1. Introduction
Reliability modelling and analysis of complex systems have been always an important topic in engineering sciences. Degradation-based modelling of failure time , as a fundamental process, has been a consistent method for analyzing the lifetime of complex systems in quite many practical situations (see, e.g., Nikulin et al. (2010), Pham (2011) Pellettier et al. (2017) and Chen et al. (2017) for a monograph on this topic). The items that are deteriorating with time having an observable process of the degradation can be entertained by a stochastic degradation model. To attain and produce high reliability systems as requested by the majority of consumers it is necessary to detect weaker systems. The association of failure time and the degradation process may not be deterministic and further investigation for the purpose of obtaining the distribution of levels of degradation and their impact on the failure time is warranted.
The stochastic process-deriven degradation model according to Albabtain et al. (2020) is considered to model the lifetime of a system. It is assumed that the stochastic process fluctuates around monotone sample paths. In the traditional definition it is assumed that the failure of an item corresponds with the time when degradation exceeds the predetermined threshold level
. Suppose that the degradation process is
with a monotonically increasing sample path as is frequently encountered in practice. The time-to-failure is denoted by
T. Then
T is the first passage time to the threshold
i.e.,
. The corresponding distribution function of the failures is denoted by
and the implied survival function is denoted by
. We also denote by
and
the distribution and density functions of
respectively. We have
If
possesses a monotonically decreasing sample path then the time-to-failure
T is the first passage time to the threshold
i.e.,
. We obtain
Degradation models vary markedly across the fields of reliability modeling. In this section, the dynamic degradation-based model for analysis of failure time data which has been recently introduced by Albabtain et al. (2020) is reviewed. The methodology behind their model is applied in situations where a unit exhibits stochastic behaviour along the time during which the degradation is taking place and there is no certain value for the degradation amount upon which the unit is failed. The pliable aspect of the dynamic degradation-based failure time model is revealed when the failure of the unit under the degradation process is assumed to follow a stochastic rule in contrast to the traditional definition that the failure of the unit is considered to be deterministic once the degradation amount reaches a pre-determined threshold.
Suppose that the amount of degradation at time
t is denoted by
with pdf
and cdf
It is considered as a postulate for increasing (decreasing) degradation paths that
for all
. In the pervious literature, it was assumed that for a given threshold value
a system under degradation fails as soon as
. This defines a stopping rule for
T to be determined so that
. This definition of the failure time was developed by Albabtain et al. (2020) so that an existing stochastic rule about the effect of degradation over time illustrates the process of the failure of the item. The failure time
T under this modified setting has the sf
where
is limit of a conditional probability given, at the level of degradation
by
To fulfill the degradation model, for an increasing (resp. decreasing) degradation path the bivariate function S must satisfy the following conditions:
- (i)
For all and for all .
- (ii)
For any fixed is decreasing in .
- (iii)
For any fixed is decreasing (resp. increasing) in .
The conditions (i)-(iii) guarantee that in (2.1) stands as a valid survival function. The model (2.1) is a dynamic failure-time model because the construction of the model is modified depending on how the survival rate of the item under degradation at a certain point of time may be influenced by the amount of the degradation. This influence is entertained by the formation of the function S.
The selection of S depends firstly on the knowledge of engineer or operator who controls the performance of the system. For example, when system degrades with time hardly (severe) then may be a proper choice. For a less severe degradation process, may be more appropriate. However, if there is no information how the system degrades with time then everything depends on failure time data (observations on T) and a model selection strategy can be accomplished, i.e., make some choices candidate and select the best one of them using some possible model selection criterions such as AIC, BIC, ... measures.
It is assumed that data on
are not achievable for all
as usually the stochastic process
is partially observed referred to well-known sources of degradation models. To move forward along the line of reputable statistical survival models a common feature for
can be adopted so that
which is the characteristic of the proportional hazard rate model whenever
is a survival function in
t for any
in which
. The function
may depend on some parameters. The baseline probability (survival rate)
measures the probability of survival of the system at the time
t at which the amount of degradation is zero. In the sequel, we may need to suppose that
is itself a survival function in
. For
the exponential distribution may always be a good choice so that
describes a no-ageing behaviour of the system under degradation. The Lomax distribution with survival function given by
is also another choice for the baseline survival rate.
