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Discrete Dynamics in Nature and Society
Volume 2012 (2012), Article ID 238641, 12 pages
http://dx.doi.org/10.1155/2012/238641
Research Article

Life Behavior of a System under Discrete Shock Model

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

Received 21 June 2012; Revised 31 July 2012; Accepted 6 August 2012

Academic Editor: M. De la Sen

Copyright © 2012 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.

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, [19] 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 𝑗1, define 𝑌𝑗=𝐵𝑗,𝐼𝑗=10,𝐼𝑗=0,(1.1) where the random variables 𝐼𝑗 and 𝐵𝑗 are independent in each time period. The random variable 𝐵𝑗 is strictly positive, and {𝐵𝑗,𝑗1} 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: 𝑊𝑘=min𝑛𝑛𝑗=1𝑌𝑗,>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. 𝑍𝑘,𝑚𝑊=min𝑘,𝑇𝑚,(1.3) where 𝑇𝑚=min𝑛𝑀𝑛>𝑚,(1.4) 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 variable 𝑊𝑘 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 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(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 𝑃𝑊𝑘>𝑛=𝑃{𝑆(𝑛)𝑘}=𝐹𝑌𝑛(𝑘),(2.1) where 𝑆(𝑛)=𝑌1++𝑌𝑛, and 𝐹𝑌𝑛 denotes the 𝑛-fold convolution of 𝐹𝑌 with itself, 𝐹𝑌(𝑥)=𝑃{𝑌𝑥}. By conditioning on the claim occurrence, one obtains 𝐹𝑌(𝑥)=1𝑝+𝑝𝐹𝐵(𝑥),(2.2) where 𝐹𝐵(𝑥)=𝑃{𝐵𝑥}.

Theorem 2.1. For 𝑛1, 𝑃𝑊𝑘>𝑛=𝑝min(𝑘,𝑏𝑢)𝑏=1𝑃𝑊𝑘𝑏𝑓>𝑛1𝐵𝑊(𝑏)+(1𝑝)𝑃𝑘,>𝑛1(2.3) and 𝑃{𝑊𝑘>0}=1, where 𝑏𝑢 is the endpoint of the support of 𝑓𝐵.

Proof. From (2.1), it follows that 𝑃𝑊𝑘>𝑛=𝐹𝑌𝐹𝑌𝑛1(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 𝐸𝑊𝑘=1𝑝+min(𝑘,𝑏𝑢)𝑏=1𝐸𝑊𝑘𝑏𝑓𝐵(𝑏),(2.5) with 𝐸(𝑊0)=1/𝑝.

Proof. Using (2.1), 𝐸𝑊𝑘=𝑛=0𝑃𝑊𝑘=>𝑛𝑛=0𝑃{𝑆(𝑛)𝑘}=𝑛=0𝐹𝑌𝑛(𝑘)=1+𝐹𝑌𝑊𝐸𝑘.(2.6)
Thus, the proof follows from (2.2) since 𝐸𝑊𝑘𝑊=1+(1𝑝)𝐸𝑘+𝑝𝐹𝐵𝑊𝐸𝑘.(2.7)

Example 2.3. Let 𝐵 have a geometric distribution with pmf 𝑓𝐵(𝑏)=(1𝛼)𝛼𝑏1,𝑏=1,2,. Then under the conditions of Proposition 2.2, 𝐸𝑊𝑘=1𝑝+(1𝛼)𝑘𝑏=1𝛼𝑏1𝐸𝑊𝑘𝑏,(2.8) with 𝐸(𝑊0)=1/𝑝.

2.1. Related Characteristics

For 𝑘>0, we define new random variables as follows: 𝑁𝑊𝑘=𝑊𝑘𝑗=1𝐼𝑗,𝑆𝑊𝑘=𝑊𝑘𝑗=1𝑌𝑗=𝑁(𝑊𝑘)𝑗=1𝐵𝑗.(2.9)

It is clear that the random variables 𝑁(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. For 𝑚1, 𝑃𝑁𝑊𝑘==𝑚𝑛=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(𝑛1,𝑚1,𝑘)+(1𝑝)𝑃(𝑛1,𝑚,𝑘),(2.11) for 𝑛𝑚 and 𝑃(𝑛,𝑚,𝑘)=0 for 𝑛<𝑚, and 𝑅(𝑛,𝑚,𝑘)=𝑝min(𝑘,𝑏𝑢)𝑏=1𝑅(𝑛1,𝑚1,𝑘𝑏)𝑓𝐵(𝑏)+(1𝑝)𝑅(𝑛1,𝑚,𝑘),(2.12) for 𝑛𝑚 and 𝑅(𝑛,𝑚,𝑘)=0 for 𝑛<𝑚.

