Consider a simple reliability system periodically bombarded by random hard shocks of magnitudes taking place at respective times . Although some of these shocks are harmless, just one of them (extreme) is fatal. Their effect is binary with respect to a given threshold H, because with , there is no impact felt by the system, but with , the system is knocked down. We can regard a kth shock harmless if , meaning that it is absorbed by one of its components such as suspension. When the system is unable to absorb a shock, which is when its magnitude exceeds H, the system fails.
1.3. N-Critical Shock System with Aging
We further embellish the above setting to model a more realistic but also analytically more challenging system. Assume that besides the hits exerted upon the system, the system also ages, so that without shocks alone, it deteriorates and eventually becomes inoperable. Suppose aging is formalized by a continuous, monotone increasing, deterministic function of time, such that (or under some other provisions to warrant crossing of some D by on a compact interval) and suppose is a sustainability threshold so that when it is crossed by , the system will be ordered to stop or suspend its operation.
In many practical situations, aging and shocks combined will cause an underlying system to fail sooner than just through aging or shocks alone. It is common that a homeowner’s insurance company will cancel someone’s policy after N large claims, or an automobile insurance company will drop its customer for N instances of traffic violations or small car accidents. On the other hand, an insurance company can drop an insured individual for reasons other than claims or violations. For example, dozens of insurance companies pulled out of the state of Florida because of recent major storms, and thus out of fear of being overwhelmed by more forthcoming claims. Also, quite a few automobile insurers significantly raised their premiums in 2023 forcing many customers to look for alternative providers. Among other reasons named by those insurers, such a policy was driven by an increasingly high volume of medical bills pertaining to car accidents in states like Florida, correlated with more intense traffic (possibly due to a record number of move-ins to Florida by residents from other states, especially California, Illinois, and New York that took place in the recent years). This situation can be identified with aging. Thus, an insured homeowner’s policy can be terminated due to N large claims (N critical shocks) or simply by the insurer’s relocation or high premium spikes (aging).
We notice, however, that system’s wear is observed exclusively on the occurrences of shocks at times . That is, if and is the exact epoch of time when a system’s wear function crosses its sustainability threshold, shutting off the system will not happen at exactly that time, but at a later moment closer to T, say some (the system can fail earlier at time , as described above, if the Nth critical shock occurs and instantly knocks the system down).
Here, we have an argument for delayed information. Unlike the common assumption in reliability literature, aging alone need not to have an instantaneous failure effect. In fact, a system can stay operational for a while, although without delivering a satisfactory performance. Consequently, an age-related failure is more difficult to detect and identify than is commonly assumed, let alone being hard to define it. This is because in many cases, systems give up slowly, and their glitches are not immediately manifested. Therefore, there is a degree of uncertainty on when the system malfunctions convincingly enough to be deemed inoperational. In real-world reliability systems, an age-related failure is often authenticated with a delay, for example, upon one of the easy-to-observe incidents of shocks when the system is rigorously tested. As it depicted in
Figure 2. Thus, it stands to reason to model reliability systems with partially delayed information, although such modeling is analytically more challenging.
We first define the age-related
delayed exit index (identifier)
Associated with is the delayed soft failure time .
As far as the hard shocks,
, are concerned, we stay under the same assumptions as above, namely, we assume that the system fails when at some
meaning that only upon a total of
N critical shocks exerted on an underlying device does it fail. Hence, in this case,
As noted, the system can become inoperational at some earlier observed epoch
if
. Therefore, we define the cumulative exit index
and the associated first passage time,
, at which the system fails either through aging or on a fatal shock, whichever of the two events comes first.
Figure 3 depicts the situation when the fourth critical and, thus, fatal shock
occurs earlier than the age-related failure at time
T and, thus, at time
when it could have been first observed delaying the real-time crossing. Thus,
while
.
In
Figure 4, we see three critical and five noncritical shocks. The 7th noncritical shock was observed at the epoch
when the system became inoperational due to aging, which makes the wear seen above
D. The real crossing of
D by
occurred at an earlier time
T. However, it was not detected in time. Furthermore, we see that the fourth critical shock
occurs at time
. However, the system failed earlier, at time
(on a noncritical shock), due to aging. Consequently,
.
