Service Lifetime Prediction Models 121
Service Lifetime Prediction Models
Based on Stochastic Stress and Resistance Factors
Hag-Min Kim
ABSTRACT
Some statistical decision support models for predicting the service lifetime of new devices are discussed. The importance of this problem has been recognized in the industry that makes new products or materials. Since the service lifetime of new material or systems is difficult to be determined resulting from the high variation of service lifetimes an alternative way is to execute several accelerated stress life tests and to characterize the failure events in laboratory environment. In this paper, probabilistic decision models are proposed to facilitate lifetime prediction by characterizing the functional relationships between and among such time-dependent factors as environmental stresses, material resistances to such stresses, and performance measures. This approach is motivated by the observation that the service lifetime of candidate materials is a function of performance over time; performance changes are caused by degradation mechanisms that result from the loss of resistance to the stresses experienced by the materials. Possible stochastic trends between stress and material resistance are assumed, and the performance of candidate materials is investigated. The present methodology should benefit those involved in lifetimes cost evaluations of a wide variety of materials or systems.
1. INTRODUCTION
Some decision support models for predicting the service lifetime of renewable energy devices, such as reflector materials and photovoltaic modules, are considered. Prospective system manufacturers need service lifetime data to make reasonable life cycle cost decisions. Emerging advanced multiplayer solar devices are expected to exhibit significantly increased service lifetimes
compared to existing devices. The variation in service of renewable energy devices is, however, very high. In general, it is not possible to determine the exact reasons for this variation because of unknown environmental conditions and material properties. For these reasons, the works related to the service lifetime prediction has been identified as one of the important decision support tasks (e.g., [5], [6], [7], and [8]).
Little is known about the service lifetime prediction model when both stress and resistance factors are considered at the same time. For most commercially important material, the linkage between the observed failure modes and root faults falls somewhere between a cause-and-effect relationship and a black box [9]. This paper is to develop a decision support model for predicting the service lifetime of coating materials so that the degradation mechanisms can be explained by the stress magnitude and the resistance property along with the time. In particular, probabilistic decision models are proposed that are designed to characterize the functional relationships between and among the factors of stresses, material resistance, performance, and time. This approach is termed SRPt because model intends to explain the bridge functions from stress (S) to performance (P) including resistance (R) aspects and from performance to lifetime (t) of the renewable energy devices. The ultimate goal of this approach is to predict the service lifetime of certain materials. The dependence structure among those factors makes the derivation task of service lifetime prediction models more difficult but more realistic. This paper is an extension of previous work by [7] in which relationships between stress and performance were investigated (SPt). The motivation behind SRPt model arises from the fact that the service lifetime of candidate is a function of performance over time; performance changes because of degradation mechanisms that result from material’s loss of resistance to the imposed stresses. Accelerated environment tests were conducted so as to estimate the empirical results of the proposed models.
The remainder of the paper is organized as follows. The next section develops the theoretical foundations for service lifetime event and the probability of failure when S and R is considered at the same time. Suitable models for three different cases were considered; (1) when both S and R are independent each other and independent of time; (2) when S and R are dependent each other, but they are each time-dependent; and (3) when R and S are dependent upon each other, as well as time-dependent. The decision support models analyzed in this study is more generally applicable to the audience who are seeing for the service lifetime of new devices. The middle sections of this paper discuss the numerical results from the experiments that accelerated stresses are continuously imposing on new materials and the resistance measures of samples were collected. Then, the service lifetime based on the regression-on-time model estimates is proposed. The last sections of this paper discuss the results and offer conclusions. The models discussed in this paper are further considered as an integrated approach for the problem of service lifetime prediction decision. Several models suitable for SRPt analysis are identified and reviewed. Coupled with an investigation of the-time-to-failure distribution associated with materials of interest, applying these methodologies to real-world data sets can provide accurate prediction of service lifetimes.
2. THEORETICAL MODEL
Consider a simple failure model for calculating the probability of failure. The model consists of two random variables, namely, stress (load) S and resistance R. S is defined as a stress that tends to produce a failure of a component, a device or a material. The load may be also defined as a mechanical load, an environmental factor, temperature, electrical current, and etc. Further, resistance is defined as the ability of a component, a device, or material to accomplish its required function satisfactorily without a failure when subjected to an external stress [5]. It is assumed that the load is univariate unless the vector notation S=(s1+s2+….+st) is used for multiple stresses. Similar notation is used for the resistance R. Consider a case where both random variables R and S are time independent (later a time-dependent model is considered). Failure can be defined as the event that the resistance R is lower than the stress random variable S:
{ Failure } : = { R < S } (1)
The units of the random variables S and R should be the same in order to define the failure event of
the specimen. If R and S are independent and identically distributed (i.i.d.) random variables, then
the probability of failure, Pr{F}, can be calculated by evaluating the convolution integral
Pr{F} = ∫ R,S fs (x) · FR (x) dx, (2)
where
fs(x) = the pdf (probability density function) of the stress S;
FR(x) = the cdf (cumulative density function) of resistance R.
This formulation can also be found in [10]. Figure 1 shows a model where R and S are time dependent. The stress increases over time (for example, because of cumulative effects), and resistance decreases (because of the propagation of a failure mechanism resulting from the applied stress). In this case, the time-to-failure is the first passage time at which R(t) decreases below S(t). In durability terms, failure occurs when the resistance R(t) becomes less than the stress S(t) in the observation period (0, t). That is,
Pr {F} = Pr { R(t) < S(t) , during (0, t)}. (3)
Thus, the probability of failure is also time dependent, which means that the failure probability at certain time changes over time. In real systems, S may be cyclic; in laboratory environments, S may be held constant. This time dependency of R (t) and S (t) is more realistic in that many
Figure 1: Resistance and Stress behaviors with the time index
Resistance (R), R(t)
Stress (S)
S(t)
Failure time at time t
Time index t
stress variables, including weathering variables, (such as ultra-violet radiation), exhibit time dependent behavior, and the materials performance, which is a result of the time dependent resistance process, degrades over time. In this case, models should include two dependence aspects and these are time dependency as well as the dependency between the two random variables R(t) and S(t). Possible models for characterizing time dependency will be discussed in the next section.
