Life Behavior of a System under Discrete Shock Model

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.

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 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 𝑌_𝑗=𝐵^𝑗0,,𝐼𝐼_𝑗𝑗=1=0,(1.1) 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: 𝑊_𝑘=min𝑛∶^𝑛_𝑗=1𝑌_𝑗>𝑘,(1.2) 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. 𝑍_𝑘,𝑚=min𝑊_𝑘,𝑇_𝑚,(1.3) where 𝑇_𝑚=min𝑛∶𝑀_𝑛>𝑚,(1.4) where 𝑀_𝑛=max(𝑌₁,…,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.

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𝑊_𝑘>𝑛=𝑃{𝑆(𝑛)≤𝑘}=𝐹^𝑌∗𝑛(𝑘),(2.1) where 𝑆(𝑛)=𝑌₁+⋯+𝑌_𝑛, and 𝐹^𝑌∗𝑛 denotes the 𝑛-fold convolution of 𝐹_𝑌 with itself, 𝐹_𝑌(𝑥)=𝑃{𝑌≤𝑥}. By conditioning on the claim occurrence, one obtains 𝐹_𝑌(𝑥)=1−𝑝+𝑝𝐹_𝐵(𝑥),(2.2) where 𝐹_𝐵(𝑥)=𝑃{𝐵≤𝑥}.

Theorem 2.1. For 𝑛≥1, 𝑃𝑊_𝑘>𝑛=𝑝^{min(𝑘,𝑏𝑢)}_𝑏=1𝑃𝑊_𝑘−𝑏>𝑛−1𝑓_𝐵(𝑏)+(1−𝑝)𝑃𝑊_𝑘>𝑛−1,(2.3) and 𝑃{𝑊_𝑘>0}=1, where 𝑏^𝑢 is the endpoint of the support of 𝑓_𝐵.

Proof. From (2.1), it follows that 𝑃𝑊_𝑘>𝑛=𝐹_𝑌∗𝐹^{𝑌∗𝑛−1}(𝑘).(2.4)
Thus, the proof is immediate from (2.2).

Proposition 2.2. For 𝑘>0, the MTTF of the system can be computed from 𝐸𝑊_𝑘=1𝑝+^{min(𝑘,𝑏𝑢)}_𝑏=1𝐸𝑊_𝑘−𝑏𝑓_𝐵(𝑏),(2.5) with 𝐸(𝑊₀)=1/𝑝.

Proof. Using (2.1), 𝐸𝑊_𝑘=^∞_𝑛=0𝑃𝑊_𝑘>𝑛=^∞_𝑛=0𝑃{𝑆(𝑛)≤𝑘}=^∞_𝑛=0𝐹^𝑌∗𝑛(𝑘)=1+𝐹_𝑌∗𝐸𝑊_𝑘.(2.6)
Thus, the proof follows from (2.2) since 𝐸𝑊_𝑘=1+(1−𝑝)𝐸𝑊_𝑘+𝑝𝐹_𝐵∗𝐸𝑊_𝑘.(2.7)

Example 2.3. Let 𝐵 have a geometric distribution with pmf 𝑓_𝐵(𝑏)=(1−𝛼)𝛼^𝑏−1,𝑏=1,2,…. Then under the conditions of Proposition 2.2, 𝐸𝑊_𝑘=1𝑝+(1−𝛼)^𝑘_𝑏=1𝛼^𝑏−1𝐸𝑊_𝑘−𝑏,(2.8) with 𝐸(𝑊₀)=1/𝑝.