Continuing with
Figure 4, note that if a noncritical shock at time
became critical and, thus, extreme or fatal, the system would undergo the two fatalities: the age-related observed failure at time
and upon the fourth critical (thus fatal) shock at the same time implying that
. In this situation, even though the real time of the age-related fatality is at time
, it was not registered.
Our target functional is , which predicts the time of system failure and —the total number of critical shocks at . Using , we can find the total material damage to the system. The latter would make sense to predict, for example, if some components will remain intact after , thereby reducing the overhaul costs compared to the total replacement.
1.4. Analysis of Related Literature
To the best of our knowledge, the above setting is new, and the methods of the underlying system are novel and original. Note that some papers assume that the aging function is stochastic, such as the gamma or Brownian motion process. While the gamma process is a useful vehicle for modeling the aging function, in our case, this feature impedes the analytical tractability of the results and, thus, deviates from our objectives.
Apparently, an
N-critical shocks system
was first introduced by Cha and Finkelstein [
1] in 2011 (the shock process was assumed to be nonhomogeneous Poisson). Wu et al. [
2] in 2022 (with no aging included) resurrected this model under the assumption that shocks arrive according to a Markov renewal process.
In 2012, Jiang et al. [
3] studied a variant of such a system with aging, soft shocks (cumulative shock model), and hard shocks. There are three thresholds,
, of which
is “critical”. It takes just one shock of a magnitude above
to knock the system down. However, once
N shocks cross
(but not
), the threshold
is downgraded to
, so that it now takes one
-critical shock (that is, of a lesser magnitude) to knock the system down. Moreover, aging, along with soft shocks, takes its course, and if the aging curve crosses some
D, the system soft fails, unless it fails earlier due to an extreme shock. Now, we see that while this system is not exactly an N-critical shock system, it carries some elements of the N-critical shock protocol.
Dshalalow and White [
4] in 2022 studied a mixed system with aging, soft, and critical shocks. The aging process was defined as linear with a deterministic slope and it was combined with soft shocks that accelerated aging, and such a cumulative aging process sooner or later crossed a sustainability threshold. The projection of such a crossing point was referred to as a soft failure. After this random point, say
, the system was shut off. The system could also fail if it was hit by one of the critical shocks, namely, by the
Nth critical shock, say at the instant
. Thus, the system fails at time
.
Related to the N-critical shocks system is a class of so-called run shock models in which an underlying system is knocked down by a total of N critical shocks landed in the system consecutively, rather than in an arbitrary order as in N-critical shock models. We note that a run shock model seems to be more rigid compared to an N-critical shock model, as the former seems to apply to more restricted classes of real-world reliability systems. Furthermore, the N-critical shock setting does not exclude consecutive runs of shocks and is, thus, more relaxed.
An apparent constraint of run models can be seen in the following situation. Assume that a system is hit by a run of
consecutive critical shocks followed by one noncritical shock, and then is followed by another run of
consecutive critical shocks and one noncritical shock, and so on. It seems likely that it takes a while (if ever) to come up against a run of
N critical shocks before the system becomes inoperational, whereas in an N-critical shocks setting, a failure would have been detected much earlier. As argued in Mallor and Omey [
5] when applied to insurance claims, only a series of
N consecutive claims deemed large enough will raise flags. It seems as if
N large claims in any order would be sufficiently concerning.
A run shock system was first introduced and studied in 2001 by Mallor and Omey [
5] followed by Mallor and Santos [
6] in 2003. Another run shock system, combined with other features, was studied by Lyu et al. [
7]. More recently in 2019, Eryilmaz and Tekin [
8] studied a run shock system that fails when the magnitudes of
N consecutive critical shocks cross a threshold
or the magnitude of one single extreme shock crosses threshold
, whichever of the two events comes first. Further embellishments of a run shock model were introduced in 2018 by Gong et al. [
9] and in 2022 by Wen et al. [
10]. Here, we see two critical thresholds,
, and two positive integers,
. The system fails if
consecutive critical shocks above
or
consecutive critical shocks above
land in the system, whichever comes first. An even more complex run model with modifications was considered by Poursaeed [
11] in 2021.
