# Random Walk Analysis in a Reliability System under Constant Degradation and Random Shocks

^{*}

^{†}

## Abstract

**:**

## 1. Introduction

#### 1.1. Background

#### 1.2. Pertinent Literature

#### 1.2.1. Shock Models

#### 1.2.2. Cumulative Shock Models

**Theorem**

**1.**

**Theorem**

**2**

**.**The functional ${\Phi}_{\mu}$ satisfies the formula

#### 1.2.3. Extreme Shock Models

#### 1.2.4. $\delta $-Shock Models

#### 1.2.5. Run Shock Models

#### 1.2.6. Mixed Shock Models

#### 1.2.7. Shock and Degradation Models

- (a)
- Soft failures are caused by continuous wear degradation and vocational non-fatal and extreme shocks that arrive in accordance with a marked Poisson process $\mathcal{S}={\sum}_{j=1}^{\infty}\left({X}_{j},{W}_{j}\right){\epsilon}_{{t}_{j}}$ of rate $\lambda $ for its support counting measure ${\sum}_{j=1}^{\infty}{\epsilon}_{{t}_{j}}$ with position independent marking and with marks ${X}_{j}$’s and ${W}_{j}$’s independent of each other. A soft failure occurs when the cumulative degradation process (due to wear and periodic shocks ${X}_{j}$’s) crosses a fixed threshold $H.$ Additionally, $N\left(t\right)={\sum}_{j=1}^{\infty}{\epsilon}_{{t}_{j}}\left[0,t\right]$;
- (b)
- Hard failures caused by extreme shocks ${W}_{j}$ (also referred to as shock loads) affect a different unit of the system and that causes a catastrophic failure. Only in this case does the system fail if any such shock exceeds some level D.

#### 1.3. Our Model

#### 1.3.1. Our Methodology

#### 1.3.2. Paper’s Layout

## 2. Fluctuation Analysis of the Linear Degradation Process with Shocks

**Theorem**

**3.**

**Proof.**

## 3. Results for a Dual-Exponential Shock Process

**Proposition**

**1.**

**Proof.**

**Corollary**

**1.**

**Proof.**

**Corollary**

**2.**

**Proof.**

**Proposition**

**2.**

**Proof.**

**Corollary**

**3.**

**Proof.**

**Corollary**

**4.**

**Proof.**

## 4. Comparison with Stochastic Simulation

## 5. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Abbreviations

General Acronyms | |

i.i.d. | independent and identically distributed |

LST | Laplace-Stieltjes transform |

probability distribution function | |

probability density function | |

PGF | probability-generating function |

r.v. | random variable |

Notation in the Literature Review | |

$\mathcal{S}$ | $={\sum}_{k=1}^{\infty}{X}_{n}{\epsilon}_{{t}_{n}}$ soft shocks marked point process in cumulative shock model |

$\mathcal{N}$ | $={\sum}_{k=1}^{\infty}{\epsilon}_{{t}_{n}}$ associated support counting measure of shocks’ arrivals |

$S\left(t\right)$ | $=\mathcal{S}[0,t]$ the number of shocks in time interval $[0,t]$ |

${\epsilon}_{a}$ | Dirac point mass (unity measure) |

${X}_{n}$ | magnitudes of soft shocks |

${t}_{n}$ | the time of the nth shock |

${\delta}_{n}$ | $={t}_{n}-{t}_{n-1}$ |

${S}_{n}$ | $={\sum}_{k=1}^{n}{X}_{n}$ |

$Y\left(t\right)$ | pure degradation process |

${W}_{n}$ | magnitudes of hard shocks |

$\mathcal{W}$ | $={\sum}_{j=1}^{\infty}{W}_{j}{\epsilon}_{{t}_{j}}$ marked point measure of hard shocks |

$\delta $ | lower threshold in $\delta $-shock policy models |

x | failure threshold |

M | soft failure threshold |

D | hard failure threshold |

$\nu \left(x\right)$ | number of soft shocks until soft failure with respect to threshold x |

$\xi \left(x\right)$ | number of hard shocks until hard failure with respect to threshold x |

A | $=[0,M)$ the set that process $\mathcal{S}$ escapes |

Notation in Our Model | |

$\nu $ | number of the first shock where degradation exceeds M |

${t}_{\nu -1}$ | pre-failure time |

${t}_{\nu}$ | soft failure time (the escape from set A) |

${S}_{\nu -1}$ | pre-failure cumulative damage |

${S}_{\nu}$ | cumulative damage to the system at the failure |

${\Phi}_{\nu}$ | $=E{\xi}^{\nu}{u}^{{S}_{\nu -1}}{v}^{{S}_{\nu}}{e}^{-\vartheta {t}_{\nu -1}-\theta {t}_{\nu}}$ |

${\mathcal{D}}^{k}$ | $\mathcal{D}$ operator |

${\gamma}_{0}(z,\theta )$ | $=E{z}^{{X}_{0}}{e}^{-{t}_{0}\theta}$ |

$\gamma (z,\theta )$ | $=E{z}^{{X}_{1}}{e}^{({t}_{1}-{t}_{0})\theta}$ |

${I}_{0}$ | modified Bessel function of order zero |

$\mathcal{A}$ | $={\sum}_{k=1}^{\infty}\left({X}_{k}+a{\delta}_{k}\right){\epsilon}_{{t}_{k}}$ soft shocks and degradation marked point process |

$\mathcal{A}[0,t]$ | $=A\left(t\right)$ cumulative degradation process including shocks |

