On some properties of shock processes in a ‘natural’ scale
Author’s Accepted Manuscript On some properties of shock processes in a ‘natural’ scale Ji Hwan Cha, Maxim Finkelstein
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To appear in: Reliability Engineering and System Safety Received date: 26 May 2015 Revised date: 1 September 2015 Accepted date: 10 September 2015 Cite this article as: Ji Hwan Cha and Maxim Finkelstein, On some properties of shock processes in a ‘natural’ scale, Reliability Engineering and System Safety, http://dx.doi.org/10.1016/j.ress.2015.09.009 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting galley proof before it is published in its final citable form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
On some properties of shock processes in a ‘natural’ scale Ji Hwan Cha1 Department of Statistics, Ewha Womans University Seoul, 120-750, Rep. of Korea. e-mail: [email protected] and Maxim Finkelstein Department of Mathematical Statistics, University of the Free State 339 Bloemfontein 9300, South Africa. e-mail: [email protected] and ITMO University, 49 Kronverkskiy pr., St. Petersburg, 197101, Russia
Abstract. We consider shocks modeling in a ‘natural’ scale which is a discrete scale of natural numbers. A system is subject to the shock process and its survival probability and other relevant characteristics are studied in this scale. It turns out that all relations for the probabilities of interest become much easier in the new scale as compared with the conventional chronological time scale. Furthermore, it does not matter what type of the point process of shocks is considered. The shock processes with delays and the analogue of a shot-noise process are discussed. Another example of the application of this concept is presented for systems with finite number of components described by signatures. Keywords: shock process; Poisson process; renewal process; discrete distribution; Shotnoise process; signature 1. Introduction Various shock models have been intensively studied in the literature (see, e.g., Finkelstein and Cha (2013) and references therein) and applied to various reliability topics. For instance, Caballé at al. (2015), Montoro-Cazorla et al. (2009), Kenzin and Frostig (2009), MontoroCazorla and Pérez-Ocón (2012), van der Weide, M.D. Pandey (2011) and Ruiz-Castro (2014) have considered applications to optimal maintenance modeling. See also Finkelstein (2007), Finkelstein and Marais (2010), Song et al. (2014), Jiang et al. (2015), Noorossana and SabriLaghaie (2015), Ismihan and Murat (2015) for general shock-based reliability modelling and analysis. In reliability and safety studies and applications, the most popular model is, probably, the so called, extreme shock model, where each shock can result in a system’s failure with the specified probability and a system survives it with the complimentary probability. This model can, in principle, account for damage accumulation as well when the probability of failure increases with each survived shock (Finkelstein and Cha, 2013). It should be noted that survival probabilities of systems subject to shock processes for extreme shock model can be obtained explicitly only for the Poisson process of shocks. Even for the renewal processes of shocks, everything becomes more cumbersome and asymptotic or approximate methods 1
Corresponding Author 1
should be used for the corresponding calculations. However, probabilistic reasoning can be substantially simplified when we change our usual chronological time scale to the scale defined by the number of shocks occurrences. Of course, this can be made only for systems that allow for this ‘transformation’. Thus, in the current paper, instead of a continuous chronological time scale [0, ∞) , we will use the discrete one Ν = [1,2,...] for ‘times’ to failure of systems. For illustration of our claim, consider first, a system subject to the orderly (without multiple occurrences) process of shocks in chronological time scale. Assume for simplicity that shocks present the only cause of its failure. In reliability applications, we are usually interested in the probability of survival of a system in [0, t ) . Denote this probability by P (t ) . The simplest model is when an operating system is subject to the homogeneous Poisson process of shocks (with the constant rate λ ) and survives each shock with probability q and fails with the complementary probability p = 1 − q . In this case, it is well known that the probability of survival of a system in [0, t ) is P (t ) = exp{− pλt} ,
(1)
whereas for the nonhomogeneous Poisson process (NHPP) with rate λ (t ) and time dependent p (t ), q (t ) , this formula turns to (Block et al.,1985; Finkelstein, 2008):
t P(t ) = exp− ∫ p (u )λ (u )du . 0
(2)
