Mathematical Model of the Software Development Process with Hybrid Management Elements

Reliable schedule-risk estimation in hybrid software development lifecycles is strategically important for organizations adopting AI in software engineering. This study addresses that need by transforming routine process telemetry (CI/CD, SAST, traceability) into explainable, quantitative predictions of completion time and rework. This paper introduces an integrated probabilistic model of the hybrid software development lifecycle that combines Generalized Evaluation and Review Technique (GERT) network semantics with I-AND synchronization, explicit artificial-intelligence (AI) interventions, and a fuzzy treatment of epistemic uncertainty. The model embeds two controllable AI nodes–an AI Requirements Assistant and AI-augmented static code analysis, directly into the process topology and applies an analytical reduction to a W-function to obtain iteration-time distributions and release-success probabilities without resorting solely to simulation. Epistemic uncertainty on critical arcs is represented by fuzzy intervals and propagated via Zadeh’s extension principle, while aleatory variability is captured through stochastic branching. Parameter calibration relies on process telemetry (requirements traceability, static-analysis signals, continuous integration/continuous delivery, CI/CD, and history). A validation case (“system design → UX prototyping → implementation → quality assurance → deployment”) demonstrates practical use: large samples of process trajectories are generated under identical initial conditions and fixed random seeds, and kernel density estimation with Silverman’s bandwidth is applied to normalized histograms of continuous outcomes. Results indicate earlier defect detection, fewer late rework loops, thinner right tails of global duration, and an approximately threefold reduction in the expected number of rework cycles when AI is enabled. The framework yields interpretable, scenario-ready metrics for tuning quality-gate policies and automation levels in Agile/DevOps settings.