Proof. By conditioning on 𝑊𝑘, 𝑃𝑁𝑊𝑘==𝑚𝑛=𝑚𝑃𝑁(𝑛)=𝑚,𝑊𝑘.=𝑛(2.13)
The probability 𝑄(𝑛,𝑚,𝑘)=𝑃{𝑁(𝑛)=𝑚,𝑊𝑘=𝑛} can be written as follows: 𝑄𝑁(𝑛,𝑚,𝑘)=𝑃(𝑛)=𝑚,𝑊𝑘𝑁>𝑛1𝑃(𝑛)=𝑚,𝑊𝑘>𝑛.(2.14) Thus, we need to get recurrences for 𝑃(𝑛,𝑚,𝑘)=𝑃{𝑁(𝑛)=𝑚,𝑊𝑘>𝑛1} and 𝑅(𝑛,𝑚,𝑘)=𝑃{𝑁(𝑛)=𝑚,𝑊𝑘>𝑛}. By conditioning on the values of 𝐼𝑛, 𝑃𝑁(𝑛,𝑚,𝑘)=𝑃(𝑛)=𝑚,𝑊𝑘>𝑛1=𝑃𝑛𝑗=1𝐼𝑗=𝑚,𝑛1𝑗=1𝑌𝑗𝑘=𝑝𝑃𝑛1𝑗=1𝐼𝑗=𝑚1,𝑛1𝑗=1𝑌𝑗𝑘+(1𝑝)𝑃𝑛1𝑗=1𝐼𝑗=𝑚,𝑛1𝑗=1𝑌𝑗𝑘=𝑝𝑃(𝑛1,𝑚1,𝑘)+(1𝑝)𝑃(𝑛1,𝑚,𝑘).(2.15) On the other hand, 𝑅(𝑛,𝑚,𝑘)=𝑃𝑁(𝑛)=𝑚,𝑊𝑘>𝑛=𝑃𝑛𝑗=1𝐼𝑗=𝑚,𝑛𝑗=1𝑌𝑗𝑘=𝑃𝑛𝑗=1𝐼𝑗=𝑚,𝑛𝑗=1𝐼𝑗𝐵𝑗𝑘,𝐼𝑛=1+𝑃𝑛𝑗=1𝐼𝑗=𝑚,𝑛𝑗=1𝐼𝑗𝐵𝑗𝑘,𝐼𝑛=0=𝑃𝑛1𝑗=1𝐼𝑗=𝑚1,𝑛1𝑗=1𝐼𝑗𝐵𝑗𝑘𝐵𝑛𝑃𝐼𝑛=1+𝑃𝑛1𝑗=1𝐼𝑗=𝑚,𝑛1𝑗=1𝐼𝑗𝐵𝑗𝑃𝐼𝑘𝑛=0=𝑝min(𝑘,𝑏𝑢)𝑏=1𝑃𝑁(𝑛1)=𝑚1,𝑊𝑘𝑏𝑓>𝑛1𝐵(𝑏)+(1𝑝)𝑃𝑁(𝑛1)=𝑚,𝑊𝑘,>𝑛1(2.16) for 𝑛𝑚. Thus, the proof is completed.

Before proceeding with the distribution of 𝑆(𝑊𝑘), it should be noted that the random variable 𝑆(𝑛)=𝑁(𝑛)𝑗=1𝐵𝑗=𝑛𝑗=1𝑌𝑗 denotes the total shock up to time 𝑛 and 𝑃{𝑆(𝑛)=𝑠}=𝑝min(𝑠,𝑏𝑢)𝑏=1𝑃{𝑆(𝑛1)=𝑠𝑏}𝑓𝐵(𝑏)+(1𝑝)𝑃{𝑆(𝑛1)=𝑠},(2.17) for 0<𝑠𝑛 and 𝑃{𝑆(𝑛)=0}=(1𝑝)𝑛.