Obviously, when λ (t ) = λ and p (t ) = p , (2) turns to (1). Even the simplest analog of the survival probability (1) for a renewal process already cannot be obtained in a similar simple form and, usually, computational methods, bounds or asymptotic methods are used for obtaining the corresponding probability P(t ) (Kalasnikov, 1997). The exact relation can be obviously formally written in the form of the infinite series as ∞
P(t ) = ∑ q k (G ( k ) (t ) − G ( k +1) (t )) k =0 ∞
= ∑ pq k −1 (1 − G ( k ) (t )) ,
(3)
k =1
where G (t ) is the baseline distribution for a renewal process, G ( k ) (t ) is the k-fold convolution of G (t ) with itself, G (1) (t ) ≡ G (t ), G ( 0) (t ) ≡ 1 and G ( k ) (t ) − G ( k +1) (t ) is the probability of k renewals in [0, t ) . When p → 0 (3) is characterized by the following asymptotic formula when p → 0 (Kalasnikov, 1997)
pt P (t ) = exp− (1 + o(1)) , µ
(4)
∞
where we assume that µ = ∫ G (u )du < ∞ . 0
Equations (1)-(4) are derived in a conventional time scale and define the corresponding probabilistic characteristics in a chronological time t . However, there are a lot of settings when we are not actually interested in survival in real time. It is well known that reliability indices can be functions not necessarily of time (as usually) but of other monotone quantities. For instance, the growing crack in a material can be described by its length s , whereas a random variable S is the length of a crack at which a failure occurs. Therefore, the cdf of the
2
time to failure can be parameterized accordingly as F ( s) = P ( S ≤ s) . Similar with automobiles, where S can represent a random mileage to failure (see, e.g., Gersbakh and Kordonsky (1997) and Finkelstein (2004) on alternative scales). Similar to the above two examples with a continuous alternative scale, we can consider the corresponding number of shocks in a general extreme shock model as a new alternative discrete scale. Thus, a random variable N S , which is the number of shocks till the system’s failure and its discrete distribution F (k ) = P ( N S ≤ k ) will be of interest. See also the relevant discussion in Shaked et al. (1995) and Lai and Xie (2006). By a shock we understand an external ‘point’ event that can result in a system’s failure. Usually shocks are described by a random magnitude. However, our description in this paper employs the probability of failure under a shock that is an aggregated characteristic, which already takes into account the shock’s magnitude. There are numerous practical examples of shocks effecting operating systems. In electrical systems, the peaks of voltage over a threshold can be considered as shocks. Each shock of this kind can result in a failure of a system, whereas when the fluctuations of voltage are within normal bounds, they are ‘harmless’. Hackers attack on computer systems or random missile attacks in warfare can be also considered as shocks, as well as earthquakes, lightning strikes, etc. There are two main advantages of our ‘time-free” approach as compared with the conventional case of the time scale a. All relations for the probabilities of interest become much easier, b. It does not matter what kind of a shock process is considered: only the number of shocks is relevant! The importance of the second claim is obvious and it is hard to overestimate it. We will illustrate the first statement with the simplest setting when each shock ‘kills’ a system with probability p and a system survives with probability q, which describes the extreme shock model. It is obvious that (1) (or (3)) corresponds now to the simplest power function P (k ) = q k , k = 1,2,... ,
(5)
whereas the discrete distribution of ‘time’ to failure is given by the following mass and cumulative distribution functions as k
f (k ) = pq k −1 , k = 1,2,...; F (k ) = ∑ f (i ) = 1 − P (k ) = 1 − q k ,
(6)
1
respectively. However, as was already emphasized, (5) and (6) do not depend on the type of arriving shock process, whereas (1) is true only for the HPP and (2) holds only for the NHPP. Thus we see that the probabilities of interest in the new scale are described by the corresponding discrete distributions. Ageing properties can be also formulated in a simpler way in the new scale, as only aging properties of the corresponding discrete distributions matter, whereas, e.g., in (2), the distribution F (t ) = 1 − P (t ) will be, for instance, IFR when λ (t ) p (t ) is increasing (non-decreasing). Therefore, ageing properties in the latter case obviously also depend on the rate of the arriving shock process. Thus, some properties of discrete distributions and, specifically, the discrete failure rate will play an important role in what follows. As the discrete failure rate is a controversial (in a way) characteristic, in the next section, we would like to discuss some of its properties relevant for our further presentation. Note that the new scale is not a universal one; it should be used in the justified situations when properties in chronological time are not so relevant. For instance, in a continuous case, for the warranties based on mileage, it does not matter how long in time this mileage has been accumulated.