${A}_{n}$ | $={\sum}_{k=1}^{n}{X}_{k}+a{\delta}_{k}$ |

${\delta}_{k}$ | $={t}_{k}-{t}_{k-1}$ |

$\nu $ | $=inf\{n\in \mathbb{N}:{A}_{n}\ge M\}$ |

M | failure threshold |

${\tau}_{\nu}$ | failure time if it occurs upon constant degradation |

${t}_{\nu}$ | failure time if it occurs upon a shock |

$\gamma (\alpha ,\theta )$ | $=E{e}^{-\alpha ({X}_{k}+a{\delta}_{k})-\theta {\delta}_{k}}$ |

${\Phi}_{\nu}$ | $=E{e}^{-\alpha {A}_{\nu -1}-\beta {S}_{\nu}-\vartheta {t}_{\nu -1}-\theta {t}_{\nu}}$ |

$\mathcal{L}$ | Laplace transform |

${\mathcal{L}}^{-1}$ | inverse Laplace transform |

p | failure threshold $(p>0)$ |

$\Phi \left(x\right)$ | standard Gaussian PDF |

$\left[Exp\right(\lambda \left)\right]$ | equivalence class of all exponential distributions with parameter $\lambda $ |

$\left[Geo\right(p\left(x\right)\left)\right]$ | equivalence classs of all geometric random variables with parameter $p\left(x\right)$ |

$\left[Ga\right(\alpha ,\beta \left)\right]$ | equivalence class of all gamma distributions with parameters $(\alpha ,\beta )$ |

$\left[N(\mu ,{\sigma}^{2})\right]$ | equivalence class of all Gaussian distributions with parameters $(\mu ,{\sigma}^{2})$ |

## Appendix A

#### Appendix A.1. Simulation Code

**Figure A1.**The simulatePath function accepts numerical inputs for each parameter of the process: a, $\lambda $, $\mu $, and M. It then simulates one path of the process and returns sampled values of $\nu $, ${A}_{\nu -1}$, ${S}_{\nu}$, ${t}_{\nu -1}$ and ${\tau}_{\nu}$ along with a flag indicating whether a degradation failure or shock failure occurred.

`import numpy as`^{~}np`def simulatePath(a, lam, mu, M):``# initialize outputs``failureTime = 0``failureDamage = 0``failureIndex = 0``degradationFailure = False``# simulate the process``while failureDamage < M:``# save A_j-1``oldDamage = failureDamage``# save t_j-1``oldTime = failureTime``# compute waiting time before the next shock``waitingTime = np.random.exponential(1/mu)``# add degradation between shocks``failureDamage += a ∗ waitingTime``# if degradation causes damage to reach M...``if failureDamage >= M:``# compute tau_nu``failureTime += (M - oldDamage)/a``# set S_nu (total damage) to M``failureDamage = M``# mark degradation as the cause of the failure``degradationFailure = True``# exit the loop``break``# else, add the shock``else:``# add the waiting time``failureTime += waitingTime``# add the shock damage``failureDamage += np.random.exponential(1/lam)``# add 1 to the shock counter``failureIndex += 1``# gather the output values into a tuple``outputs = (failureIndex, oldDamage, failureDamage,``oldTime, failureTime, degradationFailure)``# return values nu, A_nu-1, S_nu, t_nu-1, tau_nu, flag for``# failure type``return outputs`

`simulatePath(a, lam, mu, M)`

`def degradationProbability(a, lam, mu, M):``term = a ∗ lam + mu ∗ np.exp(-(lam + mu / a) ∗ M)``return term / (a ∗ lam + mu)``def failureTimeMean(a, lam, mu, M):``term = lam ∗ M + 1 - degradationProbability(a, lam, mu, M)``return term / (a ∗ lam + mu)``def failureTimeVariance(a, lam, mu, M):``dTerm = a ∗ lam + mu``eTerm = np.exp(-(lam + mu / a) ∗ M)``term1 = 2 ∗ lam ∗ mu ∗ M / dTerm ∗∗ 3``term2 = mu ∗ (mu - 4 ∗ a ∗ lam) / dTerm ∗∗ 4``term3 = 2 ∗ mu``term3 ∗= 2 ∗ a ∗∗ 2 ∗ lam + (a ∗ lam) ∗∗ 2 ∗ M - 2 ∗ mu ∗∗ 2 ∗ M``term3 ∗= eTerm / (a ∗ dTerm ∗∗ 4)``term4 = 2 ∗ mu ∗∗ 2 ∗ eTerm ∗∗ 2 / dTerm ∗∗ 4``return term1 + term2 + term3 - term4``def failureDamageMean(a, lam, mu, M):``return M + (1 - degradationProbability(a, lam, mu, M)) / lam``def failureDamageVariance(a, lam, mu, M):``dTerm = a ∗ lam + mu``eTerm = np.exp(-(lam + mu / a) ∗ M)``term1 = 2 ∗ a ∗ mu ∗ (1 - eTerm) / (lam ∗ dTerm ∗∗ 2)``term2 = mu ∗∗ 2 ∗ (1 - eTerm ∗∗ 2) / (lam ∗ dTerm) ∗∗ 2``return term1 + term2`

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**Figure 2.**Predicted and empirical probabilities of degradation failures with limiting probabilities.

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**MDPI and ACS Style**

Dshalalow, J.H.; White, R.T.
Random Walk Analysis in a Reliability System under Constant Degradation and Random Shocks. *Axioms* **2021**, *10*, 199.
https://doi.org/10.3390/axioms10030199

**AMA Style**

Dshalalow JH, White RT.
Random Walk Analysis in a Reliability System under Constant Degradation and Random Shocks. *Axioms*. 2021; 10(3):199.
https://doi.org/10.3390/axioms10030199

**Chicago/Turabian Style**

Dshalalow, Jewgeni H., and Ryan T. White.
2021. "Random Walk Analysis in a Reliability System under Constant Degradation and Random Shocks" *Axioms* 10, no. 3: 199.
https://doi.org/10.3390/axioms10030199