# Life Behavior of a System under Discrete Shock Model

Authors: Serkan Eryilmaz
Journal: Discrete Dynamics in Nature and Society
 (2012)

---

## Abstract

We study the life behavior of a system which is subjected
to shocks of random magnitudes over discrete time periods. We obtain the
survival function and mean time to failure of the system assuming that the
sizes of the shocks follow a discrete probability distribution under cumulative
and mixed shock models.


---

## Body

## 1. Introduction

There are various engineering systems which are subjected to shocks of random magnitudes at random times. The shock models can be classified in different ways. According to the cumulative shock model, the system breaks down because of a cumulative effect of shocks, while in an extreme shock model the system fails because of one single shock with large magnitude. See, for example, [1–9] for various problems on shock models.Most of the studies on shock models focus on the evaluation of system failure time in a continuous setup, that is, the shocks arrive according to a renewal process, and the times between successive shocks have a continuous probability distribution. Some results on discrete case are in [3, 7, 10].Consider a system which is subjected to periodic random shocks. A shock occurs with probabilityp in each period n=1,2,…. The period should be understood as hour, day, and so forth. The magnitude of the shock which occurs in period j is a random variable denoted by Bj. Assume that such a system fails if and only if the sum of the magnitudes of cumulative shocks exceed, the level k for k>0. Let Ij be a binary random variable representing the shock occurrences that is, Ij=1 if a shock occurs in period j and Ij=0, otherwise. For j≥1, define
(1.1)Yj={Bj,Ij=10,Ij=0,
where the random variables Ij and Bj are independent in each time period. The random variable Bj is strictly positive, and {Bj,j≥1} is a sequence of independent identically distributed (i.i.d.) random variables with cumulative distribution function (c.d.f.) and probability mass function (p.m.f.) fB.Thus, under the cumulative shock model, the failure time of the system can be defined by the following waiting time random variable:(1.2)Wk=min{n:∑j=1nYj>k},
for k>0.In the case of a mixed shock model, a system fails if either a single shock with a large magnitude occurs or the sum of cumulative shocks exceeds the critical level. Thus, in this case the time to failure of the system is defined by the following compound waiting time random variable.(1.3)Zk,m=min(Wk,Tm),
where
(1.4)Tm=min{n:Mn>m},
where Mn=max(Y1,…,Yn) for k,m>0.Such models can also be applied to insurance, replacing shock with claim and magnitude of the shock with claim amount. In this case, a period can be seen as a week, month, and so forth, and the random variableWk represents the waiting time until the cumulative sum of claim amounts exceeds the level k. Similarly, the random variable Tm is the waiting time until the first extreme claim size falls above the level m.The present paper is organized as follows. In Section2, we derive recurrence formulae for the survival function and the mean time to failure (MTTF) of the system under the cumulative shock model. We also study two related characteristics N(Wk) and S(Wk) which represent, respectively, the number of shocks and the total shock that the system is subjected up to time when the system fails. Section 3 includes the results for mixed shock model.


## 2. Cumulative Shock Model

In the following, we derive two popular reliability characteristics: survival function and mean time to failure of the system under the cumulative shock model.It is clear that(2.1)P{Wk>n}=P{S(n)≤k}=FY*n(k),
where S(n)=Y1+⋯+Yn, and FY*n denotes the n-fold convolution of FY with itself, FY(x)=P{Y≤x}. By conditioning on the claim occurrence, one obtains
(2.2)FY(x)=1-p+pFB(x),
where FB(x)=P{B≤x}.Theorem 2.1.
Forn≥1,
(2.3)P{Wk>n}=p∑b=1min(k,bu)P{Wk-b>n-1}fB(b)+(1-p)P{Wk>n-1},
and P{Wk>0}=1, where bu is the endpoint of the support of fB.Proof.
From (2.1), it follows that
(2.4)P{Wk>n}=FY*FY*n-1(k).
Thus, the proof is immediate from (2.2).Proposition 2.2.
Fork>0, the MTTF of the system can be computed from
(2.5)E(Wk)=1p+∑b=1min(k,bu)E(Wk-b)fB(b),
with E(W0)=1/p.Proof.
Using (2.1),
(2.6)E(Wk)=∑n=0∞P{Wk>n}=∑n=0∞P{S(n)≤k}=∑n=0∞FY*n(k)=1+FY*E(Wk).
Thus, the proof follows from (2.2) since
(2.7)E(Wk)=1+(1-p)E(Wk)+pFB*E(Wk).Example 2.3.
LetB have a geometric distribution with pmf fB(b)=(1-α)αb-1,b=1,2,…. Then under the conditions of Proposition 2.2,
(2.8)E(Wk)=1p+(1-α)∑b=1kαb-1E(Wk-b),
with E(W0)=1/p.