2. Stepwise survival rate at interval degradation levels
In proceeding literature the correspondence between the randomness in degradation and randomness in the implied lifetime distribution has been assumed to be strong and direct so that failure occurs when the test item’s degradation level reaches at a pre-determined threshold value (
). In such a case the resulting lifetime distribution follows from (
3) if
for
and
for
. However, the equation (2.1) stands valid as an sf of time to failure of an item under degradation when
in some time
t and degradation
w. The model (
3) add the possibility to undertake situations where the deterioration of an item is not only due to the degradation. In real problems, as the time is elapsed and even the amount of degradation is not altered, the item under degradation is being ageing likewise. Therefore, the lifetime of a device subject to degradation may decrease as the degree of degradation increases. Therefore, the device is frailer at relatively high degradation levels so that a given threshold for the degree of degradation can easily be considered a deterministic rule for device failure. However, intervals of degradation levels can be set to develop a more dynamic time-to-failure degradation model.
Let us consider a degradation process with increasing degradation path and assume that
where
for
are survival rates of a unit subject to degradation when
, respectively, as the value
w takes, lies in
where
for every
such that
and
. Note that
throughout the paper. The degradation points that are adjacent to each other may induce a same amount of probability of failure, in the way that the survival rate at degradation level
takes the form
where
is the indicator function of the set
A and
. It is assumed that
do not depend on
w.
For instance in a multiplicative degradation model with increasing mean degradation path, we accept it as postulate that the probability for failure is not altered for degradation amounts in predetermined intervals, and as the degradation exceeds the last point (the greatest value) on each interval, the probability for failure is increased. For example, for high reliability products which 100 percent of them survive before the degradation level reaches and as degradation reaches , 10 percent of products fail, and the remaining 90 percent survives before the degradation level reaches and all of them fail as soon as the degradation level reaches time-to-failure is modeled upon choosing .
By using (
3) and taking
and
we get
Note that if
for every
and
where
is the threshold for degradation in the standard model, then
, i.e., (
5) reduces to (
1). The degradation process of a life unit does not always refer to products with high reliability, where gradual failure is foreseen. It also refers to situations where sudden failures are possible, with the probability of such failures increasing as the degree of degradation increases. The model (
5) may contribute effectively in such situations. Let us suppose that
is the first passage time of the stochastic process
to the value of
. By convention,
and
. If we denote by
T the time-to-failure of the device degrading over time, then
It is necessary that (
5) and (
8) have to be valid survival functions for time-to-failure
T. For example,
for all
and further, when
for every
then (
5) defines a valid SF.
We can also consider a degradation process with decreasing degradation path and assume that
where
for
are survival rates of a unit subject to degradation when
, respectively, as the value
w lies in
where
for every
such that
and
. The survival rate at degradation level
is
where
. By appealing to (
3) when
and
we can get
In this case if
and
for every
and
where
is the threshold for degradation in the standard model, then
, i.e., (
5) reduces to (
2). Let us assume that
is the first passage time of the stochastic process
to the value of
. By convention,
and
. The time-to-failure of the device is the random variable
T, and
The following lemma is essential in deriving future results. It shows that the SF of T, in the degradation model with increasing degradation path, is a convex transformation of as and . Further, the SF of T, in the degradation model with decreasing degradation path, is a convex transformation of as and .
Lemma 1. Let , the degradation process,
(i) stochastically increases with t. Then, .
(ii) stochastically decreases with t. Then, .
Proof. We first prove (i). From (
5), we can write
where
,
and
. The proof of (i) is complete. Now, let us prove (ii). In spirit of (
8), one obtains
where
,
and
. □
In the context of standard families of degradation models studied by Bae et al. (2007) we develop the failure-time model (
3) under the multiplicative degradation model.
The general multiplicative degradation model is stated as
where
is the mean degradation path and
X is random variation around
having PDF
, CDF
and SF
. If the mean degradation path is considered as a monotonically increasing function, then we develop
under the multiplicative degradation model (
10) Note that
, thus, it is deduced from Lemma 1(i) that
The PDF of
T, time-to-failure under degradation model
10 when
is increasing in
(
for all
), having SF (
11) is obtained as follows:
The failure rate associated with the SF given in (
11) is then derived as
If the mean degradation path
is a monotonically decreasing function, then the time-to-failure is denoted by
with SF
. This SF can be obtained in the setting of the multiplicative degradation model (
10). We see that
. Therefore, using Lemma 1(ii) we get
The PDF of
, time-to-failure under degradation model
10 when
is decreasing in
(
, for all
), having SF (
14) is revealed to be:
The failure rate of
T with the SF given in (
14) is
3. Stochastic ordering results
In this section we study some stochastic ordering properties of time-to-failure distributions of two devices under multiplicative degradation model. It is acknowledged in industrial sciences that products may have different qualities, some of which are more reliable whereas the others fails earlier. The extent each subject under degradation resists not to fail can be evaluated by
’s and
’s in the models (
5) and (
8), respectively (see, e.g., Lemma 1).