Somewhat related to our system are
-shock models. A
-shock policy applies to a system that fails when the time lag between two consecutive shocks becomes less than some fixed
. A
-shock policy is often implemented whenever shock damages are hard to observe. A
-shock model was first introduced in 1999 by Li et al. [
12] followed by Li and Kong [
13] in 2007 with the same basic assumptions whereby the authors studied the asymptotic behavior when
. Another basic
-shock model from the same period was analyzed by Tang and Lam [
14].
Then, we see some modifications and embellishments in the
-policy like in work of Parvardeh and Balakrishnan [
15] from 2015 where their system fails when
the time lag between two consecutive
-critical shocks is less than a
, or
the magnitude of any single
shock is larger than an
whichever comes first. Eryilmaz [
16] in 2012 studied a significantly upgraded variant of the
-system. The author combined the idea of run shocks and
-shocks in one system that fails when the time lags of each of
N consecutive shocks is less than
. Kus, Tuncel, and Eryilmaz [
17], in 2022, studied a mixed shock model, which combines
-shock and extreme shock models, that is, the system fails if
the time lag between two consecutive
-critical shocks is less than a
, or
a single extreme shock crosses a threshold
whichever of the two comes first. A further embellishment comes under the assumption of matrix-exponential distributions for intershock times. Among other things, the authors showed that the lifetime (time-to-failure) of the system does not have a matrix-exponential distribution, but it is approximated by a matrix-exponential distribution.
An interesting embellishment of Eryilmaz’s 2012
-run model was introduced by Jiang [
18] in 2020. Such a system has
N different failure thresholds
. If the time lag between two consecutive critical shocks lies in
(
)
the system is subject to the
ith failure type. The
Nth type is irreparable and the whole system needs replacement. The first
types allow repair.
Most recently, Roozegar et al. [
19] in 2023 combined
-shock and extreme shock models. The system fails when
k interarrival times of noncritical shocks (each in interval
land between two (nonconsecutive) critical shocks, that is, with magnitudes above
H,
, the time lag between two consecutive critical shocks is less than
, whichever comes first. Other mixed and complex
-shock models were studied by Lorvand et al. [
20] in 2020, Wu [
21] 2022, and Doostmoradi [
22] of 2023.
1.5. Our Results and Techniques Used
Our present system is different from all existing reliability models in the following way. First, it combines an N-critical shock model (only a few such models are in existence) and an aging process under an arbitrary monotone that increases the deterministic function . The crossing of a sustainability threshold D by occurs at some point observed with some random delay, where system failure can be verified, unless the system fails earlier after being hit by N critical shocks (i.e., those with magnitudes above some H), which can alternate with noncritical shocks in any order. Only after the Nth critical shock is landed in the system does the system fail. Therefore, there is a competition (or game) between two failure processes with soft versus hard failure and delayed information about the soft crossing.
We apply tools of discrete and continuous operational calculus. This is novel in the reliability literature and was introduced by Dshalalow [
23] in his earlier papers and has undergone further development throughout the years. The introduction of discrete operational calculus (including the
-operator) was motivated by the needs of fluctuation theory that have arisen from random walks (cf. Dshalalow and White [
24]) and applied to queuing, stochastic games, and finance, to name a few. A similar utility of continuous operational calculus (driven by the Laplace–Carson transform) was also enhanced and included in applications to continuous-valued processes.
The objective of our methods is to bear fully explicit formulas, which offer numerous benefits over nonclosed forms (while algorithms are also useful, closed expressions offer many beneficial alternatives). We demonstrate the analytical tractability of our formulas on special cases and by using straightforward computations.
In
Section 2, we start with the formalism of special cases solved using probabilistically straightforward tools and compare it with the same results obtained by means of discrete operational calculus. In
Section 3, we formalize the general reliability problem and introduce various functionals of the time-to-failure, the overall damage to the system, and pre-time-to-failure moments, among other things, under the assumption that the stream of shocks forms an independent and stationary increment process with position-dependent marking. The main result is carried in Theorem 1.
Section 4 and
Section 5 deal with the results for the model under special assumptions, one of which is that the input stream of shocks is a marked Poisson process.
Section 6 deals with the distributions of
or
assuming that one of them comes first. In
Section 7, we discuss the case when the threshold
D and, thus, the time
T of crossing are random.