### 2.1. Related Characteristics

Fork>0, we define new random variables as follows:
(2.9)N(Wk)=∑j=1WkIj,S(Wk)=∑j=1WkYj=∑j=1N(Wk)Bj.It is clear that the random variablesN(Wk) and S(Wk) represent, respectively, the number of shocks and the total shock that the system is subjected up to time when the system fails. These two characteristics might be useful for improvement purposes and can be effectively used in optimal system design.Theorem 2.4.
Form≥1,
(2.10)P{N(Wk)=m}=∑n=m∞Q(n,m,k),
where Q(n,m,k)=P(n,m,k)-R(n,m,k), and P(n,m,k) and R(n,m,k) can be computed recursively from
(2.11)P(n,m,k)=pP(n-1,m-1,k)+(1-p)P(n-1,m,k),
for n≥m and P(n,m,k)=0 for n<m, and
(2.12)R(n,m,k)=p∑b=1min(k,bu)R(n-1,m-1,k-b)fB(b)+(1-p)R(n-1,m,k),
for n≥m and R(n,m,k)=0 for n<m.Proof.
By conditioning onWk,
(2.13)P{N(Wk)=m}=∑n=m∞P{N(n)=m,Wk=n}.
The probabilityQ(n,m,k)=P{N(n)=m,Wk=n} can be written as follows:
(2.14)Q(n,m,k)=P{N(n)=m,Wk>n-1}-P{N(n)=m,Wk>n}.
Thus, we need to get recurrences for P(n,m,k)=P{N(n)=m,Wk>n-1} and R(n,m,k)=P{N(n)=m,Wk>n}. By conditioning on the values of In,
(2.15)P(n,m,k)=P{N(n)=m,Wk>n-1}=P{∑j=1nIj=m,∑j=1n-1Yj≤k}=pP{∑j=1n-1Ij=m-1,∑j=1n-1Yj≤k}+(1-p)P{∑j=1n-1Ij=m,∑j=1n-1Yj≤k}=pP(n-1,m-1,k)+(1-p)P(n-1,m,k).
On the other hand,
(2.16)R(n,m,k)=P{N(n)=m,Wk>n}=P{∑j=1nIj=m,∑j=1nYj≤k}=P{∑j=1nIj=m,∑j=1nIjBj≤k,In=1}+P{∑j=1nIj=m,∑j=1nIjBj≤k,In=0}=P{∑j=1n-1Ij=m-1,∑j=1n-1IjBj≤k-Bn}P{In=1}+P{∑j=1n-1Ij=m,∑j=1n-1IjBj≤k}P{In=0}=p∑b=1min(k,bu)P{N(n-1)=m-1,Wk-b>n-1}fB(b)+(1-p)P{N(n-1)=m,Wk>n-1},
for n≥m. Thus, the proof is completed.Before proceeding with the distribution ofS(Wk), it should be noted that the random variable S(n)=∑j=1N(n)Bj=∑j=1nYj denotes the total shock up to time n and
(2.17)P{S(n)=s}=p∑b=1min(s,bu)P{S(n-1)=s-b}fB(b)+(1-p)P{S(n-1)=s},