2.1. Related Characteristics

For 𝑘>0, we define new random variables as follows: 𝑁𝑊_𝑘=^𝑊𝑘_𝑗=1𝑆𝑊_𝑘𝐼_𝑗,=^𝑊𝑘_𝑗=1𝑌_𝑗=^{𝑁(𝑊𝑘)}_𝑗=1𝐵_𝑗.(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, 𝑃𝑁𝑊_𝑘=𝑚=^∞_𝑛_=𝑚𝑄(𝑛,𝑚,𝑘),(2.10) 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 𝑃(𝑛,𝑚,𝑘)=𝑝𝑃(𝑛−1,𝑚−1,𝑘)+(1−𝑝)𝑃(𝑛−1,𝑚,𝑘),(2.11) for 𝑛≥𝑚 and 𝑃(𝑛,𝑚,𝑘)=0 for 𝑛<𝑚, and 𝑅(𝑛,𝑚,𝑘)=𝑝^{min(𝑘,𝑏𝑢)}_𝑏=1𝑅(𝑛−1,𝑚−1,𝑘−𝑏)𝑓_𝐵(𝑏)+(1−𝑝)𝑅(𝑛−1,𝑚,𝑘),(2.12) for 𝑛≥𝑚 and 𝑅(𝑛,𝑚,𝑘)=0 for 𝑛<𝑚.

Proof. By conditioning on 𝑊_𝑘, 𝑃𝑁𝑊_𝑘=𝑚=^∞_𝑛_=𝑚𝑃𝑁(𝑛)=𝑚,𝑊_𝑘=𝑛.(2.13)
The probability 𝑄(𝑛,𝑚,𝑘)=𝑃{𝑁(𝑛)=𝑚,𝑊_𝑘=𝑛} can be written as follows: 𝑄(𝑛,𝑚,𝑘)=𝑃𝑁(𝑛)=𝑚,𝑊_𝑘>𝑛−1−𝑃𝑁(𝑛)=𝑚,𝑊_𝑘>𝑛.(2.14) Thus, we need to get recurrences for 𝑃(𝑛,𝑚,𝑘)=𝑃{𝑁(𝑛)=𝑚,𝑊_𝑘>𝑛−1} and 𝑅(𝑛,𝑚,𝑘)=𝑃{𝑁(𝑛)=𝑚,𝑊_𝑘>𝑛}. By conditioning on the values of 𝐼_𝑛, 𝑃(𝑛,𝑚,𝑘)==𝑃𝑃𝑁_𝑛(𝑛)=𝑚,𝑊_𝑘>𝑛−1_𝑗=1𝐼_𝑗=𝑚,^𝑛−1_𝑗=1𝑌_𝑗≤𝑘=𝑝𝑃^𝑛−1_𝑗=1𝐼_𝑗=𝑚−1,^𝑛−1_𝑗=1+(1−𝑝)𝑃_𝑛−1𝑌_𝑗≤𝑘_𝑗=1𝐼_𝑗=𝑚,^𝑛−1_𝑗=1=𝑝𝑃(𝑛−1,𝑚−1,𝑘)+(1−𝑝𝑌)_𝑗𝑃≤(𝑛𝑘−1,𝑚,𝑘).(2.15) On the other hand, 𝑅(𝑛,𝑚,𝑘)=𝑃𝑁(𝑛)=𝑚,𝑊_𝑘>𝑛=𝑃^𝑛_𝑗=1𝐼_𝑗=𝑚,^𝑛_𝑗=1=𝑃_𝑛𝑌_𝑗≤𝑘_𝑗=1𝐼_𝑗=𝑚,^𝑛_𝑗=1+𝑃_𝑛𝐼_𝑗𝐵_𝑗≤𝑘,𝐼_𝑛=1_𝑗=1𝐼_𝑗=𝑚,^𝑛_𝑗=1=𝑃_𝑛−1𝐼_𝑗𝐵_𝑗≤𝑘,𝐼_𝑛=0_𝑗=1𝐼_𝑗=𝑚−1,^𝑛−1_𝑗=1𝐼_𝑗𝐵_𝑗≤𝑘−𝐵_𝑛𝑃𝐼^𝑛+𝑃_𝑛−1=1_𝑗=1𝐼_𝑗=𝑚,^𝑛−1_𝑗=1=𝑝_{min(𝑘,𝑏𝑢)}𝐼_𝑗𝐵_𝑗≤𝑘𝑃𝐼_𝑛=0_𝑏=1+(1−𝑝𝑃)𝑃𝑁𝑁(𝑛(𝑛−−1)1=)=𝑚𝑚−,𝑊1,_𝑘𝑊_𝑘−𝑏>𝑛−1𝑓_𝐵(𝑏)>𝑛−1,(2.16) 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=∑^𝑛𝑗=1𝑌_𝑗 denotes the total shock up to time 𝑛 and 𝑃{𝑆(𝑛)=𝑠}=𝑝^{min(𝑠,𝑏𝑢)}_𝑏=1𝑃{𝑆(𝑛−1)=𝑠−𝑏}𝑓_𝐵(𝑏)+(1−𝑝)𝑃{𝑆(𝑛−1)=𝑠},(2.17) for 0<𝑠≤𝑛 and 𝑃{𝑆(𝑛)=0}=(1−𝑝)^𝑛.

