7.4). Thus, given t t≥0,
PMt=u t=e− t tu/u! u=01 (4.57) For usage that manifests as damage due to shocks of random magnitude, such as the landing gear of an airplane, the ‘compound Poisson process’ is a meaningful model. Specifically, if Nt t≥0, denotes the number of shocks (or landings) inflicted on a unit by time t, and if Nt t≥0is described by a non-homogenous Poisson process with mean value function t, thenMt, the cumulative damage (or equivalently, the usage) at timetisMt=Nt$
i=1
Xi, where Xiis the damage to the unit caused by thei-th shock. If all theXis are assumed to be independent and have a distribution that is identical to that ofX, where the distribution ofXhas a density at xgiven asfXx, then the random variableMthas density atuof the form
e− t j=1
tj
j! fXju (4.58)
the stochastic process Mt t≥0is known as acompound Poisson process. The quantity fXjuis the density atuof the random variableX1+X2+ · · · +Xj; it is known as thej-fold convolutionoffXu.
The Poisson and the compound Poisson processes have independent increments (section 4.7.4) and are therefore appropriate if the future of Mt is not influenced by its past. Furthermore the number of increments of a Poisson process in a finite interval of time are finite. Thus they are meaningful for describing usage such as damage due to shocks (which are intermittent). By contrast ‘gamma processes’ (section 4.7.6) have an infinite number of increments in a finite
interval of time, and are therefore suitable for describing wear caused by continuous use. The structure of a gamma process is given below.
Suppose that at is non-decreasing and left continuous in t, with a0=0, and b=0, a positive constant. Then the processMt t≥0is said to be agamma processwith a shape functionatand a scale parameterb, if for anyt≥s≥0, andM0≡0,
(i) Mthas independent increments, and
(ii) Mt−Mshas a gamma distribution with a scale parameter 1/b, and a shape parameter at−as.
Whenatis linear int Mt t≥0will become a ‘Lévy Process’; this process is appropriate for describing wear caused by a continuous use of the item. Lévy processes are discussed in section 7.2.2. Like damage, wear (which can be observed and measured) is a proxy for usage.
When usage is intermittent so is the wear; when such is the caseMt t≥0can be described by Çinlar’s (1972), ‘Markov additive process’ (Singpurwalla and Wilson, 1998). Markov additive processes are overviewed in section 7.3.4, wherein an illustration involving random usage is also given.
Describing the Effect of Usage on Time to Failure
Suppose that the item in question has a propensity to fail even when it does not experience any usage. This could happen due to a deterioration in the item’s resistance to failure because of natural causes. Letr0tbe the failure rate of the item were it not to be subjected to any usage;
r0tis known as thebaseline failure rate. Usage modifiesr0tby increasing it; we assume that this modification is additive so thatrt, the failure rate of the distribution ofT, is of the form
rt=r0t+Mt (4.59)
where >0 is a constant. The model of (4.59) is known as an additive hazards model; it suggests that each unit of use increases the failure rate ofT by the same amount. In actuality, such a model may or may not be true; all the same, it is considered here for illustrative purposes – also see Section 7.5.2. Let
t= t 0
Muduand0t= t 0
r0udu
Then, by the exponential formula of reliability (section 4.3) conditional on Mt=u and t=,
fTMt ttu=r0t+uexp−0t+
this is the conditional density ofT att, given Mtand t. Averaging out with respect to tgivenMt, enables us to obtain the conditional density ofT att, givenMt. The details are in (4.6) of Singpurwalla and Wilson (1998), who go on to show that whenMt t≥0is the Poisson process of (4.57), the conditional density ofT att, givenMt=uis of the form:
r0t+ue−0t t
0
e−t−st tds
u
(4.60)
wheres=dsd sis the intensity function of the process.
FAILURE DISTRIBUTIONS WITH MULTIPLE SCALES 123 4.9.2 A Failure Model Indexed by Two Scales
In section 4.9.1, we discussed several possibilities forfMtuandfTMtu, (4.60) being an example of the latter. The purpose of this section is to see an illustration of how the above two can be put together to arrive atfTUt u, the joint density att u ofT andU. Suppose now that the usage process is the non-homogenous Poisson process of (4.57), and that the additive hazards model of (4.59) is invoked. Then,fTMttuwill be given by (4.60) and withfMtu given by (4.57), we have
fTUt u=r0t+u
u! e−0t− t t
0
se−t−sds
u
fort≥0 andu=012
A simplification of the above is achieved if r0t=r, a constant and t=t, for some constant >0; that is,Mt t≥0is a homogenous Poisson process. When such is the case our bivariate failure model, with timet≥0 as one scale, and usageu=012 , as the other scale becomes
fTUt u=r+u u!
1−e−t
u
e−r+t (4.61)
The marginal distributions ofT andU are PT > t=exp
−r+t+
1−e−t and PU=u= ur+u
&u
i=0+r+i
Results analogous to (4.61) for other processes discussed in section 4.9.1 tend to be cumber- some; they entail approximations and simulations. More details, with an example involving an application of the ideas given here to a problem entailing the specification of a warranty for traction motors of electric locomotives are in Singpurwalla and Wilson (1998).
I conclude this section by citing the work of Lawless and his colleagues on the role of multi- indexed failure models in the context of statistical analysis of product warranty data; an overview is in Lawless (1998).