Analysis and development of Markov models of three-phase systems with partial combination of services based on Z-transforms

This paper presents a new Markov model for analyzing the performance and stability of a three-phase discrete service system with partial phase overlap, uniquely integrating the Needham-Schroeder authentication protocol with queueing theory and Z-transforms. This interdisciplinary approach addresses a critical gap by evaluating the system's efficiency and reliability under real-world loads, complementing traditional cryptographic security analysis. We rigorously derive the system of balance equations for various states and apply Z-transforms to obtain explicit analytical expressions for the phase state probabilities x(k) and y(k). Numerical results obtained under 100% system utilization (ρ=1) demonstrate the model’s ability to achieve a stable stationary distribution, with converging probabilities (e.g., for buffer size k=7, x(7)=42/67 and y(7)=21/67). These findings confirm the effectiveness of the proposed methodology for quantitatively assessing complex buffered systems, providing critical insights for designing resilient and secure network protocols in critical infrastructure.