Let and denote two probability vectors assigning to a couple of devices working under a multiplicative degradation model with increasing mean degradation path. We suppose that P and are associated with with random lifetimes T and , respectively, such that and also where for . In a similar manner, let and denote other probability vectors related to a pair of devices working under a multiplicative degradation model with decreasing mean degradation path. It is assumed that and are associated with with random lifetimes and , respectively, such that and also where for . Suppose that is the underlying degradation model. We impose a partial order condition among P and or/and conditions on distribution of X (random variation around ) such that some stochastic orders between T and are procured. Further, we find some conditions on and and other conditions on distribution of X such that several stochastic orders between and are fulfilled.
There are some concepts in applied probability that we need to introduce them before developing our stochastic comparison results. The following definition is due to Karlin (1968).
Definition 2.
The function w, as a transformation on , is said to be totally positive of order 2, , [reverse regular of order 2, ] in , if and
for all and for all where and are two subsets of .
It is plain to verify that the [] property of w, as a transformation on , is equivalent to being non-decreasing [non-increasing] in i whenever by considering the conventions when and if .
The following lemma from Karlin (1968) known as general composition theorem (or basic composition formula) will be frequently used across the paper.
Lemma 3.
(i) (discrete case): Let g be TP in and also let w be TP (respectively, RR) in where . Then, the function , given by
(i) (continuous case): Let is in and let is (respectively, ) in , where and are two subsets of . Then,
The following definition proposes some class of functions.
Definition 4. Suppose that w, as a transformation of non-negative values, is a non-negative function. Then, w is said to have
(i) one-sided scaled-ratio increasing (decreasing), OSSRI (OSSRD), property if is increasing (decreasing) in for every .
(ii) two-sided scaled-ratio increasing (decreasing), TSSRI (TSSRD), property if is increasing (decreasing) in for every with and .
From Definition 4, it is apparent that if and also , then from assertion (ii) the ratio is increasing (decreasing) in x for every . Equivalently, this realizes that is increasing (decreasing) in x for all . Therefore, every w with having TSSRI (TSSRD) property will also fulfill the OSSRI (OSSRD) property.
Remark 5. The properties in Definition 4(i) can be applied to produce reliability classes of lifetime distributions. It can be observed that X has increasing proportional likelihood ratio (IPLR) property if, and only if, has the OSSRD property and also X has decreasing proportional likelihood ratio (DPLR) property if, and only if, has the OSSRI property (see, Romero and Díaz (2001) for definitions of IPLR and DPLR). It can also be seen that X has increasing proportional hazard rate (IPHR) property if, and only if, has the OSSRD property and, in parallel, X has decreasing proportional hazard rate (DPHR) property if, and only if, has the OSSRI property (see, Belzunce et al. (2002) for IPHR and DPHR properties). It can also be verified that X has decreasing proportional reversed failure rate (DPRFR) property if, and only if, has the OSSRD property and, further, X has increasing proportional failure rate (IPRFR) property if, and only if, has the OSSRI property (see, Oliveira and Torrado (2014) for DPRFR and IPRFR classes).
In applied probability, stochastic orders among random variables have been a useful approach for comparison of reliability of systems (see, e.g., Müller and Stoyan (2002), Osaki (2002), Shaked and Shanthikumar (2007) and Belzunce et al. (2015)). Stochastic orderings are considered a basic tool for making decisions under uncertainty (see, for instances, Mosler (1991) and Li and Li (2013)).
Let us assume that T and are random variables with absolutely continuous CDFs and , SFs and and PDFs and , respectively. We suppose that T and have hazard rate functions and , reversed hazard rate functions and , respectively. Then:
Definition 6. We say that T is smaller than or equal with in the
- (i)
likelihood ratio order (denoted as ) if is increasing in .
- (ii)
hazard rate order (denoted as ) if is increasing in or equivalently, for all .
- (iii)
reversed hazard rate order (denoted as ) if is increasing in or equivalently, for all .
- (iv)
usual stochastic order (denoted as ) if for all .
As given in Shaked and Shanthikumar (2007) we have:
It is, furthermore, well-known that
Two compare T and , according to the usual stochastic order, one sufficient conditions is found to be the well-known majorization order as given in the next definition. Majorization is a partial order relation of two probability vectors with same dimension inducing that the elements in one vector are less spread out or more nearly equal than the elements in another vector. The majorization order makes an elegant framework to compare two probability vectors (see, e.g., Marshall et al. (1965)).