for 0<s≤n and P{S(n)=0}=(1-p)n.Theorem 2.5.
Fors>k,
(2.18)P{S(Wk)=s}=∑n=1∞Q*(n,s,k),
where
(2.19)Q*(n,s,k)=p∑b=s-kmin(s,bu)P{S(n-1)=s-b}fB(b).Proof.
By the definition ofS(Wk),
(2.20)P{S(Wk)=s}=∑n=1∞P{S(n)=s,Wk=n}.
For s>k,
(2.21)Q*(n,s,k)=P{S(n)=s,Wk=n}=P{S(n)=s,S(n-1)≤k,S(n)>k}=P{S(n)=s,S(n-1)≤k}=P{S(n-1)+Yn=s,S(n-1)≤k}=pP{S(n-1)=s-Bn,S(n-1)≤k}+(1-p)P{S(n-1)=s,S(n-1)≤k}.
The proof follows by conditioning on Bn and noting that P{S(n-1)=s,S(n-1)≤k}=0 for s>k.The following result readily follows from the definitions ofN(Wk) and S(Wk) and Wald’s equation.Proposition 2.6.
Fork>0,
(2.22)E(N(Wk))=pE(Wk),E(S(Wk))=pE(Wk)E(B).In Table1 we compute MTTF=E(Wk), E(N(Wk)), and E(S(Wk)) whenever the shock size random variable B has a geometric distribution with mean E(B)=1/(1-α). From Table 1 we observe that an increase in k leads to an increase in MTTF of the system. If the probability of observing a shock in a period increases, then the MTTF decreases. We also observe that MTTF is proportional to p. Therefore, for the same shock size distribution the expected number of shocks E(N(Wk)) and expected total shock E(S(Wk)) remain the same for different values of p.Table 1
E
(
W
k
)
,
E
(
N
(
W
k
)
), and E(S(Wk)) for geometric shock size distribution.
p
=
0.1
E
(
B
)
=
8
p
=
0.05
E
(
B
)
=
8
k
E
(
W
k
)
E
(
N
(
W
k
)
)
E
(
S
(
W
k
)
)
E
(
W
k
)
E
(
N
(
W
k
)
)
E
(
S
(
W
k
)
)
5
16.25
1.625
13
32.5
1.625
13
10
22.50
2.250
18
45
2.250
18
15
28.75
2.875
23
57.5
2.875
23
20
35.00
3.500
28
70
3.500
28
p
=
0.1
E
(
B
)
=
5
p
=
0.05
E
(
B
)
=
5
k
E
(
W
k
)
E
(
N
(
W
k
)
)
E
(
S
(
W
k
)
)
E
(
W
k
)
E
(
N
(
W
k
)
)
E
(
S
(
W
k
)
)
5
20
2
10
40
2
10
10
30
3
15
60
3
15
15
40
4
20
80
4
20
20
50
5
25
100
5
25