Theorem 2.5. For 𝑠>𝑘, 𝑃𝑆𝑊𝑘==𝑠𝑛=1𝑄(𝑛,𝑠,𝑘),(2.18) where 𝑄(𝑛,𝑠,𝑘)=𝑝min(𝑠,𝑏𝑢)𝑏=𝑠𝑘𝑃{𝑆(𝑛1)=𝑠𝑏}𝑓𝐵(𝑏).(2.19)

Proof. By the definition of 𝑆(𝑊𝑘), 𝑃𝑆𝑊𝑘==𝑠𝑛=1𝑃𝑆(𝑛)=𝑠,𝑊𝑘.=𝑛(2.20) For 𝑠>𝑘, 𝑄𝑆(𝑛,𝑠,𝑘)=𝑃(𝑛)=𝑠,𝑊𝑘=𝑛=𝑃{𝑆(𝑛)=𝑠,𝑆(𝑛1)𝑘,𝑆(𝑛)>𝑘}=𝑃{𝑆(𝑛)=𝑠,𝑆(𝑛1)𝑘}=𝑃𝑆(𝑛1)+𝑌𝑛=𝑠,𝑆(𝑛1)𝑘=𝑝𝑃𝑆(𝑛1)=𝑠𝐵𝑛,𝑆(𝑛1)𝑘+(1𝑝)𝑃{𝑆(𝑛1)=𝑠,𝑆(𝑛1)𝑘}.(2.21) The proof follows by conditioning on 𝐵𝑛 and noting that 𝑃{𝑆(𝑛1)=𝑠,𝑆(𝑛1)𝑘}=0 for 𝑠>𝑘.

The following result readily follows from the definitions of 𝑁(𝑊𝑘) and 𝑆(𝑊𝑘) and Wald’s equation.

Proposition 2.6. For 𝑘>0, 𝐸𝑁𝑊𝑘𝑊=𝑝𝐸𝑘,𝐸𝑆𝑊𝑘𝑊=𝑝𝐸𝑘𝐸(𝐵).(2.22)

In Table 1 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/(1𝛼). 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.

tab1
Table 1: 𝐸(𝑊𝑘),𝐸(𝑁(𝑊𝑘)), and 𝐸(𝑆(𝑊𝑘)) for geometric shock size distribution.

3. Mixed Shock Model

For 𝑘𝑚, 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>𝑚1 and 𝑛1, 𝑃𝑍𝑘,𝑚>𝑛=𝑝min(𝑚,𝑏𝑢)𝑏=1𝑃𝑍𝑘𝑏,𝑚𝑓>𝑛1𝐵𝑍(𝑏)+(1𝑝)𝑃𝑘,𝑚,>𝑛1(3.1) and 𝑃{𝑍𝑘,𝑚>0}=1, where 𝑦𝑏=𝑥=0 if 𝑥>𝑦.

Proof. For 𝑛1, 𝑃𝑍𝑘,𝑚𝑊>𝑛=𝑃𝑘>𝑛,𝑇𝑚>𝑛=𝑃𝑛𝑗=1𝑌𝑗𝑘,𝑌1𝑚,,𝑌𝑛.𝑚(3.2)
By conditioning on the values of 𝐼𝑛, 𝑃𝑍𝑘,𝑚>𝑛=𝑃𝑛𝑗=1𝐼𝑗𝐵𝑗𝑘,𝑌1𝑚,,𝑌𝑛𝑚,𝐼𝑛=1+𝑃𝑛𝑗=1𝐼𝑗𝐵𝑗𝑘,𝑌1𝑚,,𝑌𝑛𝑚,𝐼𝑛=0=𝑃𝑛1𝑗=1𝐼𝑗𝐵𝑗𝑘𝐵𝑛,𝑌1𝑚,,𝑌𝑛1𝑚,𝐵𝑛𝑃𝐼𝑚𝑛=1+𝑃𝑛1𝑗=1𝐼𝑗𝐵𝑗𝑘,𝑌1𝑚,,𝑌𝑛1𝑃𝐼𝑚𝑛.=0(3.3) By conditioning on 𝐵𝑛, 𝑃𝑍𝑘,𝑚>𝑛=𝑝min(𝑘,𝑚,𝑏𝑢)𝑏=1𝑃𝑛1𝑗=1𝐼𝑗𝐵𝑗𝑘𝑏,𝑌1𝑚,,𝑌𝑛1𝑓𝑚𝐵𝑛(𝑏)+(1𝑝)𝑃𝑛1𝑗=1𝐼𝑗𝐵𝑗𝑘,𝑌1𝑚,,𝑌𝑛1.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>𝑚1, the MTTF of the system under mixed shock model can be computed from 𝐸𝑍𝑘,𝑚=1𝑝+min(𝑚,𝑏𝑢)𝑏=1𝐸𝑍𝑘𝑏,𝑚𝑓𝐵(𝑏),(3.5) with 𝐸(𝑍0,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/(1𝛼).