Theorem 2.5. For 𝑠>𝑘, 𝑃𝑆𝑊_𝑘=𝑠=^∞_𝑛=1𝑄^∗(𝑛,𝑠,𝑘),(2.18) where 𝑄^∗(𝑛,𝑠,𝑘)=𝑝^{min(𝑠,𝑏𝑢)}_𝑏=_𝑠−𝑘𝑃{𝑆(𝑛−1)=𝑠−𝑏}𝑓_𝐵(𝑏).(2.19)

Proof. By the definition of 𝑆(𝑊_𝑘), 𝑃𝑆𝑊_𝑘=𝑠=^∞_𝑛=1𝑃𝑆(𝑛)=𝑠,𝑊_𝑘=𝑛.(2.20) For 𝑠>𝑘, 𝑄^∗(𝑛,𝑠,𝑘)====𝑃𝑃𝑃𝑃{{𝑆𝑆𝑆𝑆((((𝑛𝑛𝑛𝑛)))−===1𝑠𝑠𝑠),,,+𝑆𝑆𝑊𝑌((_𝑘𝑛𝑛_𝑛=−−𝑛11))≤≤𝑘𝑘,}𝑆(𝑛)>𝑘}=𝑝𝑃𝑆(𝑛−1)=𝑠=−𝑠𝐵,_𝑛𝑆(𝑛−1)≤𝑘+(1−𝑝)𝑃{𝑆(𝑛−1)=,𝑠𝑆,𝑆(𝑛(−𝑛−1)1≤)≤𝑘𝑘}.(2.21) The proof follows by conditioning on 𝐵_𝑛 and noting that 𝑃{𝑆(𝑛−1)=𝑠,𝑆(𝑛−1)≤𝑘}=0 for 𝑠>𝑘.

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, 𝐸𝐸𝑆𝑁𝑊_𝑘𝑊_𝑘=𝑝𝐸𝑊_𝑘,=𝑝𝐸𝑊_𝑘𝐸(𝐵).(2.22)

In Table 1 we compute 𝑀𝑇𝑇𝐹=𝐸(𝑊_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 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.

Table 1 

𝐸(𝑊_𝑘),𝐸(𝑁(𝑊_𝑘)), and 𝐸(𝑆(𝑊_𝑘)) for geometric shock size distribution.

3. 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.

Theorem 3.1. For 𝑘>𝑚≥1 and 𝑛≥1, 𝑃𝑍_𝑘,𝑚>𝑛=𝑝^{min(𝑚,𝑏𝑢)}_𝑏=1𝑃𝑍_{𝑘−𝑏,𝑚}>𝑛−1𝑓_𝐵(𝑏)+(1−𝑝)𝑃𝑍_𝑘,𝑚>𝑛−1,(3.1) and 𝑃{𝑍_𝑘,𝑚>0}=1, where ∑^{𝑦𝑏=𝑥}=0 if 𝑥>𝑦.

