Dependability Analysis Tool Based on Multi-Dimensional Stochastic Noisy Model for Cloud Computing with Big Data

This paper focuses on a big data on cloud computing environment by using open source software such as Open Stack and Eucalyptus because of the unification management of data and low cost. We propose a new approach to software dependability assessment based on stochastic differential equation modelling and jump diffusion process modelling in order to consider the interesting aspect of the numbers of components, cloud applications, and users. Moreover, we discuss the determination of an optimum software maintenance time minimizing the total expected software cost. In particular, we develop the three-dimensional AIR application for reliability and cost optimization analysis based on the proposed method. Then, we show numerical performance of the developed AIR application to evaluate the method of software reliability assessment for the big data on cloud computing


Introduction
The cloud computing with big data are used as the next-generation software service paradigm. The cloud computing service is connected by many mobile devices. Recently, the mobile clouds based on cloud service become known as the next-generation software service paradigm. Then, the installer software developed by the third-party developers indirectly effect on the reliability of a mobile device by using such mobile clouds. Especially, OSS (Open Source Software) serves as key components of critical infrastructures in our society. The OSS project contains special features so-called software composition by which many distributed components are developed in all parts of the world. However, the poor handling of quality problem and customer support has limited the progress of OSS, because the OSS development cycle has no specific testing phase in order to detect and remove software faults introduced in the development process. Also, the mobile OSS known as one of OSS has been gaining a lot of attention in the embedded system area, i.e., Android (Open Handset Alliance, 2016), Busy Box (Andersen, 2016), Firefox OS (Firefox OS, 2016), etc. Therefore, it is difficult for many companies to assess the reliability in mobile clouds, because a mobile OSS includes the different software versions, the vulnerability issues, the opened source code, the security hole, etc. of the developed AIR application to evaluate the method of software reliability assessment for the big data on cloud computing.

Wiener Process Modeling for Reliability Assessment Considering the Cloud Computing
By using Ito's formula (Wong, 1971;Arnold 1974;Yamada et al., 1994;Mikosch, 1998), we have proposed the following stochastic equation model Yamada, 2015a, 2015b): where ( ) is the software fault-detection rate at operation time and a non-negative function, ( ), means the amount of changes of requirements specification.
As the noise-by-noise sample path for each factor, we can represent as the following equations. First, the sample path in terms of fault factor is given as Second, the sample path in terms of network factor is given as Therefore, the cumulative numbers of detected faults are obtained as follows: In the proposed model, we assume that the parameter 1 depends on the parameter resulting from the failure-occurrence phenomenon. Similarly, we assume that the parameter 2 depends on the parameter resulting from the network environment of cloud computing.

Jump-Diffusion Modeling for Reliability Assessment Considering the Big Data
Similarly, as the noise-by-noise sample path for each factor, we can represent as the following equations. First, the sample path in terms of fault factor is given as where ( ) is a Poisson point process with parameter at operation time . Also, ( ) is the number of occurred jumps, the jump rate. ( ), 1 ( ), 2 ( ), and are assumed to be mutually independent. Moreover, is -th jump range.
Second, the sample path in terms of network factor is given as By using Ito's formula (Arnold, 1974;Wong, 1971), the solution of the former equation can be obtained as follows: The proposed model in Eq. (7) includes the noise with jump term . The cloud managers can assess several characteristics of the cloud with big data by using the size and shape of the noises with jump term, because the proposed model can totally comprehend the provisioning process, the change of users, the change of cloud applications, the indirectly effects as the results from the huge-complicated data in cloud computing with big data as the noise.

