2. Cumulative Shock Model

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Volume 2012, Article ID 238641,12pages doi:10.1155/2012/238641

Research Article

Life Behavior of a System under Discrete Shock Model

Serkan Eryilmaz

Department of Industrial Engineering, Atilim University, Incek, 06836 Ankara, Turkey

Correspondence should be addressed to Serkan Eryilmaz,[email protected] Received 21 June 2012; Revised 31 July 2012; Accepted 6 August 2012

Academic Editor: M. De la Sen

Copyrightq2012 Serkan Eryilmaz. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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.

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,1–9for 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 in3,7,10.

Consider a system which is subjected to periodic random shocks. A shock occurs with probabilitypin each periodn1,2, . . .. The period should be understood as hour, day, and so forth. The magnitude of the shock which occurs in periodjis a random variable denoted byBj. Assume that such a system fails if and only if the sum of the magnitudes of cumulative shocks exceed, the levelkfork >0. LetIjbe a binary random variable representing the shock occurrences that is,I_j1 if a shock occurs in periodjandI_j0, otherwise. Forj≥1, define

Yj

B_j, I_j1

0, Ij0, 1.1

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where the random variables Ij and Bj are independent in each time period. The random variable B_j is strictly positive, and {B_j, j ≥ 1} is a sequence of independent identically distributed i.i.d. random variables with cumulative distribution function c.d.f. and probability mass functionp.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:

Wkmin

⎧⎨

⎩n: n j1

Yj> k

⎫⎬

⎭, 1.2

fork >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

Zk,mminWk, Tm, 1.3

where

T_mmin{n:M_n> m}, 1.4

whereMnmaxY1, . . . , Ynfork, 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 variable Wk represents the waiting time until the cumulative sum of claim amounts exceeds the levelk. Similarly, the random variableT_mis the waiting time until the first extreme claim size falls above the levelm.

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 characteristicsNW_kandSW_kwhich represent, respectively, the number of shocks and the total shock that the system is subjected up to time when the system fails. Section3includes 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

P{Wk> n}P{Sn≤k}F_Y^∗nk, 2.1

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whereSn Y1· · ·Yn, andF_Y^∗ndenotes then-fold convolution ofFY with itself,FYx P{Y ≤x}. By conditioning on the claim occurrence, one obtains

F_Yx 1−ppF_Bx, 2.2

whereFBx P{B≤x}.

Theorem 2.1. Forn≥1,

P{W_k> n}p^mink,b

u

b1

P{W_k−b> n−1}f_Bb 1−p

P{W_k> n−1}, 2.3

andP{W_k>0}1, whereb^uis the endpoint of the support off_B. Proof. From2.1, it follows that

P{W_k> n}F_Y∗F_Y^∗n−1k. 2.4

Thus, the proof is immediate from2.2.

Proposition 2.2. Fork >0, the MTTF of the system can be computed from

EWk 1

p ^mink,b

u

b1

EWk−bfBb, 2.5

withEW0 1/p.

Proof. Using2.1,

EWk

∞ n0

P{Wk> n}^∞

n0

P{Sn≤k}^∞

n0

F_Y^∗nk 1FY ∗EWk. 2.6

Thus, the proof follows from2.2since

EW_k 1

1−p

EW_k pF_B∗EW_k. 2.7

Example 2.3. LetBhave a geometric distribution with pmff_Bb 1−αα^b−1, b 1,2, . . ..

Then under the conditions of Proposition2.2,

EWk 1

p 1−α^k

b1

α^b−1EWk−b, 2.8

withEW0 1/p.

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2.1. Related Characteristics

Fork >0, we define new random variables as follows:

NW_k ^W^k

j1

I_j,

SW_k ^W^k

j1

Y_j^NW^k

j1

B_j.

2.9

It is clear that the random variablesNWk and SWk 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,

P{NW_k m} ^∞

nm

Qn, m, k, 2.10

where Qn, m, k Pn, m, k −Rn, m, k, and Pn, m, k and Rn, m, k can be computed recursively from

Pn, m, k pPn−1, m−1, k 1−p

Pn−1, m, k, 2.11

forn≥mandPn, m, k 0 forn < m, and

Rn, m, k p^mink,b

u

b1

Rn−1, m−1, k−bfBb 1−p

Rn−1, m, k, 2.12

forn≥mandRn, m, k 0 forn < m.