3
An important example of the alternative discrete scale is when a piece of equipment operates in cycles and the observation is the number of cycles completed before the failure. In this case, we can ‘interpret’ one cycle of usage as a shock and, therefore, the probability of failure under a shock, e.g., in the model (5)-(6), p can be equivalently ‘interpreted’ as the probability of failure on the corresponding cycle. Thus, our formulation based on shock modelling considered in this paper can be generally applied to such systems. Furthermore, this setting describes a rather broad class of technical systems that are used intermittently (Shaked et al., 1995) and its importance for reliability practice is hard to overestimate. For the meaningful discussion of applications of discrete distributions see Bracquemond et al (2001). This note is organized as follows. In Section 2 we discuss some properties of the discrete failure rate. In Section 3 some well-known in the chronological scale shock-based setting are ‘translated’ to the discrete scale. Section 4 deals with shocks affecting the operation of a monotone system of n components. Finally, some remarks are given in Section 5.
2. Preliminaries: discrete failure rate As we are dealing with discrete distributions, we will briefly present now some well-known facts on the discrete failure rate in the manner useful for further presentation and discussion. We think that there is nothing wrong with the notion of the discrete failure rate except the term itself, as its meaning is slightly different from that in the case of continuous random variables, which sometimes can result in confusion. Moreover, some alterations of the classical definition were suggested to avoid this confusion (see the next section). For the sake of comparison, denote by Tc the lifetime random variable with absolutely continuous cdf F (t ) = P(Tc ≤ t ) and the pdf f (t ) = F ′(t ) , the failure rate λ (t ) = f (t ) / F (t ) . When ∆(t ) is sufficiently small, obviously Pr(t < Tc ≤ t + ∆t | Tc > t ) ≈ λ (t )∆t ,
(7)
whereas λ (t ) does not have the meaning of probability itself. It is well known that the series system of n independent components with failure rates λi (t ), i = 1,2,..., n has the failure rate n
λS (t ) = ∑ λ i (t ) .
(8)
1
The analog of (7) for this system with the time to failure denoted by TS will be n
n
1
1
Pr(t < TS ≤ t + ∆t | TS > t ) ≈ 1 − ∏ (1 − λi (t )∆t ) ≈ ∆t ∑ λ i (t ) ,
(9)
where the second approximation comes from the fact that ∆t → 0 . Therefore, the probability of a series system to fail in (t , t + ∆t ] , on condition that it did not fail before is approximaten
ly ∆t ∑ λ i (t ) when ∆t is small. But this is not the case for discrete distributions, where 1
∆t = 1 . That is why the analogue of (8) and, therefore, the additive property of failure rates in the series system do not hold in this case (without additional assumption that λ i (t ), i = 1,2,..., n are sufficiently small). However, we should not be worried about that. It is just the term ‘failure rate’ defined in a traditional way that sometimes can lead to confusion (Lai and Xie, 2006):
4
Pr(Td = k ) Pr(Td > k − 1)
r ( k ) = Pr(Td = k | Td > k − 1) = =
f (k ) F ( k − 1) − F (k ) = , F ( k − 1) F ( k − 1)
where Td is now a discrete lifetime with support in {1,2,....} and the probability mass function f ( k ) and the corresponding cdf, F (k ) . Just to stress our point about certain differences between the continuous and discrete cases we will call r (k ) in this paper, the ‘risk function’, although we do not suggest to rename this term generally, as it is widely used in the literature. Indeed, it is the risk (probability) of failure of an item at k on condition that it did not fail before. The following product formula is obvious (Lai and Xie, 2006). k
P( k ) ≡ F ( k ) = ∏ (1 − r (i )) .