tab2
Table 2: 𝐸(𝑍𝑘,𝑚) for geometric shock size distribution.

Theorem 3.3. For 𝑛1, 𝑃𝑁𝑍𝑘,𝑚==𝑛𝑠=𝑛[],𝑝𝑈(𝑛1,𝑠1,𝑘,𝑚)+(1𝑝)𝑈(𝑛,𝑠1,𝑘,𝑚)𝑈(𝑛,𝑠,𝑘,𝑚)(3.6) where 𝑈(𝑛,𝑠,𝑘,𝑚)=𝑝min(𝑚,𝑏𝑢)𝑏=1𝑈(𝑛1,𝑠1,𝑘𝑏,𝑚)𝑓𝐵(𝑏)+(1𝑝)𝑈(𝑛,𝑠1,𝑘,𝑚).(3.7)

Proof. By conditioning on 𝑍𝑘,𝑚, 𝑃𝑁𝑍𝑘,𝑚==𝑛𝑠=𝑛𝑃𝑁(𝑠)=𝑛,𝑍𝑘,𝑚.=𝑠(3.8) It is clear that 𝑃𝑁(𝑠)=𝑛,𝑍𝑘,𝑚𝑁=𝑠=𝑃(𝑠)=𝑛,𝑍𝑘,𝑚𝑁>𝑠1𝑃(𝑠)=𝑛,𝑍𝑘,𝑚>𝑠.(3.9) By the definition of 𝑍𝑘,𝑚, 𝑈(𝑛,𝑠,𝑘,𝑚)=𝑃𝑁(𝑠)=𝑛,𝑍𝑘,𝑚>𝑠=𝑃𝑠𝑗=1𝐼𝑗=𝑛,𝑠𝑗=1𝑌𝑗𝑘,𝑌1𝑚,,𝑌𝑠𝑚=𝑝𝑃𝑠1𝑗=1𝐼𝑗=𝑛1,𝑠1𝑗=1𝑌𝑗𝑘𝐵𝑠,𝑌1𝑚,,𝑌𝑠1𝑚,𝐵𝑠𝑚+(1𝑝)𝑃𝑠1𝑗=1𝐼𝑗=𝑛,𝑠1𝑗=1𝑌𝑗𝑘,𝑌1𝑚,,𝑌𝑠1𝑚=𝑝min(𝑚,𝑏𝑢)𝑏=1𝑃𝑠1𝑗=1𝐼𝑗=𝑛1,𝑠1𝑗=1𝑌𝑗𝑘𝑏,𝑌1𝑚,,𝑌𝑠1𝑓𝑚𝐵(𝑏)+(1𝑝)𝑃𝑠1𝑗=1𝐼𝑗=𝑛,𝑠1𝑗=1𝑌𝑗𝑘,𝑌1𝑚,,𝑌𝑠1𝑚=𝑝min(𝑚,𝑏𝑢)𝑏=1𝑃𝑁(𝑠1)=𝑛1,𝑍𝑘𝑏,𝑚𝑓>𝑠1𝐵(𝑏)+(1𝑝)𝑃𝑁(𝑠1)=𝑛,𝑍𝑘,𝑚.>𝑠1(3.10) On the other hand, 𝑃𝑁(𝑠)=𝑛,𝑍𝑘,𝑚>𝑠1=𝑃𝑠𝑗=1𝐼𝑗=𝑛,𝑠1𝑗=1𝑌𝑗𝑘,𝑌1𝑚,,𝑌𝑠1𝑚=𝑝𝑃𝑁(𝑠1)=𝑛1,𝑍𝑘,𝑚>𝑠1+(1𝑝)𝑃𝑁(𝑠1)=𝑛,𝑍𝑘,𝑚>𝑠1=𝑝𝑈(𝑛1,𝑠1,𝑘,𝑚)+(1𝑝)𝑈(𝑛,𝑠1,𝑘,𝑚).(3.11) Thus the proof is completed.

