# Reliability Analysis Based on a Jump Diffusion Model with Two Wiener Processes for Cloud Computing with Big Data

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Related Research

- ► In the case of a mobile device, the network access devices are frequently used by many types of software installed via the installer software.
- ► By using the installer software, the various types of third-party software are installed via the network.
- ► In the case of open source software, the weakness of reliability and security becomes a significant problem with respect to a computer network.

- ► Cloud computing has a particular maintenance phase, such as the provisioning processes.
- ► Big data as the result of many and complicated data from using the Internet cause system-wide failures because of the complexity of data management.
- ► The various mobile devices are connected via the network to the cloud service.
- ► The data storage areas for cloud computing are reconfigured via the various mobile devices.

## 3. Model Description

#### 3.1. Wiener Process Modeling

_{1}and σ

_{2}are positive constants representing a magnitude of the irregular fluctuation, v

_{1}(t) and v

_{2}(t) standardized Gaussian white noise.

_{i}(t) is the i-th one-dimensional Wiener process, which is formally defined as an integration of the white noise v

_{i}(t) with respect to time t. We define the two dimensionprocesses [ω

_{1}(t), ω

_{2}(t)] as follows [24]:

_{1}depends on the parameter b resulting from the failure occurrence phenomenon. Similarly, we assume that the parameter σ

_{2}depends on the parameter c resulting from the network environment of cloud computing.

#### 3.2. Jump-Diffusion Modeling

_{i}is approximately estimated as the positive values in almost all cases, because the mean value μ keeps a large value.

_{t}(γ) is a Poisson point process with parameter γ at operation time t. Furthermore, Y

_{t}(γ) is the number of jumps that occurred, γ the jump rate. Y

_{t}(γ), ω(t) and V

_{i}are assumed to be mutually independent. Moreover, V

_{i}is the i-th jump’s range.

- ► The Brownian motion ω
_{1}represents the results from the failure-occurrence phenomenon. - ► The Brownian motion ω
_{2}represents the results from cloud computing having the unique characteristics of provisioning processes, the change of the number of log-in users, etc. - ► The jump term means the indirect effects as a result of the many and complicated data from using the Internet, causing the system-wide failures because of the complexity of data management, i.e., the system failures of DataNodeand NameNodein terms of Hadoop and NoSQLin order to manage big data, etc.

_{i}. The software managers can assess several characteristics of cloud computing by using the size and shape of noises with the jump term, because the proposed model can totally comprehend the provisioning process, the change of users, the change of cloud applications, the indirect effects as a result of the many and complicated data in cloud computing, with big data as the noise.

## 4. Parameter Estimation

#### 4.1. Method of Maximum-Likelihood

_{1}in Equation (9) is presented. Then, we assume that σ

_{2}and l are the given parameters, because σ

_{2}and l are considered as the network factors. The joint probability distribution function of the process M(t) is denoted as:

_{k}; y

_{k})(k = 1, 2, ⋯, K) is constructed as follows:

^{*}, β

^{*}, b

^{*}and ${\sigma}_{1}^{*}$ are the values making Λ in Equation (24) maximal. These can be obtained as the solutions of the following simultaneous likelihood equations [23]:

#### 4.2. Estimation of the Jump Diffusion Parameters

_{i}.

- Step 1: The initial individuals are randomly generated. Furthermore, the set of initial individuals is converted to the binary digit.
- Step 2: Two parental individuals are selected, and new individuals are produced by the crossover recombination.
- Step 3: The value of fitness is calculated from the evaluated value of each individual. The following value of fitness as the error between the estimated and the actual values is defined in this paper:$$\begin{array}{l}\underset{\theta}{\mathrm{min}}\phantom{\rule{0.2em}{0ex}}{F}_{i}(\theta ),\\ {F}_{i}={\displaystyle \sum _{i=0}^{K}}{\left\{{M}_{j}(i)-{y}_{i}\right\}}^{2},\end{array}$$
_{j}(i) is the number of detected faults at operation time i in the proposed jump diffusion model and y_{i}the number of actual detected faults. Furthermore, θ means the set of parameters γ, μ, and τ. - Step 4: Step 2 and Step 3 are continued until reaching a specific size.

## 5. Optimal Maintenance Problem

- c
_{1}: the fixing cost per fault during the operation, - c
_{2}: the maintenance cost per unit time during the operation, - c
_{3}: the maintenance cost per fault after the maintenance.

^{*}is obtained by minimizing C(t) in Equation (29). Then, we consider the optimal maintenance problem as follows:

^{*}can be estimated numerically by using the optimization algorithms.

## 6. Numerical Examples

#### 6.1. Reliability Assessment

#### 6.2. Optimal Maintenance Time

^{*}= 384.17 days. Then, the total software maintenance cost is 677:01. Moreover, we can estimate the optimal maintenance time with stability requirements by using the amount of noise in Figure 6. Then, the estimated total software cost with stability requirements is about 450.

## 7. Concluding Remarks

## Acknowledgments

## Author Contributions

## Conflicts of Interest

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**Figure 2.**The estimated sample path of cumulative numbers of detected faults in terms of fault and network factors.

**Figure 3.**The estimated sample path of cumulative numbers of detected faults in terms of the fault factor.

**Figure 4.**The estimated sample path of cumulative numbers of detected faults in terms of the network factor.

**Figure 5.**The estimated sample path of cumulative numbers of detected faults in terms of fault and network factors.

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**MDPI and ACS Style**

Tamura, Y.; Yamada, S. Reliability Analysis Based on a Jump Diffusion Model with Two Wiener Processes for Cloud Computing with Big Data. *Entropy* **2015**, *17*, 4533-4546.
https://doi.org/10.3390/e17074533

**AMA Style**

Tamura Y, Yamada S. Reliability Analysis Based on a Jump Diffusion Model with Two Wiener Processes for Cloud Computing with Big Data. *Entropy*. 2015; 17(7):4533-4546.
https://doi.org/10.3390/e17074533

**Chicago/Turabian Style**

Tamura, Yoshinobu, and Shigeru Yamada. 2015. "Reliability Analysis Based on a Jump Diffusion Model with Two Wiener Processes for Cloud Computing with Big Data" *Entropy* 17, no. 7: 4533-4546.
https://doi.org/10.3390/e17074533