(10)
1
In a continuous case, the product formula (10) will turn to a well-known exponential representation for survival probability. Alternatively, r i (t ), i = 1,2,..., n , should be sufficiently small. However, as ∆t = 1 and r i (t ) are not necessarily small in the discrete case, exponential representation does not hold for a general case and all other differences between continuous failure rate and discrete risk function stem from this fact, which is important to keep in mind. The risk function that corresponds to a geometric distribution (5)-(6) is obviously r (k ) = p, k = 1,2,... and is the analog of a constant failure rate of the exponential distribution in the continuous case. The increasing p (t ) in (2) can be, e.g., manifestation of the damage accumulation with every survived shock. This phenomenon can be modeled, e.g., via the IFR discrete Weibull distribution of the following form α
P(k ) = q k , α
α
f (k ) = q ( k −1) − q k , r (k ) = 1 − q k
α
α
−( k −1)
(11)
,
where k ≥ 1, α ≥ 1, 0 < q < 1 . When α = 1 this distribution reduces to the geometrical one. The ‘structure’ of the formulas (11) is obvious as, in accordance with (10), k
P( k ) = ∏ (1 − r (i )) = q
k
k −1
1
0
∑ iα −∑ iα
α
= qk .
1
The corresponding ‘shape’ of the risk function usually gives an indication of the desired properties of stochastic model. For a general case of an arbitrary increasing r (k ) (aging), we need to use a general relation (10). As point processes are defined by a counting measure, it is natural to illustrate our reasoning by considering some well-known settings defined via the ordinary continuous time scale in the discrete alternative time scale. This will be done in the next section.
3. Some shock-based settings in a natural scale. a. Shocks with delay. Assume first, that each shock can ‘kill’ an operating system with probability 1 not only immediately, but also with a delay described by the probability mass 5
function d ( j ), j ≥ 0 and the corresponding cdf D(i ) = ∑ j =0 d ( j ) which, for simplicity, does j =i
not depend on the number of the arriving shock. Therefore, a system can be ‘killed’ by all previous shocks as well. These delayed failures occur on arrival of subsequent shocks and can be interpreted as some latent faults in a system induced by a current shock and becoming failures under subsequent shocks.
Remark 1. Similar considerations hold for other not shock-related discrete settings. For instance, when we deal with systems operating in cycles, a similar explanation takes place. This is because there can be some latent faults in a system developing to failures at subsequent cycles. Consider a shock at ‘time’ k . Note that, for convenience, we will call k , k = 1,2,... , a discrete ‘time’. Let Fd ( k ) be the cdf of ‘time’ to failure. Denote by rd ( k ) probability that an operating system will be ‘killed’ at ‘time’ k on condition it has survived previous shocks. In accordance with the definition of the risk function, k −1
F (k − 1) − Fd (k ) F (k ) rd (k ) = d =1− d =1− Fd (k − 1) Fd (k − 1)
∏ D ( j) j =0 k −2
= 1 − D (k − 1)
∏ D ( j) j =0
= d (0) + d (1) + .... + d (k − 1) = D(k − 1) ,
(12)
which is intuitively obvious. When k → ∞ , this function tends to 1 that makes sense, as all previous shocks are inputting in the probability of failure. For instance, when D(i ) is a geometric distribution with probability of success θ , (12) simplifies to 1 − θ k −1 as D (k − 1) = θ k −1 and, therefore, in accordance with (10), k
Fd (k ) = ∏1θ i −1 = θ 0+1+...+ k −1 = θ
k ( k −1) 2
;0 < θ < 1 ,
which is a simple new IFR distribution. If we interpret the sum in (12) as an inputs of components in the corresponding virtual series system, which is tempting due to additivity, then, k −1
1 − ∏ (1 − d ( j )) ≠ D(k − 1) . j =0
Let now each shock be effective with probability p , which means that with this probability it can ‘kill’ the system (with a delay) and with complementary probability q = 1 − p it is harmless. Denote by Fdp (k ) the probability of surviving k consecutive shocks (not necessarily effective) and by rdp (k ) the corresponding risk function. Then, slightly modifying (12)
rdp (k ) = 1 −
Fdp (k ) Fdp (k − 1)
=1−
qFdp (k − 1) + pD (k − 1) Fdp (k − 1) Fdp (k − 1)
= p ( d (0) + d (1) + .... + d (k − 1)) = pD ( k − 1) .