Proof. By conditioning onWk,

P{NW_k m} ^∞

nm

P{Nn m, W_kn}. 2.13

The probabilityQn, m, k P{Nn m, Wkn}can be written as follows:

Qn, m, k P{Nn m, Wk> n−1} −P{Nn m, Wk> n}. 2.14

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Thus, we need to get recurrences forPn, m, k P{Nn m, Wk > n−1}andRn, m, k P{Nn m, W_k> n}. By conditioning on the values ofI_n,

Pn, m, k P{Nn m, Wk> n−1}

P

⎧⎨

⎩ n j1

I_jm,ⁿ⁻¹

j1

Y_j≤k

⎫⎬

⎭pP

⎧⎨

⎩ n−1

j1

I_jm−1,ⁿ⁻¹

j1

Y_j≤k

⎫⎬

⎭

1−p P

⎧⎨

⎩ n−1

j1

I_jm,ⁿ⁻¹

j1

Y_j ≤k

⎫⎬

⎭ pPn−1, m−1, k

1−p

Pn−1, m, k.

2.15

On the other hand,

Rn, m, k P{Nn m, W_k> n}P

⎧⎨

⎩ n

j1

I_jm,ⁿ

j1

Y_j≤k

⎫⎬

⎭ P

⎧⎨

⎩ n

j1

I_jm,ⁿ

j1

I_jB_j≤k, I_n1

⎫⎬

⎭ P

⎧⎨

⎩ n j1

I_jm,ⁿ

j1

I_jB_j≤k, I_n0

⎫⎬

⎭ P

⎧⎨

⎩ n−1

j1

Ijm−1,ⁿ⁻¹

j1

IjBj≤k−Bn

⎫⎬

⎭P{In1}

P

⎧⎨

⎩

n−1

j1

Ijm,ⁿ⁻¹

j1

IjBj≤k

⎫⎬

⎭P{In0}

p^mink,b

u

b1

P{Nn−1 m−1, Wk−b> n−1}fBb

1−p

P{Nn−1 m, W_k> n−1},

2.16

forn≥m. Thus, the proof is completed.

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Before proceeding with the distribution ofSWk, it should be noted that the random variableSn _Nn

j1 Bj_n

j1Yjdenotes the total shock up to timenand

P{Sn s}p^mins,b

u

b1

P{Sn−1 s−b}fBb 1−p

P{Sn−1 s}, 2.17

for 0< s≤nandP{Sn 0} 1−pⁿ. Theorem 2.5. Fors > k,

P{SWk s}^∞

n1

Q^∗n, s, k, 2.18

where

Q^∗n, s, k p^mins,b

u

bs−k

P{Sn−1 s−b}f_Bb. 2.19

Proof. By the definition ofSWk,

P{SWk s}^∞

n1

P{Sn s, Wkn}. 2.20

Fors > k,

Q^∗n, s, k P{Sn s, Wkn}

P{Sn s, Sn−1≤k, Sn> k}

P{Sn s, Sn−1≤k}

P{Sn−1 Yns, Sn−1≤k}

pP{Sn−1 s−B_n, Sn−1≤k}

1−p

P{Sn−1 s, Sn−1≤k}.

2.21

The proof follows by conditioning onB_nand noting thatP{Sn−1 s, Sn−1≤k}0 for s > k.

The following result readily follows from the definitions ofNWkand SWkand Wald’s equation.

Proposition 2.6. Fork >0,

ENW_k pEW_k,

ESWk pEWkEB. 2.22

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Table 1:EWk, ENWk, andESWkfor geometric shock size distribution.

p0.1 EB 8 p0.05 EB 8

k EWk ENWk ESWk EWk ENWk ESWk

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

p0.1 EB 5 p0.05 EB 5

k EWk ENWk ESWk EWk ENWk ESWk

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

In Table 1 we compute MTTF EW_k, ENW_k, and ESW_k whenever the shock size random variable B has a geometric distribution with mean EB 1/1−α.