We take and as two vectors of real numbers such that and denote increasing arrangement of values of and values of respectively, where is the ith smallest value among and is the ith smallest value among , for .
Definition 7. It is said that is majorized by , written as whenever and for every
In the sequel of the paper, we will assume that
T and
are two random variables denoting time-to-failure under the dynamic multiplicative degradation model
where
is an increasing function with SFs
The corresponding PDFs are derived as
We will also suppose that
and
are two random variables denoting time-to-failure under the multiplicative degradation model
where
is a decreasing function with SFs
The associated PDFs are obtained as
We utilize the following technical lemma.
Lemma 8.
- (i)
Let be a set of real numbers satisfying . If is non-decreasing in , then
- (ii)
Let be real numbers. If is non-increasing for , then
The next result discusses sufficient conditions for stochastic comparison of T and and also stochastic ordering of and according to the usual stochastic order.
Theorem 9.
- (i)
Let and be two probability vectors satisfying and such that . Then, .
- (ii)
Let and be two probability vectors with and such that . Then, .
Proof. Firstly, we prove assertion (i). Note that for any
,
By appealing to Eq. (
11) and since
for every
and also from (
17),
for every
, as
we thus by rearranging
1 the elements in sigma in Eq. (
11) conclude that
Let us take
which, by (
17), is a non-increasing function in
Since
, thus
for all
. Therefore, from Lemma 8(ii),
is non-negative, which means that
We now prove assertion (ii). For each fixed
, we have:
By applying Eq. (
14) and since
for every
and also from (
18),
for every
, when
we thus by rearranging
2 the elements of sigma in Eq. (
14) can get
We set
which by (
18), is a non-decreasing function in
Since
, thus
for all
and
. Hence, an application of Lemma 8(i) yields
is non-negative, which means that
The proof is complete. □
Remark 10. The result of Theorem 9 indicates that the usual stochastic order between T and and also that of and do not depend on the distribution of random variation X. The conditions imposed to get in Theorem 9(i) consist of an order relation among ’s (i.e., ) and the same order relation among ’s (i.e., ) and a condition of majorization order of P and . The probability vector () which majorizes the other probability vector (P) will lead to a more reliable product under multiplicative degradation model with increasing The order relations and are valid assumptions in practical works. This is because in a multiplicative degradation model with increasing , as the time t is elapsed, the degradation amount is increased and, therefore, the probability for failure is correspondingly grown. Note that the first elements of P and are associated with smaller amounts of . The conditions found to obtain in Theorem 9(ii) are, firstly, an order relation of ’s (i.e., ) and an analogues order relation of ’s (i.e., ) and, secondly, the majorization order of and Π. The probability vector (Π) which majorizes the other probability vector () will lead to a less reliable product under multiplicative degradation model with decreasing The order constraints and are also valid assumptions in practical situations. This is because in a multiplicative degradation model with decreasing , as the time t is elapsed, the factor for degradation is decreased and, therefore, the probability for failure of the product is correspondingly going up. Notice that the first elements of Π and are associated with smaller amounts take.
The following theorems present some conditions to make the order between time-to-failure random variables in the dynamic multiplicative degradation model with increasing mean degradation path (Theorem 11(i)) and the dynamic multiplicative degradation model with decreasing mean degradation path (Theorem 11(ii)).
Theorem 11.
- (i)
Let and be two probability vectors so that is non-decreasing in . If is OSSRD (OSSRI), then .
- (ii)
Let and be two probability vectors so that is non-decreasing in . If is OSSRI (OSSRD), then .
Proof. To prove (i) it suffices to demonstrate that
is non-decreasing (non-increasing) in
. Set
, for
and
, for
and also
. Therefore,
if, and only if,
is
(
) in
. Note that, by assumption,
is non-decreasing in
hence,
is
in
and also since
is OSSRD (OSSRI) and
is non-decreasing in
, thus, for every
is non-decreasing (non-increasing) in
. This means
is
(
) in
By Lemma 3(i),
is
(
) in
, and this completes the proof of (i). To prove (ii) one needs to show that
is non-decreasing (non-increasing) in
. We take
, for
and
, for
and also set
. Thus,
if, and only if,
is
(
) in
. From assumption,
is non-decreasing in
hence,
is
in
and also since
is OSSRI (OSSRD) and
is non-increasing in
, thus, for every
is non-decreasing (non-increasing) in
. This means
is
(
) in
By Lemma 3(i),
is
(
) in
, which validates the proof of (ii). □
The following theorem presents conditions to make the order between time-to-failure random variables in the dynamic multiplicative degradation model with increasing mean degradation path .