6
= p (1 − D (k − 1))
The numerator can be interpreted as follows: qFdp (k − 1) is the survival probability when the k-th shock is non-effective and pD (k − 1) Fdp (k − 1) is the survival probability when it is effective multiplied by the probability that the delay is larger than k − 1 , which is D (k − 1) . Similar results hold when the probability of effective shocks is number-dependent: pi i ≥ 1 , where pi is the probability that the i th shock is effective. In this case, rd { pi } (k ) = 1 −
=1−
Fd { pi } ( k ) Fd { pi } ( k − 1)
qk Fd { pi } ( k − 1) + Fd { pi } ( k − 1)( pk (1 − d (0)) − pk −1d (1) − .... − p1d ( k − 1)) Fd { pi } ( k − 1)
= pk d (0) + pk −1d (1) + .... + p1d ( k − 1). These equations are much simpler than those in an ordinary time scale even for the simplest case of the NHPP (see Cha and Finkelstein, 2013), whereas, as stated above, we are not concerned with the type of the arrival process of shocks in the discrete scale. b. ‘Discrete shot noise’. Another important example that can be also interpreted as a generalized delay model is the classical shot noise process developed for the chronological time. It is defined in the ordinary time scale by the corresponding intensity process (Lemoine and Wenocur, 1986) as N (t )
λt = β ∑ D j h(t − T j ) ,
(13)
1
where {N (t ), t ≥ 0} is the point process of shocks with arrival times T j , j = 1,2,..., , D j are i.i.d. random variables and h(t ) is a decreasing function, whereas β > 0 is a coefficient of proportionality. Thus the effect of the previous shocks at times T j ≤ t is accumulated with the corresponding attenuation. We will see now, that (13) in the discrete case should be modified accordingly. The reason for that is again the difference between the failure rate function in continuous time and the risk function in discrete scale. The meaningful equation (13) has an additive nature and we already know that the sum of risk functions of components in a series system does not define a risk function of this system. For the methodological reason, consider first, the simplified version of (13), when D j ≡ 1, j = 1,2..., β = 1 N (t )
λt = ∑ h(t − T j ) .
(14)
1
Relationship (14) still defines the intensity process, as arrival times of shocks are random. What will be the discrete scale analogue of this simplified version of the shot-noise process? As ‘arrival times’ in the discrete scale are deterministic, this analogue is already also deterministic, thus defining the ‘ordinary’ risk function and not the corresponding stochastic process, i.e., k
rsn (k ) = ∑ h(k − j ) .
(15)
j =1
It should be noted that very specific artificial conditions should be imposed at this stage on the function h(k ) in order (15) to be the risk function for some discrete distribution (i.e., rsn ( k ) < 1, k = 1,2,...; ∑1 rsn ( k ) = ∞ ). However, there is another more natural way to deal with ∞
7
this problem. We can define the discrete analog of (14) and of the general case (13) using (as an intermediate tool) instead of the risk function r (k ) the, so-called, alternative failure rate of a discrete random variable (see, e.g., Roy and Gupta (1992), Xie et al. (2002)). The alternative discrete failure rate function, which eliminates some problems that appear when using the risk function (e.g., restoring additive property for the series systems), is defined by
λ (k ) = − ln
F (k ) , k = 1,2,... . F ( k − 1)
(16)
As
λ (k ) = − ln(1 − r (k )) or r (k ) = 1 − exp(−λ (k )) ,
(17)
the alternative failure rate can be considered as a suitable transformation of r ( k ) . Similar to (13), assume now that the alternative failure rate process describing the ‘discrete shot noise process’ is given by k
λk = β ∑ D j h(k − j ), k = 1,2,... .
(18)
j =1
In fact, (18) is suggested to be the analog of (13) in the discrete case. When h(i ) ≡ 1, i = 0,1,... we have a ‘pure’ accumulation, whereas for the decreasing h( j ) , similar to (13), λk models it with the corresponding attenuation. Now, using these initial considerations we want to come back to the risk function as we think that probabilistically it can be better motivated than the alternative failure rate. Using (17) and (18) we can define the corresponding discrete risk process as k
where M D (⋅) is the mgf of the random variable D j . Therefore, the corresponding risk function rsn ( k ) is given by
rsn ( k ) =
Fsn (k − 1) − Fsn (k ) = 1− Fsn ( k − 1)
k
k
j =1 k −1
m= j k −1
∏ M D (−β ∑ h(m − j )) ∏M
D
( − β ∑ h(m − j ))
j =1
,
m= j
whereas for the specific case when β = 1 and D j ≡ 1, j = 1,2,... , we have
rsn (k ) =
Fsn (k − 1) − Fsn (k ) =1− Fsn (k − 1)
k
m
m=1 k −1
j =1 m
m=1
j =1
∏ exp{−∑ h(m − j )} ∏ exp{−∑ h(m − j )}
k
= 1 − exp{−∑ h(k − j )}. j =1
It is important to note that survival probability in (21) has a reasonably simple structure that can be used in applications.