From Table1we observe that an increase inkleads 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 top. Therefore, for the same shock size distribution the expected number of shocksENWkand expected total shockESWkremain the same for diﬀerent values ofp.

3. Mixed Shock Model

Fork ≤m, the mixed shock model is same as the cumulative shock model. Thus we assume thatk > m. The following is a recursive equation for the survival probability of the system under mixed shock model.

Theorem 3.1. Fork > m≥1 andn≥1,

P{Z_k,m> n}p^minm,b

u

b1

P{Z_k−b,m> n−1}f_Bb 1−p

P{Z_k,m> n−1}, 3.1

andP{Z_k,m>0}1, where_y

bx0 ifx > y.

Proof. Forn≥1,

P{Z_k,m> n}P{W_k> n, T_m> n}P

⎧⎨

⎩ n j1

Y_j ≤k, Y1≤m, . . . , Y_n≤m

⎫⎬

⎭. 3.2

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By conditioning on the values ofIn,

P{Zk,m> n}P

⎧⎨

⎩ n j1

IjBj≤k, Y1 ≤m, . . . , Yn≤m, In1

⎫⎬

⎭ P

⎧⎨

⎩ n

j1

IjBj≤k, Y1≤m, . . . , Yn≤m, In0

⎫⎬

⎭ P

⎧⎨

⎩

n−1

j1

IjBj≤k−Bn, Y1≤m, . . . , Yn−1≤m, Bn≤m

⎫⎬

⎭P{In1}

P

⎧⎨

⎩ n−1

j1

IjBj≤k, Y1≤m, . . . , Yn−1≤m

⎫⎬

⎭P{In0}.

3.3

By conditioning onB_n,

P{Z_k,m> n}p^mink,m,b

u

b1

P

⎧⎨

⎩

n−1

j1

I_jB_j ≤k−b, Y1≤m, . . . , Y_n−1≤m

⎫⎬

⎭f_B_nb

1−p P

⎧⎨

⎩

n−1

j1

I_jB_j≤k, Y1≤m, . . . , Y_n−1≤m

⎫⎬

⎭.

3.4

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

EZk,m 1

p^minm,b

u

b1

EZk−b,mfBb, 3.5

withEZ0,m 1/p, where_y

bx0 ifx > y.

In Table2, using Proposition3.2, we compute the MTTF of the system under mixed shock model when the shock size random variableBhas a geometric distribution with mean EB 1/1−α.

Theorem 3.3. Forn≥1,

P{NZ_k,m n}^∞

sn

pUn−1, s−1, k, m 1−p

Un, s−1, k, m−Un, s, k, m , 3.6

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Table 2:EZk,mfor geometric shock size distribution.