Optimal Maintenance Problem
Considering the existing optimal software release problems (Yamada andOsaki, 1985, 1987), we define the following cost parameters: Second, the sample path in terms of network factor is given as

Multi-Dimensional AIR Application for Reliability Analysis
The specification requirement of the reliability and optimization analysis tool for big data on cloud computing are shown as follows: a) This tool should be operated by clicking the mouse button and typing on the keyboard to input the data through GUI system. In particular, the user experience design is adopted as the important element of our tool. Moreover, the threedimensional space is used in the proposed tool in order to represent multidimensional stochastic jump diffusion processes model. b) Open source Apache Flex SDK (Flex.org, 2016) should be used to implement the program. This tool is developed as a stand-alone Adobe AIR application on Windows, UNIX, and Mac OS operating system. Also, this tool operates as Web application. Moreover, the three-dimensional space is implemented by illustrating several three-dimensional graphs.
c) The method of maximum-likelihood and Genetic Algorithm (GA) are used as the estimation of unknown parameters in our model Yamada, 2015a, 2015b).
d) This tool treats the proposed multi-dimensional stochastic processes model considering the cloud computing with big data, and illustrate the cumulative numbers of detected faults at arbitrary time , the mean time between software failures, the number of remaining faults, and the total expected software cost as software dependability and optimization assessment measures.
It is known the following items as the elements of user experience design.
"The elements of user experience design".
Visual Design; Information Architecture; Information; Structuring, Organization and Labeling; Finding and Managing; Interaction Design; Usability; Accessibility; Human-Computer Interaction.
We develop the dynamic reliability analysis tool based on "Visual Design" and "Interaction Design" by using the animation effects of Flex. As an example, we use the following source code as the "Visual Design" and "Interaction Design".

Numerical Performance Illustration
We focus on Hadoop (The Apache Software Foundation, 2016) and Open Stack (The Open Stack project, 2016) in order to evaluate the performance of our models. In this paper, we show numerical examples of software reliability assessment by using the data sets for Hadoop of database software and Open Stack of cloud software. The data sets used in this paper are collected in the bug tracking systems on the websites of Hadoop and Open Stack open source projects.
We show the main screen of the developed dynamic reliability analysis tool in Fig. 1. The items of several reliability assessment measures are shown in the left side of screen. Also, the right side of screen shows the data edit function.
The sample path of the estimated number of detected faults for the fault and n etwork factors in Eq. (7) is shown in Fig. 2. From Fig. 2, we can confirm that the jump noise of network factor becomes large in the early operating phase of cloud computing. Also, we found that the noise becomes large from 100 to 200 days. On the other hand, we can confirm that the noise of fault factor becomes small in all operating phase of cloud computing. The developed tool will be useful to assess the reliability of the characteristics of big data on cloud computing.  (12) is shown in Fig. 5. From Fig. 5, we can confirm that the noise of network factor becomes large in the early operation phase of cloud computing. On the other hand, we can confirm that the noise of fault factor becomes small in all operation phase of cloud computing. Therefore, we find th at the cloud computing environment in this case keeps in stable condition in terms of the reliability. Moreover, we find that the optimum maintenance time is given as * =367days considering the noise. Then, the total expected software cost is 345.
The developed cost optimization tool is described by using several noise parameters based on the Wiener and jump diffusion processes. From the mentioned results above, the software managers will be able to visually confirm the stability of cloud computing by using the sample path of the proposed model. In particular, the developed tool can assess the reliability and evaluate cost optimization in terms of multi -factors for big data on cloud computing.

Concluding Remarks
In this paper, we have focused on the big data on cloud computing. In particular, we have developed the cost optimization analysis tool in order to consider the characteristics of cloud computing under big data. Then, we have proposed the method of reliability assessment based on the jump diffusion process model incorporating the interaction among 3V's model on big data. Moreover, we have developed the three-dimensional AIR application based on the proposed method. Additionally, we have presented several performance illustrations of the developed AIR application and the proposed method for the actual data. Furthermore, it is important for software managers to assess the reliability and cost optimization for the big data on cloud computing. We have shown the estimated cumulative numbers of detected faults and total expected software cost considering the cloud computing. Thereby, we have found that the developed AIR application can assess integrated reliability considering both software failure and network traffic.
In case of considering the effect of external factors on entire system in the development of software reliability assessment methods for cloud computing, it is necessary to grasp the deeplyintertwined factors. In this paper, we have shown that the proposed method can grasp such deeply-intertwined factors by assuming 3V's model of big data. Also, we have analyzed actual data to show numerical performance of software reliability assessment for the cloud computing. Moreover, we have formulated the total expected software cost considering the characteristics of big data on cloud computing. Furthermore, we have found that the developed tool can evaluate the optimum software maintenance time for the big data on cloud computing.