Example 1. Let D j in our shot noise process (18) follow the exponential distribution with mean 1 / η . Furthermore, assume that the function h( j ) exponentially decreases with j , i.e., h( j ) = exp{−κj} , κ > 0 . Under this setting, it is easy to see that M D (t ) = η /(η − t ) , t < η , and, accordingly, k
F sn (k ) = ∏ j =1
η k
η + β ∑ exp{−κ (m − j )}
, k = 1,2,...
m= j
and k
∏
η
k −1 + − − exp{ ( m j )} η β κ η + β ∑ exp{−κ (m − j )} ∑ ∏ m= j j =1 m= j . rsn (k ) = 1 − k −1 = 1 −η k k η η + β ∑ exp{−κ (m − j )} ∏ k −1 ∏ j =1 m= j j =1 η + β ∑ exp{−κ (m − j )} k −1
k
j =1
m= j
The graphs for F sn (k ) and rsn (k ) are given in Figures 1 and 2, respectively, for β = 1 , η = 10 and κ = 0.1 .
9
[Figure 1 About Here]
[Figure 2 About Here]
As can be seen from Figure 2, rsn (k ) is monotonically increasing.
c. Multiplicative model. It is well-known in reliability and survival analysis that the proportional hazards (multiplicative) model in its simplest continuous-time form is defined as
λ (t ) = zλb (t )
(22)
where λb (t ) is the baseline failure rate and z is a constant. This model describes the impact of environment in a well-justified way. When, e.g., z > 1 , the environment is more severe. What is the analog of (22) in the discrete case? The environment in this case is modeled by shocks and the meaning of parameter z is similar to that in the continuous one. When, e.g., z > 1 , the environment is more severe, which means that probabilities of failures at each shock are larger than the baseline case (the magnitude of each shock is larger). Keeping in mind the probabilistic meaning of the risk function r (k ) , it is reasonable to assume that the model (22) can be modified in the discrete case as r ( k ) = zrb ( k ), k = 1,2,...
(23)
under the condition sup{zrb ( k ), k = 1,2,...} < 1 , which can be absolutely reasonable in most practical situations, as usually, rb ( k ) << 1 . However, if this is not the case, then again the alternative failure rate for a discrete distribution can be easily employed (see (16)-(17)). Indeed, if we define the discrete multiplicative model via this function, which is obviously an assumption, as
λ ( k ) = zλb ( k ), k = 1,2,... , then coming back to the corresponding risk function, alternatively to (23) we get r ( k ) = 1 − exp(− zλb (k )), k = 1,2,.... The implications of this kind of modeling in practice are still to be explored.