3. SERVICE LIFETIME PREDICTION MODELS
Once the probability model for failure is established the next step is to discuss how to predict the service lifetime of a specimen. Three cases are discussed by considering the dependence among R, S, and t: (1) R and S are independent of each other, and each random variable is time-independent (i.e., coindependent); (2) R and S are independent of each other, but they are each time-dependent; (3) R and S are dependent upon each other, as well as time-dependent (i.e., codependent).
3.1 Models Where R and S are Independent of Each Other
Suppose the random variables R and S are measurable in equivalent units during the observation period (0,t), but they are coindependent. In this case, the analysis is relatively straightforward. The probability of failure in (0,t) can be obtained by evaluating the convolution integral of Equation (2). If R and S happen to be exponentially distributed, Pr{F} can be easily calculated. Let f R(ｒ)= λe-λr be the pdf of R with λ,r>0,and further f S(s)=μe-μs be the pdf of with μ,s>0. Let X = R – S be a random variable obtained by subtracting the stress value from the resistance. Then, the Laplace transformation of x is given by
(4)
where
= expected value (mean value) of the random variable of ,
= parameters that specify the mean resistance rate and the mean
stress rate, respectively when both of S and R are exponentially distributed.
Utilizing the partial fraction technique in the Laplace transformation, one can obtain the marginal distributions of x for each negative and positive part:
(5)
The probability of failure is obtained when the level of stress exceeds the material’s resistance (i.e.,
X=0) in Equation (5), which is given by
(6)
Then, Pr{F} can be represented as the ratio of mean resistance rate to the sum of the mean stress and resistance rates. In this case, as the mean level of stress increases and the mean level of resistance decreases, the probability of failure increases. Furthermore, Pr{F} can not be influenced by the variation of R and S because of their memoryless property.
If R and S have parametric distributions other than the exponential case, Pr {F} can also be obtained. However, the calculation may not be simple because the closed-form solution for the Laplace transformation is not easily obtained for the non-exponential case, and approximations are necessary. For brevity, the approximation techniques are not enumerated herein (see [2]), but the calculation can be carried out using inverse forms of the Laplace transformation of the random variable X = R – S. These models are quite similar to models used in mechanical interference theory [8]. According to the interference theory of reliability, a component fails when the applied stress exceeds the component’s strength. The method is quite simple, but it requires the assumption of time homogeneity, (i.e., the probability density functions for R and S do not depend on time ). However, stress as well as resistance is often time-dependent. For example, the environmental stress variables such as temperature, humidity, and ultraviolet radiation exhibit time dependence. Similarly, the resistance of a component or system nerally degrades over time, which also indicates time dependency. Another disadvantage of using this model is that it is very difficult to predict the service lifetime of candidate materials when the date is right censored (0,t) because no observed lifetime information is available. Thus, this case is of limited predictability and other models must be considered.
3.2 Models Where R(t) and S(t) are Dependent
Suppose both random variables, R and S, are measurable in equivalent units and are mutually independent, but each is time dependent. Let R(t) be the resistance function, which may be represented as a functional form of time. Similarly, let S(t) be the stress function of time. It is necessary to develop models for characterizing the time dependence of R (t) and S (t). Then, the probability of failure can be obtained by evaluating Equation (3). The idea is very simple, but the calculation is not a trivial task. Further, if the failure event does not happen during the observation period, the forecasting of two time series models is necessary to carry out the calculation of the probability of failure (i.e., Pr{R(t) < S(t)}).
Let {Rt;t>0} be the observed stochastic resistance process and {St;t>0} be the stochastic stress process with discrete time space t. Let Xt = Rt – St, t=1,2,3,…,n, be the difference time series at time t. Then, the service lifetime is defined as
Lifetime : = min{t: Xt = Rt - St < 0 , t>0 } (7)
where
min{t} = the first time at which the event t occurs, i.e., the first time when the
stress first exceeds the resistance R.
The lifetime of candidate devices in Equation (7) is the first passage time such that the stochastic process {X,} decreases below 0. Then, the probability of failure at time t, Pr {Ft} can be written as
Pr {Ft} : = Pr {Failure at time t } = Pr { Xt < 0}. (8)
Therefore, the foregoing presentation proceeds with the analysis of the stochastic process {Xt ; t>0}, which is also time dependent. Some possible combinations for the two input processes { Rt ; ; t>0} and {St ;; t>0}, and thus the resulting behavior of the process { Xt ; t>0}, can be postulated in the following table:
TABLE 1: Possible stochastic trends for Stress and Resistance
Cases / {St;;t>0},stress process / {Rt;;t>0},
resistance process / { Xt = Rt-St ; t>0}
the difference
process
I
/ Constant / Constant / ConstantII / Constant / Decreasing / Decreasing
III / Increasing / Constant / Decreasing
IV / Increasing / Decreasing / Decreasing
V / Random / Constant / Random
VI / Random / Random / Random
VII / Random / Decreasing / Decreasing
All models proposed in Table 1 assume that the {Xt ; t>0} is a non-increasing function of time because most devices degrade over time. For example, Figure 1 depicts the situation listed in Case IV in TABLE 1. Therefore, the best way to decide which When there exists some stochastic trend for {Xt ; t>0}, two approaches can be considered.: the first is a time-series approach; and the other is regression-on-time approach.