# Renyi Entropy of the Residual Lifetime of a Reliability System at the System Level

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## Abstract

**:**

## 1. Introduction

## 2. Renyi Entropy of the Residual Lifetime

**Theorem**

**1.**

**Proof.**

**Theorem**

**2.**

**Proof.**

**Example**

**1.**

- (i)
- Let X follow the uniform distribution in $[0,1].$ From Equation (11), we immediately obtain$${H}_{\alpha}\left({{T}_{t}}^{1,5}\right)=log(1-t)+\frac{1}{1-\alpha}log{\int}_{0}^{1}{g}_{V}^{\alpha}\left(u\right)du,\phantom{\rule{4pt}{0ex}}t>0.$$We see that ${H}_{\alpha}\left({{T}_{t}}^{1,5}\right)$ is decreasing in $t.$ We note that the uniform distribution has the IFR property, and therefore ${H}_{\alpha}\left({{T}_{t}}^{1,5}\right)$ decreases as t increases, as we expected based on Theorem 2.

- (ii)
- Think about a Pareto type II with the SF$$S\left(t\right)={(1+t)}^{-2},\phantom{\rule{4pt}{0ex}}t>0.$$It is not hard to see that$${H}_{\alpha}\left({{T}_{t}}^{1,5}\right)=log\left(\frac{1+t}{2}\right)+\frac{1}{1-\alpha}log{\int}_{0}^{1}{u}^{\alpha -1}{g}_{V}^{\alpha}\left(u\right)du,\phantom{\rule{4pt}{0ex}}t>0.$$It is obvious that the RE of ${H}_{\alpha}\left({{T}_{t}}^{1,5}\right)$ is increasing in terms of $t.$ Thus, the uncertainty of the conditional lifetime ${{T}_{t}}^{1,5}$ increases as t increases. We recall that this distribution has the DFR property.

- (iii)
- Let us suppose that X has a Weibull distribution with the shape parameter k and with the SF$$S\left(t\right)={e}^{-{t}^{k}},\phantom{\rule{4pt}{0ex}}k,t>0.$$Through some manipulation, we obtain$${H}_{\alpha}\left({{T}_{t}}^{1,5}\right)=-logk+\frac{1}{1-\alpha}log{\int}_{0}^{1}{\left({t}^{k}-logu\right)}^{(1-\frac{1}{k})(\alpha -1)}{u}^{\alpha -1}{g}_{V}^{\alpha}\left(u\right)du,\phantom{\rule{4pt}{0ex}}t>0.$$It is not a facile assignment to acquire a plain statement for the above relation, and therefore we computed it numerically. In Figure 1, we framed the entropy of ${{T}_{t}}^{1,5}$ in terms of the time t for values of $\alpha =0.2$ and $\alpha =2$ as well as $0<k<1$, which has the DFR property. As expected from Theorem 2, it is evident that ${H}_{\alpha}\left({{T}_{t}}^{1,5}\right)$ increases in $t.$ In Figure 2, we plotted the entropy of ${{T}_{t}}^{1,5}$ with respect to time t for values of $\alpha =0.2$ and $\alpha =2$ along with $k\ge 1$, which has the IFR property. As expected from Theorem 2, it is evident that ${H}_{\alpha}\left({{T}_{t}}^{1,5}\right)$ decreases in $t.$

**Theorem**

**3.**

**Proof.**

**Theorem**

**4.**

**Proof.**

## 3. Renyi Entropy Comparison

**Theorem**

**5.**

**Proof.**

**Example**

**2.**

**Theorem**

**6.**

- (i)
- If ${f}_{t}\left({S}_{t}^{-1}\left(u\right)\right)$ increases in u for all $t>0,$ then ${H}_{\alpha}({T}_{1,t}^{1,n})\ge {H}_{\alpha}({T}_{2,t}^{1,n})$ for all $\alpha >0$.
- (ii)
- If ${f}_{t}\left({S}_{t}^{-1}\left(u\right)\right)$ decreases in u for all $t>0,$ then ${H}_{\alpha}({T}_{1,t}^{1,n})\le {H}_{\alpha}({T}_{2,t}^{1,n})$ for all $\alpha >0$.

**Proof.**

**Example**

**3.**

## 4. Bounds for the Renyi Entropy of the Residual Lifetime

**Theorem**

**7.**

**Proof.**

**Theorem**

**8.**

**Proof.**

**Example**

**4.**

## 5. Conclusions

## Author Contributions

## Funding

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**The exact values of ${H}_{\alpha}\left({{T}_{t}}^{1,5}\right)$ with respect to t for the Weibull distribution for values of $\alpha =0.2$ and $\alpha =2$ when $0<k<1$.

**Figure 2.**The exact values of ${H}_{\alpha}\left({{T}_{t}}^{1,5}\right)$ with respect to t for the Weibull distribution for values of $\alpha =0.2$ and $\alpha =2$ when $k\ge 1$.

**Figure 3.**The exact values of ${H}_{\alpha}({T}_{t}^{X,1,4})$ (blue color) and ${H}_{\alpha}({T}_{t}^{Y,1,4})$ (red color) with respect to t for values of $\alpha =0.2$ and $\alpha =2$.

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**MDPI and ACS Style**

Mesfioui, M.; Kayid, M.; Shrahili, M. Renyi Entropy of the Residual Lifetime of a Reliability System at the System Level. *Axioms* **2023**, *12*, 320.
https://doi.org/10.3390/axioms12040320

**AMA Style**

Mesfioui M, Kayid M, Shrahili M. Renyi Entropy of the Residual Lifetime of a Reliability System at the System Level. *Axioms*. 2023; 12(4):320.
https://doi.org/10.3390/axioms12040320

**Chicago/Turabian Style**

Mesfioui, Mhamed, Mohamed Kayid, and Mansour Shrahili. 2023. "Renyi Entropy of the Residual Lifetime of a Reliability System at the System Level" *Axioms* 12, no. 4: 320.
https://doi.org/10.3390/axioms12040320