Life Behavior of a System under Discrete Shock Model

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 probability p 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


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 probability p 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 B j .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 I j be a binary random variable representing the shock occurrences that is, I j 1 if a shock occurs in period j and I j 0, otherwise.For j ≥ 1, define where the random variables I j and B j 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 function p.m.f.f B .Thus, under the cumulative shock model, the failure time of the system can be defined by the following waiting time random variable: 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 where where M n max Y 1 , . . ., Y n 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 variable W k represents the waiting time until the cumulative sum of claim amounts exceeds the level k.Similarly, the random variable T m is the waiting time until the first extreme claim size falls above the level m.
The present paper is organized as follows.In Section 2, 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 W k and S W k 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.

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 3 , where b u is the endpoint of the support of f B .
Proof.From 2.1 , it follows that Thus, the proof is immediate from 2.2 .
Proposition 2.2.For k > 0, the MTTF of the system can be computed from 5 Thus, the proof follows from 2.2 since

Related Characteristics
For k > 0, we define new random variables as follows: 2.9 It is clear that the random variables N W k and S W k 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.For m ≥ 1, for n ≥ m and P n, m, k 0 for n < m, and Proof.By conditioning on W k ,

2.13
The probability Q n, m, k P {N n m, W k n} can be written as follows:

2.15
On the other hand, for n ≥ m.Thus, the proof is completed.
Before proceeding with the distribution of S W k , it should be noted that the random variable S n N n j 1 B j n j 1 Y j denotes the total shock up to time n and Theorem 2.5.For s > k, where

2.19
Proof.By the definition of S W k ,

2.21
The proof follows by conditioning on B n and noting that The following result readily follows from the definitions of N W k and S W k and Wald's equation.Proposition 2.6.For k > 0,  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 W k and expected total shock E S W k remain the same for different values of p.

Mixed Shock Model
For k ≤ 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.
Proof.For n ≥ 1, By conditioning on the values of I n ,

3.3
By conditioning on B n ,

3.4
Thus, the proof is completed.
The following result can be proved similar to Proposition 2.2, and hence its proof is omitted.

Proposition 3.2. For k > m ≥ 1, the MTTF of the system under mixed shock model can be computed from
In Table 2, 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 .
Theorem 3.3.For n ≥ 1, where Proof.By conditioning on Z k,m , It is clear that

3.10
On the other hand,

3.11
Thus the proof is completed.
Before the derivation of the distribution of S Z k,m , we note the following recursion which will be useful in the sequel:

3.14
Proof.By the definition of S Z k,m ,

3.15
For s > k,

3.17
Thus, the proof is completed.

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 is p, 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 probability p.As a future work, the case in which the shock occurrence indicators form a Markov chain can also be considered.

Table 1 :
E W k , E N W k , and E S W k for geometric shock size distribution.
k , E N W k , and E S W k whenever the shock size random variable B has a geometric distribution with mean E B 1/ 1 − α .From Table

Table 2 :
E Z k,m for geometric shock size distribution.