4. Shocks and signatures In the previous sections, we have discussed discrete distributions and corresponding shock models on the infinite support and now we will deal with a finite support in a meaningful reliability interpretation. The purpose of this section is the illustration of how rather complex problems that cannot be reasonably described probabilistically in the ordinary time scale are effectively treated in the discrete scale. Following the reasoning of Finkelstein and Gertsbakh (2015a), consider first, a monotone system of n binary, identical, statistically independent, non-repairable components. It is well known that in this case, the structure function of a system can be defined by the values of the discrete distribution ( f1 , f 2 ..., f n ) that is called a signature, where f i is the probability that
10
the system failure takes place at the instant of the ith consecutive failure of a systems’ component (Samaniego, 2007; Gertsbakh and Spungin, 2009). Using this characteristic and employing the i.i.d. property we can define the distribution of a lifetime T of our system to be denoted as FS (t ) . Denote the cdf of each component by G (t ) , and let Gi (t ), i = 1,2,..., n be the cdf of the i-th order statistic that corresponds to G(t). Since the components failures occur in accordance with the arrival times described by distributions Gi (t ), i = 1,2,..., n . n
FS (t ) = Pr(T ≤ t ) = ∑ f i Gi (t ) ,
(24)
1
However, the meaning of ( f1 , f 2 ..., f n ) is more general, which will be illustrated in what follows. In fact, f = ( f1 , f 2 ..., f n ) is the probability mass function with the corresponding cdf and survival functions F ( x) = ∑1 f i , x = 1,2,..., n , x
(25)
F ( x) ≡ 1 − F ( x) = ∑x+1 f i , x = 1,2,..., n . n
In line with the previous sections, consider now a system (network) subject to some orderly (without multiple occurrences) external shock process {N (t ), t ≥ 0} , where N (t ) is the number of shocks in [0, t ) and assume that shocks is the only cause of failures of the components and the system, i.e., a system is absolutely reliable without shocks. Assume that each shock with equal probabilities ‘kills’ only one of the operating at the time of its occurrence components. Let the signature of a system f , which is a time-invariant characteristic, be given. Obviously, the distribution of ‘time’ to failure in the new scale (number of shocks) is given just by the cdf (25), which is remarkable due to its simplicity. Let us obtain now for comparison this distribution in the usual chronological time scale (Finkelstein and Gertsbakh, 2015a). The ordered failures of the components are governed now not by the order statistics concept as in (24), but by the orderly point process of arriving shocks, where shocks (and, therefore, the failures) are ordered in time automatically. Remark 2. In fact, it follows from the above reasoning that we can consider the probability mass function f (k ) = F (k − 1) − F (k ), k = 1,2,... of the previous sections as the signature of a system subject to a shock process in a general extreme shock model. For instance, for the α
α
Weibull distribution f (k ) = q ( k −1) − q k which reduces to f (k ) = q ( k −1) − q k = q ( k −1) p for the simplest geometrical distribution case described by equation (1). The difference is that for the shock models f (k ) is usually given via the risk function r (k ) , whereas for monotone systems, the signature should be obtained via the structure of as system, which is often not so simple especially when the number of components is not very small. The corresponding survival function can be obtained as the probability that a system survives all first n − 1 shocks in [0, t ) (the nth shock, by definition, kills the system): n −1
FS (t ) = Pr(T > t ) = ∑ Pi (t )(1 − F (i )),
(26)
0
where Pi (t ) denotes the probability of i shocks in [0, t ) and the truncation in the upper index of summation is due to 1 − F (i ) = 0, i = n, n + 1,... .
11
Equation (26) has a straightforward interpretation. Indeed, we are looking at the point process of shocks affecting a system and (26) presents the total probability of survival in this case. We now must specify the point process. The computations, in accordance with (26) can be well-performed for the NHPP (specifically for the HPP). When the point process is renewal with the cdf of the inter-arrival time R(t ) , the system cdf can be (alternatively to (3)) written as (Finkelstein and Gertsbakh, 2015a) FS (t ) = Pr(T ≤ t ) , ∞
n
= ∑ P(T ≤ t | N (t ) = k ) P( N (t ) = k ) = ∑ ( f1 + f 2 + ... + f k ) P ( N (t ) = k ) + P( N (t ) > n) 1
1 n
= ∑ f i R (i ) (t ) 1
is the i-fold convolution of R(t ) with itself R (1) (t ) = R(t ) . Specifically, for the HPP with rate λ , distribution R ( i ) (t ) becomes Erlangian of order i , R (1) (t ) = 1 − exp{−λt} . Thus we see that, as our time-free, approach does not depend on the type of the point process of shocks, it significantly simplifies derivations where applicable (see (25)). Consider now a meaningful generalization of the previous setting when each shock with probability p ‘kills’ one of the components and with probability 1 − p has no effect (Finkelstein and Gersbakh, 2015b). We call the first type of shocks the effective shocks. We observe all shocks without knowing whether they are effective or not. Using the signature of our system, the probability of a failure before (or on) the k-th shock (cdf) can be obtained as
where R
(i )
min( k ,n )
Fp ( k ) =
∑ j =1
k− j j −1 + l l q . f j p j ∑ l =0 l