## 2.1. Related Characteristics

Fork>0, we define new random variables as follows:
(2.9)N(Wk)=∑j=1WkIj,S(Wk)=∑j=1WkYj=∑j=1N(Wk)Bj.It is clear that the random variablesN(Wk) and S(Wk) represent, respectively, the number of shocks and the total shock that the system is subjected up to time when the system fails. These two characteristics might be useful for improvement purposes and can be effectively used in optimal system design.Theorem 2.4.
Form≥1,
(2.10)P{N(Wk)=m}=∑n=m∞Q(n,m,k),
where Q(n,m,k)=P(n,m,k)-R(n,m,k), and P(n,m,k) and R(n,m,k) can be computed recursively from
(2.11)P(n,m,k)=pP(n-1,m-1,k)+(1-p)P(n-1,m,k),
for n≥m and P(n,m,k)=0 for n<m, and
(2.12)R(n,m,k)=p∑b=1min(k,bu)R(n-1,m-1,k-b)fB(b)+(1-p)R(n-1,m,k),
for n≥m and R(n,m,k)=0 for n<m.Proof.
By conditioning onWk,
(2.13)P{N(Wk)=m}=∑n=m∞P{N(n)=m,Wk=n}.
The probabilityQ(n,m,k)=P{N(n)=m,Wk=n} can be written as follows:
(2.14)Q(n,m,k)=P{N(n)=m,Wk>n-1}-P{N(n)=m,Wk>n}.
Thus, we need to get recurrences for P(n,m,k)=P{N(n)=m,Wk>n-1} and R(n,m,k)=P{N(n)=m,Wk>n}. By conditioning on the values of In,
(2.15)P(n,m,k)=P{N(n)=m,Wk>n-1}=P{∑j=1nIj=m,∑j=1n-1Yj≤k}=pP{∑j=1n-1Ij=m-1,∑j=1n-1Yj≤k}+(1-p)P{∑j=1n-1Ij=m,∑j=1n-1Yj≤k}=pP(n-1,m-1,k)+(1-p)P(n-1,m,k).
On the other hand,
(2.16)R(n,m,k)=P{N(n)=m,Wk>n}=P{∑j=1nIj=m,∑j=1nYj≤k}=P{∑j=1nIj=m,∑j=1nIjBj≤k,In=1}+P{∑j=1nIj=m,∑j=1nIjBj≤k,In=0}=P{∑j=1n-1Ij=m-1,∑j=1n-1IjBj≤k-Bn}P{In=1}+P{∑j=1n-1Ij=m,∑j=1n-1IjBj≤k}P{In=0}=p∑b=1min(k,bu)P{N(n-1)=m-1,Wk-b>n-1}fB(b)+(1-p)P{N(n-1)=m,Wk>n-1},
for n≥m. Thus, the proof is completed.Before proceeding with the distribution ofS(Wk), it should be noted that the random variable S(n)=∑j=1N(n)Bj=∑j=1nYj denotes the total shock up to time n and
(2.17)P{S(n)=s}=p∑b=1min(s,bu)P{S(n-1)=s-b}fB(b)+(1-p)P{S(n-1)=s},
for 0<s≤n and P{S(n)=0}=(1-p)n.Theorem 2.5.
Fors>k,
(2.18)P{S(Wk)=s}=∑n=1∞Q*(n,s,k),
where
(2.19)Q*(n,s,k)=p∑b=s-kmin(s,bu)P{S(n-1)=s-b}fB(b).Proof.
By the definition ofS(Wk),
(2.20)P{S(Wk)=s}=∑n=1∞P{S(n)=s,Wk=n}.
For s>k,
(2.21)Q*(n,s,k)=P{S(n)=s,Wk=n}=P{S(n)=s,S(n-1)≤k,S(n)>k}=P{S(n)=s,S(n-1)≤k}=P{S(n-1)+Yn=s,S(n-1)≤k}=pP{S(n-1)=s-Bn,S(n-1)≤k}+(1-p)P{S(n-1)=s,S(n-1)≤k}.
The proof follows by conditioning on Bn and noting that P{S(n-1)=s,S(n-1)≤k}=0 for s>k.The following result readily follows from the definitions ofN(Wk) and S(Wk) and Wald’s equation.Proposition 2.6.
Fork>0,
(2.22)E(N(Wk))=pE(Wk),E(S(Wk))=pE(Wk)E(B).In Table1 we compute MTTF=E(Wk), E(N(Wk)), and E(S(Wk)) whenever the shock size random variable B has a geometric distribution with mean E(B)=1/(1-α). From Table 1 we observe that an increase in k leads to an increase in MTTF of the system. If the probability of observing a shock in a period increases, then the MTTF decreases. We also observe that MTTF is proportional to p. Therefore, for the same shock size distribution the expected number of shocks E(N(Wk)) and expected total shock E(S(Wk)) remain the same for different values of p.Table 1
E
(
W
k
)
,
E
(
N
(
W
k
)
), and E(S(Wk)) for geometric shock size distribution.
p
=
0.1
E
(
B
)
=
8
p
=
0.05
E
(
B
)
=
8
k
E
(
W
k
)
E
(
N
(
W
k
)
)
E
(
S
(
W
k
)
)
E
(
W
k
)
E
(
N
(
W
k
)
)
E
(
S
(
W
k
)
)
5
16.25
1.625
13
32.5
1.625
13
10
22.50
2.250
18
45
2.250
18
15
28.75
2.875
23
57.5
2.875
23
20
35.00
3.500
28
70
3.500
28
p
=
0.1
E
(
B
)
=
5
p
=
0.05
E
(
B
)
=
5
k
E
(
W
k
)
E
(
N
(
W
k
)
)
E
(
S
(
W
k
)
)
E
(
W
k
)
E
(
N
(
W
k
)
)
E
(
S
(
W
k
)
)
5
20
2
10
40
2
10
10
30
3
15
60
3
15
15
40
4
20
80
4
20
20
50
5
25
100
5
25