Proof. For 𝑛≥1, 𝑃𝑍_𝑘,𝑚>𝑛=𝑃𝑊_𝑘>𝑛,𝑇_𝑚>𝑛=𝑃^𝑛_𝑗=1𝑌_𝑗≤𝑘,𝑌₁≤𝑚,…,𝑌_𝑛≤𝑚.(3.2)
By conditioning on the values of 𝐼_𝑛, 𝑃𝑍_𝑘,𝑚>𝑛=𝑃^𝑛_𝑗=1+𝑃𝐼_𝑗𝑛𝐵_𝑗≤𝑘,𝑌₁≤𝑚,…,𝑌_𝑛≤𝑚,𝐼_𝑛=1_𝑗=1=𝑃_𝑛−1𝐼_𝑗𝐵_𝑗≤𝑘,𝑌₁≤𝑚,…,𝑌_𝑛≤𝑚,𝐼_𝑛=0_𝑗=1+𝑃𝐼_𝑗𝑛−1𝐵_𝑗≤𝑘−𝐵_𝑛,𝑌₁≤𝑚,…,𝑌_𝑛−1≤𝑚,𝐵_𝑛≤𝑚𝑃𝐼_𝑛=1_𝑗=1𝐼_𝑗𝐵_𝑗≤𝑘,𝑌₁≤𝑚,…,𝑌_𝑛−1≤𝑚𝑃𝐼_𝑛=0.(3.3) By conditioning on 𝐵_𝑛, 𝑃𝑍_𝑘,𝑚>𝑛=𝑝^{min(𝑘,𝑚,𝑏𝑢)}_𝑏=1𝑃^𝑛−1_𝑗=1+(1−𝑝)𝑃_𝑛−1𝐼_𝑗𝐵_𝑗≤𝑘−𝑏,𝑌₁≤𝑚,…,𝑌_𝑛−1≤𝑚𝑓_𝐵𝑛(𝑏)_𝑗=1𝐼_𝑗𝐵_𝑗≤𝑘,𝑌₁≤𝑚,…,𝑌_𝑛−1≤𝑚.(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 𝐸𝑍_𝑘,𝑚=1𝑝+^{min(𝑚,𝑏𝑢)}_𝑏=1𝐸𝑍_{𝑘−𝑏,𝑚}𝑓_𝐵(𝑏),(3.5) with 𝐸(𝑍_0,𝑚)=1/𝑝, where ∑^{𝑦𝑏=𝑥}=0 if 𝑥>𝑦.

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).

Table 2 

𝐸(𝑍_𝑘,𝑚) for geometric shock size distribution.

Theorem 3.3. For 𝑛≥1, 𝑃𝑁𝑍_𝑘,𝑚=𝑛=^∞_𝑠=𝑛[𝑝𝑈(𝑛−1,𝑠−1,𝑘,𝑚)+(1−𝑝)𝑈(𝑛,𝑠−1,𝑘,𝑚)−𝑈(𝑛,𝑠,𝑘,𝑚)],(3.6) where 𝑈(𝑛,𝑠,𝑘,𝑚)=𝑝^{min(𝑚,𝑏𝑢)}_𝑏=1𝑈(𝑛−1,𝑠−1,𝑘−𝑏,𝑚)𝑓_𝐵(𝑏)+(1−𝑝)𝑈(𝑛,𝑠−1,𝑘,𝑚).(3.7)