Theorem 12. Let and be two probability vectors such that
- (i)
is non-decreasing in . If is OSSRD (OSSRI), then we have .
- (ii)
is non-decreasing in . If is TSSRD (TSSRI), then we have .
Proof. For assertion (i) to be proved it is enough to show that
is non-decreasing (non-increasing) in
. Let us take
, for
and
, for
and also
. Thus,
if, and only if,
is
(
) in
. By assumption,
is non-decreasing in
i hence,
is
in
and further, since
is OSSRD (OSSRI) and
is non-decreasing in
, thus, for every
in domain of
i,
is non-decreasing (non-increasing) in
. This is equivalent to saying that
is
(
) in
By Lemma 3(i),
is
(
) in
, and this ends the proof of (i). For the proof of assertion (ii) one needs to prove that
is non-decreasing (non-increasing) in
. We can set
, for
and
, for
and also take
which is non-negative since
. By these notations
if, and only if,
is
(
) in
. From assumption,
is non-decreasing in
i hence,
is
in
and moreover, since
is TSSRD (TSSRI) and
is non-decreasing in
, thus, for every
is non-decreasing (non-increasing) in
. This is equivalent to
being
(
) in
On applying Lemma 3(i),
is
(
) in
, and this gives the required result in assertion (ii). □
In the context of Theorem 12, if is non-decreasing in , then is also non-decreasing in . We can use Lemma 3(i) to prove it. Let us take for and for when . Set where and . Since is non-decreasing in , thus is in and also it is straightforward to show that is in . Hence, is in , i.e., is non-decreasing in Therefore, the condition on probabilities in Theorem 14(ii) is weaker than the condition imposed on probabilities in Theorem 14(i). It is also plain to show that if is TSSRD (TSSRI) then is OSSRD (OSSRI). Therefore, the condition on random effect distribution in Theorem 14(ii) is stronger than the condition on random effect distribution in Theorem 14(i).
The theorem below presents conditions to make the order between time-to-failure random variables in the dynamic multiplicative degradation model with decreasing mean degradation path . The proof being similar to the proof of Theorem 14 has been omitted.
Theorem 13. Let and be two probability vectors such that
- (i)
is non-decreasing in . If is OSSRI (OSSRD), then we have .
- (ii)
is non-decreasing in . If is TSSRI (TSSRD), then we have .
The next result presents conditions under which the order is fulfilled by time-to-failure random variables in the dynamic multiplicative degradation model with increasing mean degradation path .
Theorem 14. Let and be two probability vectors such that
- (i)
is non-decreasing in . If is OSSRD (OSSRI), then .
- (ii)
is non-decreasing in . If is TSSRD (TSSRI), then .
Proof. The assertion (i) is established if one shows that
is non-decreasing (non-increasing) in
. Let
, for
and
, for
and also
. As a result,
if, and only if,
is
(
) in
. By assumption,
is non-decreasing in
i hence,
is
in
and further, since
is OSSRD (OSSRI) and
is non-decreasing in
, thus, for every
is non-decreasing (non-increasing) in
, which means
is
(
) in
Using Lemma 3(i),
is
(
) in
, and this provides the proof of (i). For assertion (ii) we need to demonstrate that
is non-decreasing (non-increasing) in
. Let us define
, for
and
, for
and also define
. Now,
if, and only if,
is
(
) in
. By assumption,
is non-decreasing in
i hence,
is
in
and in addition, since
is TSSRD (TSSRI) and
is non-decreasing in
, thus,
is
(
) in
By Lemma 3(i),
is
(
) in
, and this proves assertion (ii). □
In the setting of Theorem 14, if is non-decreasing in , then is non-decreasing in . Lemma 3(i) can be used to prove it. Let us set for and for when . Set where and . Since is non-decreasing in , thus is in and also is in . Thus, is in , i.e., is non-decreasing in Hence, the condition on probabilities in Theorem 12(ii) is weaker than the condition on probabilities in Theorem 12(i). Furthermore, if is TSSRD (TSSRI) then is OSSRD (OSSRI). This means that the condition on random effect distribution in Theorem 14(ii) is stronger than the condition on random effect distribution in Theorem 14(i).
The theorem given next presents conditions to make the order between time-to-failure random variables in the dynamic multiplicative degradation model with decreasing mean degradation path . The proof being similar to the proof of Theorem 14 has been omitted.
Theorem 15. Let and be two probability vectors such that
- (i)
is non-decreasing in . If is OSSRI (OSSRD), then we have .
- (ii)
is non-decreasing in . If is TSSRI (TSSRD), then we have .