Indeed, the failure that corresponds to f j occurs on the j-th effective shock. That means that there should be j − 1 effective shocks and l non-effective shocks, 0 ≤ l ≤ k − j before the failure. Given the signature f = ( f1 , f 2 ..., f n ) , can be easily obtained numerically . As our last illustration of the concept of signatures in shocks modeling, consider the monotone multistate system of n binary components that has M + 1 states: J = M , M − 1, M − 2,...,0 . In the state “ M ” all components are operable (initial state). The state “0” corresponds to the failure of a system. As previously, each shock with equal probabilities kills one of the components of a system, which gradually moves from the initial state to the state of failure. Each shock not necessarily leads to transition and therefore, n ≥ M . The case n = M means that all shocks result in transitions. We also assume that each shock can result in the transition only to the next state (no jumps more than 1 in states). The M-th transition brings the system into the failure state, whereas all other states are operable with the following ordered mean rewards per unit of ‘time’: RM > RM −1 > ... > R0 = 0. We will be interested in this application in the mean reward before the k-th transition and will obtain it via the corresponding signatures. This can be done using multivariate signatures (Gertsbakh and Spungin, 2011). However, here we will employ a simpler approach via a set of univariate signatures. Let f k = ( f1k , f 2k ...., f nk ) denote the ‘ordinary’ univariate signature that describes the structure of our system when the state M − k is ‘final’ and k = 1,2,..., M is the number of transitions to this state. When k = M , we, e.g., arrive at the state of a ‘total’ 12
failure, J = 0 . Thus each f k , k = 1,2,..., M is the signature for the corresponding binary system. Obviously, the mean ‘time’ to the k-th transition is n
LST (k ) = ∑ if i k , i =1 k
where f i is the probability that this transition will occur on the i-th shock; i=1,2,…,n. The mean holding time between transitions k and k+1 is, therefore, LST (k + 1) − LST (k ) and the corresponding expected reward before the k-th transition is k
where LST (0) = 0 and k = 1,2,..., M . Thus, we only need the set of the corresponding univariate signatures and the values of mean rewards per unit time in each operable state for obtaining R(k ) in this case. As an important application, the models considered in this section can be used for obtaining optimal maintenance schedules minimizing the long-run average cost rate. In this case, a system is replaced by a new one either on failure or after the ‘optimal k * th shock’ whichever comes first (the first 2 settings). In the last setting, a system is replaced by a new one after the ‘optimal k * th transition’ and this optimal number is obtained via LST (k ) and R (k ) . Similar topics are discussed in detail in Finkelstein and Gertsbakh (2015a,b), whereas we in this section are emphasizing only the usefulness, where applicable, of the ‘time-free’ modeling. 5. Concluding remarks
We consider some popular shock models described via the chronological time scale in the new ‘natural’ time scale, which is defined by the number of shocks experienced by a system. There are important advantages of this approach: a. All relations become much easier, as compared with the usual time scale; b. It does not matter what kind of a shock process is considered: only the number of shocks is relevant! However, the new scale is not a universal one; it should be used in the justified situations when the corresponding properties in chronological time are not so relevant. For instance, in a continuous case, for the warranties based on mileage, it does not matter how long in time this mileage has been accumulated. The example for the discrete case is when a piece of equipment operates in cycles and the observation is the number of cycles completed before the failure, which describes a rather broad class of technical systems. In this case, the probability of failure under a shock, e.g., in model (5)-(6), p can be equivalently ‘interpreted’ as the probability of failure during a cycle. As another illustration, we consider shock processes on a finite support when a system of n components is subject to shocks and each shock with equal probabilities “kills” one of the operating components. We suggest effective relations for probabilities of survival via the corresponding signatures that describe the structure of a system. Acknowledgements
The authors would like to thank the Associate Editor and referees for valuable comments and 13
suggestions. The work of the first author was supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MEST) (No. 2011-0017338). The work of the first author was also supported by Priority Research Centers Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (2009-0093827). The work of the second author was supported by the NRF (National Research Foundation of South Africa) grant IFR2011040500026. References
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Figure 1. The graph for F sn (k ) Figure 2. The graph for rsn (k )
[Figure 1]
[Figure 2]
16
Research Highlights on the paper “On some properties of shock processes in a ‘natural’ scale” ● A novel approach to systems subject to shock processes is suggested. ● Alternative to chronological time, the discrete time scale of natural numbers is considered. ● Shock processes with delays and a shot-noise process are discussed in the new alternative scale. ● As an application, n-component systems with structures described by signatures is suggested.