Proof. By conditioning on 𝑍_𝑘,𝑚, 𝑃𝑁𝑍_𝑘,𝑚=𝑛=^∞_𝑠=𝑛𝑃𝑁(𝑠)=𝑛,𝑍_𝑘,𝑚=𝑠.(3.8) It is clear that 𝑃𝑁(𝑠)=𝑛,𝑍_𝑘,𝑚=𝑠=𝑃𝑁(𝑠)=𝑛,𝑍_𝑘,𝑚>𝑠−1−𝑃𝑁(𝑠)=𝑛,𝑍_𝑘,𝑚>𝑠.(3.9) By the definition of 𝑍_𝑘,𝑚, 𝑈(=𝑛,𝑃𝑠,𝑘𝑁,𝑚()𝑠)=𝑛,𝑍_𝑘,𝑚=𝑃_𝑠>𝑠_𝑗=1𝐼_𝑗=𝑛,^𝑠_𝑗=1=𝑝𝑃_𝑠−1𝑌_𝑗≤𝑘,𝑌₁≤𝑚,…,𝑌_𝑠≤𝑚_𝑗=1𝐼_𝑗=𝑛−1,^𝑠−1_𝑗=1+(1−𝑝)𝑃_𝑠−1𝑌_𝑗≤𝑘−𝐵_𝑠,𝑌₁≤𝑚,…,𝑌_𝑠−1≤𝑚,𝐵_𝑠≤𝑚_𝑗=1𝐼_𝑗=𝑛,^𝑠−1_𝑗=1=𝑝_{min(𝑚,𝑏𝑢)}𝑌_𝑗≤𝑘,𝑌₁≤𝑚,…,𝑌_𝑠−1≤𝑚_𝑏=1𝑃^𝑠−1_𝑗=1𝐼_𝑗=𝑛−1,^𝑠−1_𝑗=1+(1−𝑝)𝑃_𝑠−1𝑌_𝑗≤𝑘−𝑏,𝑌₁≤𝑚,…,𝑌_𝑠−1≤𝑚𝑓_𝐵(𝑏)_𝑗=1𝐼_𝑗=𝑛,^𝑠−1_𝑗=1=𝑝_{min(𝑚,𝑏𝑢)}𝑌_𝑗≤𝑘,𝑌₁≤𝑚,…,𝑌_𝑠−1≤𝑚_𝑏=1+(1−𝑝)𝑃𝑃𝑁𝑁((𝑠𝑠−−11))==𝑛𝑛,−𝑍1,_𝑘,𝑚𝑍_{𝑘−𝑏,𝑚}>𝑠−1𝑓_𝐵(𝑏)>𝑠−1.(3.10) On the other hand, 𝑃=𝑁𝑃(𝑠)=_𝑠𝑛,𝑍_𝑘,𝑚>𝑠−1_𝑗=1𝐼_𝑗=𝑛,^𝑠−1_𝑗=1=𝑝𝑃𝑁(𝑠−1)=𝑛𝑌−_𝑗1≤,𝑍𝑘,𝑌_1𝑘,𝑚≤𝑚,…,𝑌_𝑠−1≤𝑚+(1−𝑝)𝑃𝑁(𝑠−1)=𝑛,𝑍>_𝑘,𝑚𝑠−1=𝑝𝑈(𝑛−1,𝑠−1,𝑘,𝑚)+(1−𝑝)𝑈>(𝑠𝑛,−𝑠1−1,𝑘,𝑚).(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: 𝑉(𝑛,𝑠,𝑚)==𝑝𝑃𝑆(𝑛)_{min(𝑚,𝑠,𝑏𝑢)}=𝑠,𝑌₁≤𝑚,…,𝑌_𝑛≤𝑚_𝑏=1𝑉(𝑛−1,𝑠−𝑏,𝑚)𝑓_𝐵(𝑏)+(1−𝑝)𝑉(𝑛−1,𝑠,𝑚).(3.12)

