#### 3.1. Reliability Modeling for Humidity Sensors Subject to Soft Failure

Figure 1a demonstrates that the soft failure of humidity sensors occurs when the total drift exceeds

D. The total drift

X_{S}(

t) includes long-term continuous drift, positive offsets, and offset reduction due to self-recovery. The long-term continuous drift is due to ageing,

X(

t), is given as

The

X(

t) may follow a linear degradation path with random coefficients or a randomized logistic degradation path. Furthermore, it may be necessary to apply a transformation to result in a linear form [

29,

30]. For illustration purposes, we used a linear degradation path to characterize the long-term continuous drift, where the parameter

β is a random variable that corresponds to normal distribution

β~

N(

μ,

σ^{2}) and where

a is a constant.

We assumed that the positive offsets caused by shocks are normally distributed, and denoted as

Y_{i} for

i = 0, 1, 2, …,

∞, and

Y_{i}~

N(

μ_{Y},

σ_{Y}). When considering self-recovery, the positive offsets distribution function is

where Φ(•) is the cumulative density function (CDF) of a standard normally distributed variable.

t_{i} represents the arrival time of the

i-th shock, and

T_{i} represents the inter-arrival time between the

i-th shock and the (

i + 1)-th shock.

The cumulative positive offset

S(

t) is given by a compound Poisson process

where

N(

t) is the number of random shocks.

Ignoring self-recovery, the cumulative positive offset S_{1}(t) can be calculated as

The difference of the cumulative positive offset between considering and ignoring self-recovery can be calculated as

The larger the number of shocks, the higher the shock frequency and the better self-recovery performance, the greater the difference of whether it considers self-recovery.

The probability of the i-th shock occurring by time t is

Furthermore, if we consider G(t) to be the CDF of ${\widehat{Y}}_{i}$ at time t, and G^{j}(t) a j convolution of G(t), then the CDF of the S(t) can be derived as

The total drift X_{S}(t) of humidity sensors can be expressed as

By using Equations (1), (3) and (6), the reliability model of humidity sensors subject to soft failure can be derived as

The reliability model in Equation (8) can be derived for a specific case with a normally distributed Y_{i} and β

#### 3.3. Reliability Modeling for Humidity Sensors Subject to MDCFPs

Hard or soft failure can cause humidity sensors to fail. The reliability function can be derived as

The reliability function can be expressed for a more specific case

As shown in Equation (13), when 0 < τ < ∞, the smaller the value of τ, the stronger the product’s self-recovery. When τ_{1} < τ_{2}, offsets caused by shocks with an inter-arrival time greater than τ_{1} can recover, which includes the offsets caused by shocks with the inter-arrival time between τ_{1} and τ_{2}. Therefore, the smaller the value of τ, the more the offsets recover, the less the degradation volume, and the higher the reliability.

Based on Equation (13), the probability density function (PDF) of the failure time is

where

φ(•) is the PDF of a standard normally distributed variable.

#### 3.4. Some Special Cases

With different parameters, the reliability model Equation (13) can be transformed into different reliability models and coincides with models with a slight difference to the previous literature.

When

τ = ∞, the reliability model Equation (13) can be transformed into a reliability model for dependent competing failure as shown in Equation (15). The model of Equation (15) ignores self-recovery as with the previous literature [

21]. As

τ = ∞ means that when the inter-arrival time of two continuous shocks is smaller than infinite, a shock can cause positive offsets to the continuous long-term drift, that is, all shocks can cause offsets.

Ignoring self-recovery (τ = ∞), the reliability is shown as

Based on Equation (15), the PDF of the failure time is derived as

When

τ = 0, the reliability model of Equation (13) is transformed into a reliability model for independent competing failure as shown in Equation (17). As

τ = 0 means that positive offsets can recover when the inter-arrival time of two continuous shocks is greater than 0, that means all offsets can recover. It also means that shocks do not cause offsets to long-term continuous drift when

τ = 0, that is, hard failure and soft failure are independent of each other. This reliability model is similar to the model used in the previous study [

32].

When soft failure and hard failure are independent (τ = 0), the reliability is shown as

Based on Equation (17), the PDF of the failure time is derived as

Reliability modeling for products that experience soft failure only concerns the performance degradation process. By setting the parameters of random shock in Equation (13) to 0 and ignoring hard failure, the reliability model of Equation (13) can be converted to the reliability model based on performance degradation.

Based on Equation (19), the PDF of the failure time is derived as

Traditional reliability theory only focuses on hard failure. By setting the parameters of soft failure in Equation (13) to 0, the reliability model of Equation (13) can be converted to the traditional reliability model, which only considers hard failure due to random shocks.

Based on Equation (21), the PDF of the failure time is derived as

The reliability model Equation (13) developed in this paper can be transformed into different reliability models seen in previous literature, as shown in

Table 1. This means that models 1, 2, 3, and 4 are special cases of the reliability model developed in this paper.