Before the derivation of the distribution of 𝑆(𝑍𝑘,𝑚), we note the following recursion which will be useful in the sequel: 𝑉𝑆(𝑛,𝑠,𝑚)=𝑃(𝑛)=𝑠,𝑌1𝑚,,𝑌𝑛𝑚=𝑝min(𝑚,𝑠,𝑏𝑢)𝑏=1𝑉(𝑛1,𝑠𝑏,𝑚)𝑓𝐵(𝑏)+(1𝑝)𝑉(𝑛1,𝑠,𝑚).(3.12)

Theorem 3.4. For 𝑠>𝑘, 𝑃𝑆𝑍𝑘,𝑚=𝑠=𝑝𝑏𝑛=1𝑢𝑏=𝑠𝑘𝑉(𝑛1,𝑠𝑏,𝑚)𝑓𝐵(𝑏),(3.13) for 𝑚<𝑠𝑘, 𝑃𝑆𝑍𝑘,𝑚=𝑠=𝑝𝑛=1min(𝑠,𝑏𝑢)𝑏=𝑚+1𝑉(𝑛1,𝑠𝑏,𝑚)𝑓𝐵(𝑏).(3.14)

Proof. By the definition of 𝑆(𝑍𝑘,𝑚), 𝑃𝑆𝑍𝑘,𝑚𝑆𝑊=𝑠=𝑃𝑘=𝑠,𝑊𝑘𝑇𝑚𝑆𝑇+𝑃𝑚=𝑠,𝑇𝑚<𝑊𝑘=𝑃𝑆𝑊𝑘=𝑠,𝑊𝑘𝑇𝑚𝑃𝑆𝑇,if𝑠>𝑘𝑚=𝑠,𝑇𝑚<𝑊𝑘,if𝑚<𝑠𝑘.(3.15) For 𝑠>𝑘, 𝑃𝑆𝑊𝑘=𝑠,𝑊𝑘𝑇𝑚=𝑛=1𝑃𝑆(𝑛)=𝑠,𝑇𝑚𝑛,𝑊𝑘==𝑛𝑛=1𝑃𝑆(𝑛)=𝑠,𝑌1𝑚,,𝑌𝑛1𝑚,𝑊𝑘==𝑛𝑛=1𝑃𝑆(𝑛)=𝑠,𝑌1𝑚,,𝑌𝑛1=𝑚,𝑆(𝑛1)𝑘𝑛=1𝑃𝑆(𝑛1)+𝑌𝑛=𝑠,𝑆(𝑛1)𝑘,𝑌1𝑚,,𝑌𝑛1𝑚=𝑝𝑏𝑛=1𝑢𝑏=𝑠𝑘𝑃𝑆(𝑛1)=𝑠𝑏,𝑌1𝑚,,𝑌𝑛1𝑓𝑚𝐵(𝑏)=𝑝𝑏𝑛=1𝑢𝑏=𝑠𝑘𝑉(𝑛1,𝑠𝑏,𝑚)𝑓𝐵(𝑏).(3.16) Similarly, for 𝑚<𝑠𝑘, 𝑃𝑆𝑇𝑚=𝑠,𝑇𝑚<𝑊𝑘=𝑛=1𝑃𝑆(𝑛)=𝑠,𝑊𝑘>𝑛,𝑇𝑚==𝑛𝑛=1𝑃𝑆(𝑛)=𝑠,𝑆(𝑛)𝑘,𝑌1𝑚,,𝑌𝑛1𝑚,𝑌𝑛=>𝑚𝑛=1𝑃𝑆(𝑛1)+𝑌𝑛=𝑠,𝑌1𝑚,,𝑌𝑛1𝑚,𝑌𝑛>𝑚=𝑝𝑛=1min(𝑠,𝑏𝑢)𝑏=𝑚+1𝑃𝑆(𝑛1)=𝑠𝑏,𝑌1𝑚,,𝑌𝑛1𝑓𝑚𝐵(𝑏)=𝑝𝑛=1min(𝑠,𝑏𝑢)𝑏=𝑚+1𝑉(𝑛1,𝑠𝑏,𝑚)𝑓𝐵(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.

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