Theorem 3.4. For 𝑠>𝑘, 𝑃𝑆𝑍_𝑘,𝑚=𝑠=𝑝^∞_𝑏𝑛=1^𝑢_{𝑏=𝑠−𝑘}𝑉(𝑛−1,𝑠−𝑏,𝑚)𝑓_𝐵(𝑏),(3.13) for 𝑚<𝑠≤𝑘, 𝑃𝑆𝑍_𝑘,𝑚=𝑠=𝑝^∞_{𝑛=1min(𝑠,𝑏}^𝑢)_𝑏=𝑚+1𝑉(𝑛−1,𝑠−𝑏,𝑚)𝑓_𝐵(𝑏).(3.14)

Proof. By the definition of 𝑆(𝑍_𝑘,𝑚), 𝑃𝑆𝑍_𝑘,𝑚=𝑠==𝑃𝑃𝑆𝑆𝑊𝑊_𝑘_𝑘=𝑠,𝑊_𝑘≤𝑇_𝑚+𝑃𝑆𝑇_𝑚=𝑠,𝑇_𝑚<𝑊_𝑘𝑃𝑆𝑇=𝑠,𝑊_𝑘≤𝑇_𝑚,if𝑠>𝑘_𝑚=𝑠,𝑇_𝑚<𝑊_𝑘,if𝑚<𝑠≤𝑘.(3.15) For 𝑠>𝑘, 𝑃𝑆𝑊_𝑘=𝑠,𝑊_𝑘≤𝑇_𝑚=^∞_𝑛=1=𝑃𝑆(𝑛)=𝑠,𝑇_𝑚≥𝑛,𝑊_𝑘=𝑛_∞_𝑛=1=𝑃𝑆(𝑛)=𝑠,𝑌₁≤𝑚,…,𝑌_𝑛−1≤𝑚,𝑊_𝑘=𝑛_∞_𝑛=1=𝑃𝑆(𝑛)=𝑠,𝑌₁≤𝑚,…,𝑌_𝑛−1≤𝑚,𝑆(𝑛−1)≤𝑘_∞_𝑛=1=𝑝_∞𝑃𝑆(𝑛−1)+𝑌_𝑛=𝑠,𝑆(𝑛−1)≤𝑘,𝑌₁≤𝑚,…,𝑌_𝑛−1≤𝑚_𝑏𝑛=1^𝑢_{𝑏=𝑠−𝑘}=𝑝_∞𝑃𝑆(𝑛−1)=𝑠−𝑏,𝑌₁≤𝑚,…,𝑌_𝑛−1≤𝑚𝑓_𝐵(𝑏)_𝑏𝑛=1^𝑢_{𝑏=𝑠−𝑘}𝑉(𝑛−1,𝑠−𝑏,𝑚)𝑓_𝐵(𝑏).(3.16) Similarly, for 𝑚<𝑠≤𝑘, 𝑃=𝑆_∞𝑇_𝑚=𝑠,𝑇_𝑚<𝑊_𝑘_𝑛=1=𝑃𝑆(𝑛)=𝑠,𝑊_𝑘>𝑛,𝑇_𝑚=𝑛_∞_𝑛=1=𝑃𝑆(𝑛)=𝑠,𝑆(𝑛)≤𝑘,𝑌₁≤𝑚,…,𝑌_𝑛−1≤𝑚,𝑌_𝑛>𝑚_∞_𝑛=1=𝑝_∞𝑃𝑆(𝑛−1)+𝑌_𝑛=𝑠,𝑌₁≤𝑚,…,𝑌_𝑛−1≤𝑚,𝑌_𝑛>𝑚_{𝑛=1min(𝑠,𝑏}^𝑢)_𝑏=𝑚+1=𝑝_∞𝑃𝑆(𝑛−1)=𝑠−𝑏,𝑌₁≤𝑚,…,𝑌_𝑛−1≤𝑚𝑓_𝐵(𝑏)_{𝑛=1min(𝑠,𝑏}^𝑢)_𝑏=𝑚+1𝑉(𝑛−1,𝑠−𝑏,𝑚)𝑓_𝐵(𝑏).(3.17) 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 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.

Acknowledgment

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