p0.1, EB 8 p0.05, EB 8

k m EZk,m EZk,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

p0.1, EB 5 p0.05, EB 5

k m EZk,m EZk,m

5 3 17.7442 35.4944

10 3 19.2442 38.4885

10 5 25.4125 50.8250

20 5 29.3396 58.6792

where

Un, s, k, m p^minm,b

u

b1

Un−1, s−1, k−b, mf_Bb 1−p

Un, s−1, k, m. 3.7

Proof. By conditioning onZk,m,

P{NZk,m n}^∞

snP{Ns n, Zk,ms}. 3.8

It is clear that

P{Ns n, Z_k,ms}P{Ns n, Z_k,m> s−1} −P{Ns n, Z_k,m> s}. 3.9

By the definition ofZk,m,

Un, s, k, m

P{Ns n, Z_k,m> s}

P

⎧⎨

⎩ s j1

I_j n,^s

j1

Y_j ≤k, Y1≤m, . . . , Y_s≤m

⎫⎬

⎭ pP

⎧⎨

⎩ s−1 j1

I_jn−1,^s−1

j1

Y_j≤k−B_s, Y1 ≤m, . . . , Y_s−1≤m, B_s≤m

⎫⎬

⎭

1−p P

⎧⎨

⎩ s−1 j1

I_jn,^s−1

j1

Y_j≤k, Y1≤m, . . . , Y_s−1≤m

⎫⎬

⎭

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p^minm,b

u

b1

P

⎧⎨

⎩ s−1 j1

I_jn−1,^s−1

j1

Y_j≤k−b, Y1≤m, . . . , Y_s−1 ≤m

⎫⎬

⎭f_Bb

1−p P

⎧⎨

⎩ s−1 j1

I_jn,^s−1

j1

Y_j≤k, Y1≤m, . . . , Y_s−1≤m

⎫⎬

⎭ p^minm,b

u

b1

P{Ns−1 n−1, Zk−b,m> s−1}fBb

1−p

P{Ns−1 n, Z_k,m> s−1}.

3.10

On the other hand,

P{Ns n, Zk,m> s−1}

P

⎧⎨

⎩ s j1

I_j n,^s−1

j1

Y_j ≤k, Y1≤m, . . . , Y_s−1≤m

⎫⎬

⎭ pP{Ns−1 n−1, Zk,m> s−1}

1−p

P{Ns−1 n, Z_k,m> s−1}

pUn−1, s−1, k, m 1−p

Un, s−1, k, m.

3.11

Thus the proof is completed.

Before the derivation of the distribution ofSZk,m, we note the following recursion which will be useful in the sequel:

Vn, s, m P{Sn s, Y1≤m, . . . , Y_n≤m}

p^minm,s,b

u

b1

Vn−1, s−b, mf_Bb 1−p

Vn−1, s, m. 3.12

Theorem 3.4. Fors > k,

P{SZk,m s}p^∞

n1 b^u

bs−k

Vn−1, s−b, mfBb, 3.13

form < s≤k,

P{SZ_k,m s}p^∞

n1 mins,b^u

bm1

Vn−1, s−b, mf_Bb. 3.14

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Proof. By the definition ofSZk,m,

P{SZk,m s}P{SWk s, Wk≤Tm}P{STm s, Tm< Wk}

P{SW_k s, W_k≤T_m}, ifs > k P{STm s, Tm< Wk}, ifm < s≤k.

3.15

Fors > k,

P{SW_k s, W_k≤T_m}^∞

n1

P{Sn s, T_m≥n, W_kn}

^∞

n1

P{Sn s, Y1≤m, . . . , Y_n−1≤m, W_kn}

^∞

n1

P{Sn s, Y1≤m, . . . , Y_n−1≤m, Sn−1≤k}

^∞

n1

P{Sn−1 Yns, Sn−1≤k, Y1≤m, . . . , Yn−1≤m}

p^∞

n1 b^u

bs−k

P{Sn−1 s−b, Y1≤m, . . . , Yn−1≤m}fBb

p^∞

n1 b^u

bs−k

Vn−1, s−b, mf_Bb.

3.16

Similarly, form < s≤k,

P{ST_m s, T_m< W_k} ^∞

n1

P{Sn s, W_k> n, T_mn}

^∞

n1

P{Sn s, Sn≤k, Y1≤m, . . . , Y_n−1≤m, Y_n> m}

^∞

n1

P{Sn−1 Yns, Y1≤m, . . . , Yn−1≤m, Yn> m}

p^∞

n1 mins,b^u

bm1

P{Sn−1 s−b, Y1≤m, . . . , Yn−1≤m}fBb

p^∞

n1 mins,b^u

bm1

Vn−1, s−b, mfBb.

3.17

Thus, the proof is completed.

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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 levelcumulative 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.

Acknowledgment

The author thanks referees for their very useful comments and suggestions, which improved the paper.

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2. Cumulative Shock Model

Research Article

Life Behavior of a System under Discrete Shock Model

Serkan Eryilmaz

1. Introduction

2. Cumulative Shock Model

3. Mixed Shock Model

4. Summary and Conclusions

Acknowledgment

References

Submit your manuscripts at http://www.hindawi.com

Mathematics

Complex Analysis

Optimization

Combinatorics

Function Spaces

The Scientific World Journal

Discrete Mathematics

Stochastic Analysis