# Probabilistic Model Checking GitHub Repositories for Software Project Analysis
## 1 Introduction  
When software projects fail, the consequences can be substantial, resulting in significant time and cost losses [1]. To minimize these risks and ensure successful project completion, effective software project management is important [2]. Failures often stem from a lack of collaboration or communication within project teams [3]. However, understanding and improving collaboration patterns in software projects remain critical. Recent complexities in collaborative processes, coupled with the dispersion of team members, require more efficient organization and comprehension of software development activities by project managers and developers [4, 5]. Since the software development process involves human activities across all stages, from design and development to testing and maintenance, managing human factors is a crucial aspect [6]. In essence, collaboration and communication among team members in the software development environment are fundamental elements for project success. In response to these challenges, GitHub has gained prominence as a platform for collaborative software development. GitHub facilitates smooth collaboration among developers by offering numerous features such as issue tracking, code review, and pull requests, streamlining version control and project management [7]. This platform provides a transparent development environment, allowing people to observe and actively contribute to the actions and trends of other developers, fostering a collaborative atmosphere [8]. Additionally, GitHub promotes open collaboration by structuring workflows based onpull requests [9]. To contribute to GitHub's open-source projects, developers must follow specific procedures. Analyzing these development processes is crucial from a project management perspective, providing insights into collaboration patterns and project trends, thereby enhancing the efficiency of development teams [10; 11]. This study aims to capture the trends and characteristics of individual projects and developers through the analysis of GitHub's development processes. GitHub, as the fundamental unit of projects, enables easy communication and contributions from diverse developers through repositories. GitHub comprises a remote repository as a central hub and local repositories for individual developers to store their work. Particularly, when multiple developers contribute simultaneously or manage version control, GitHub enhances the organization and comprehension of software development activities. The procedures involved in integrating changes from local repositories to remote repositories, including branch creation, committing changes, submitting pull requests for completed features, and accepting or rejecting merge requests, unfold over time, allowing for the observation of various patterns [12]. To handle these time-series data, we employ Discrete-Time Markov Chain (DTMC) modeling. DTMC is a mathematical model representing a system transitioning between a finite set of states at discrete time intervals [13]. This modeling approach is adept at capturing probabilistic transitions between different states, making it useful for analyzing sequential data over time. It excels in handling categorical data sequences and aids in devising optimal plans through predictive analysis in decision processes [14; 15]. Through DTMC modeling, we observe the trends in the development processes on GitHub over time. Our research analyzes three aspects: repository characteristics, branch management, and participant analysis. Repository characteristics focus on the overall activities and trends of the project. Branch management centrally addresses the aspects of distributed version control systems inherent in GitHub. Lastly, participant analysis explores individual contributions, trends, and relationships among contributors. These results are expected to be utilized for identifying areas of improvement in project management or gaining insights into the project's progress.