## 3. Mixed Shock Model

Fork≤m, the mixed shock model is same as the cumulative shock model. Thus we assume that k>m. The following is a recursive equation for the survival probability of the system under mixed shock model.Theorem 3.1.
Fork>m≥1 and n≥1,
(3.1)P{Zk,m>n}=p∑b=1min(m,bu)P{Zk-b,m>n-1}fB(b)+(1-p)P{Zk,m>n-1},
and P{Zk,m>0}=1, where ∑b=xy=0 if x>y.Proof.
Forn≥1,
(3.2)P{Zk,m>n}=P{Wk>n,Tm>n}=P{∑j=1nYj≤k,Y1≤m,…,Yn≤m}.
By conditioning on the values ofIn,
(3.3)P{Zk,m>n}=P{∑j=1nIjBj≤k,Y1≤m,…,Yn≤m,In=1}+P{∑j=1nIjBj≤k,Y1≤m,…,Yn≤m,In=0}=P{∑j=1n-1IjBj≤k-Bn,Y1≤m,…,Yn-1≤m,Bn≤m}P{In=1}+P{∑j=1n-1IjBj≤k,Y1≤m,…,Yn-1≤m}P{In=0}.
By conditioning on Bn,
(3.4)P{Zk,m>n}=p∑b=1min(k,m,bu)P{∑j=1n-1IjBj≤k-b,Y1≤m,…,Yn-1≤m}fBn(b)+(1-p)P{∑j=1n-1IjBj≤k,Y1≤m,…,Yn-1≤m}.
Thus, the proof is completed.The following result can be proved similar to Proposition2.2, and hence its proof is omitted.Proposition 3.2.
Fork>m≥1, the MTTF of the system under mixed shock model can be computed from
(3.5)E(Zk,m)=1p+∑b=1min(m,bu)E(Zk-b,m)fB(b),
with E(Z0,m)=1/p, where ∑b=xy=0 if x>y.In Table2, using Proposition 3.2, we compute the MTTF of the system under mixed shock model when the shock size random variable B has a geometric distribution with mean E(B)=1/(1-α).Table 2
E
(
Z
k
,
m
) for geometric shock size distribution.
p
=
0.1
,
E
(
B
)
=
8
p
=
0.05
,
E
(
B
)
=
8
k
m
E
(
Z
k
,
m
)
E
(
Z
k
,
m
)
5
3
14.4705
28.9410
10
3
14.8958
29.7916
10
5
18.4929
36.9858
20
5
19.4064
38.8128
p
=
0.1
,
E
(
B
)
=
5
p
=
0.05
,
E
(
B
)
=
5
k
m
E
(
Z
k
,
m
)
E
(
Z
k
,
m
)
5
3
17.7442
35.4944
10
3
19.2442
38.4885
10
5
25.4125
50.8250
20
5
29.3396
58.6792Theorem 3.3.
Forn≥1,
(3.6)P{N(Zk,m)=n}=∑s=n∞[pU(n-1,s-1,k,m)+(1-p)U(n,s-1,k,m)-U(n,s,k,m)],
where
(3.7)U(n,s,k,m)=p∑b=1min(m,bu)U(n-1,s-1,k-b,m)fB(b)+(1-p)U(n,s-1,k,m).Proof.
By conditioning onZk,m,
(3.8)P{N(Zk,m)=n}=∑s=n∞P{N(s)=n,Zk,m=s}.
It is clear that
(3.9)P{N(s)=n,Zk,m=s}=P{N(s)=n,Zk,m>s-1}-P{N(s)=n,Zk,m>s}.
By the definition of Zk,m,
(3.10)U(n,s,k,m)=P{N(s)=n,Zk,m>s}=P{∑j=1sIj=n,∑j=1sYj≤k,Y1≤m,…,Ys≤m}=pP{∑j=1s-1Ij=n-1,∑j=1s-1Yj≤k-Bs,Y1≤m,…,Ys-1≤m,Bs≤m}+(1-p)P{∑j=1s-1Ij=n,∑j=1s-1Yj≤k,Y1≤m,…,Ys-1≤m}=p∑b=1min(m,bu)P{∑j=1s-1Ij=n-1,∑j=1s-1Yj≤k-b,Y1≤m,…,Ys-1≤m}fB(b)+(1-p)P{∑j=1s-1Ij=n,∑j=1s-1Yj≤k,Y1≤m,…,Ys-1≤m}=p∑b=1min(m,bu)P{N(s-1)=n-1,Zk-b,m>s-1}fB(b)+(1-p)P{N(s-1)=n,Zk,m>s-1}.
On the other hand,
(3.11)P{N(s)=n,Zk,m>s-1}=P{∑j=1sIj=n,∑j=1s-1Yj≤k,Y1≤m,…,Ys-1≤m}=pP{N(s-1)=n-1,Zk,m>s-1}+(1-p)P{N(s-1)=n,Zk,m>s-1}=pU(n-1,s-1,k,m)+(1-p)U(n,s-1,k,m).
Thus the proof is completed.Before the derivation of the distribution ofS(Zk,m), we note the following recursion which will be useful in the sequel:
(3.12)V(n,s,m)=P{S(n)=s,Y1≤m,…,Yn≤m}=p∑b=1min(m,s,bu)V(n-1,s-b,m)fB(b)+(1-p)V(n-1,s,m).Theorem 3.4.
Fors>k,
(3.13)P{S(Zk,m)=s}=p∑n=1∞∑b=s-kbuV(n-1,s-b,m)fB(b),
for m<s≤k,
(3.14)P{S(Zk,m)=s}=p∑n=1∞∑b=m+1min(s,bu)V(n-1,s-b,m)fB(b).Proof.
By the definition ofS(Zk,m),
(3.15)P{S(Zk,m)=s}=P{S(Wk)=s,Wk≤Tm}+P{S(Tm)=s,Tm<Wk}={P{S(Wk)=s,Wk≤Tm},ifs>kP{S(Tm)=s,Tm<Wk},ifm<s≤k.
For s>k,
(3.16)P{S(Wk)=s,Wk≤Tm}=∑n=1∞P{S(n)=s,Tm≥n,Wk=n}=∑n=1∞P{S(n)=s,Y1≤m,…,Yn-1≤m,Wk=n}=∑n=1∞P{S(n)=s,Y1≤m,…,Yn-1≤m,S(n-1)≤k}=∑n=1∞P{S(n-1)+Yn=s,S(n-1)≤k,Y1≤m,…,Yn-1≤m}=p∑n=1∞∑b=s-kbuP{S(n-1)=s-b,Y1≤m,…,Yn-1≤m}fB(b)=p∑n=1∞∑b=s-kbuV(n-1,s-b,m)fB(b).
Similarly, for m<s≤k,
(3.17)P{S(Tm)=s,Tm<Wk}=∑n=1∞P{S(n)=s,Wk>n,Tm=n}=∑n=1∞P{S(n)=s,S(n)≤k,Y1≤m,…,Yn-1≤m,Yn>m}=∑n=1∞P{S(n-1)+Yn=s,Y1≤m,…,Yn-1≤m,Yn>m}=p∑n=1∞∑b=m+1min(s,bu)P{S(n-1)=s-b,Y1≤m,…,Yn-1≤m}fB(b)=p∑n=1∞∑b=m+1min(s,bu)V(n-1,s-b,m)fB(b).
Thus, the proof is completed.


## 4. Summary and Conclusions

In this paper, we studied the life behavior of a system under discrete time cumulative and mixed shock models. The probability of getting a shock in any period isp, and the shock occurrences are assumed to be independent over the periods. The size of the shock occuring in a period follows a discrete probability distribution and the system’s lifetime coincides with the waiting time random variable which represents the time until the cumulative sum of shocks exceeds a specified level (cumulative shock model). We derived recurrence formulae for the survival function and the MTTF of the system. We also obtained recurrences for the distributions and expected values of the two related quantities which represent the number of shocks and the total shock that the system is subjected until failure. The results were illustrated for the case when the shock size distribution is geometric. We have also obtained a recurrence for the survival function of the system under a mixed shock model. The assumption of discrete shock size distribution enables us to obtain recursive formulae. However, the consideration of continuous shock size distribution might be of special interest in some applications. Therefore, a possible future work can be on discrete time shock models with a continuous shock size distribution.In the model that was studied in the paper shock occurrence indicators are assumed to be independent and identical with a constant probabilityp. As a future work, the case in which the shock occurrence indicators form a Markov chain can also be considered.



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*Source: 238641-2012-09-11.xml*