Abstract

This study presents a Markov chain-based metric called the Reliability Reserve (τ), designed to estimate the time available for operator intervention during the ayran fermentation process. This indicator can be integrated into a digital twin forecast management system. The fermentation process was obtained using a DTMC (discrete-time Markov chain) and divided into five states according to pH (S1–S5). Laboratory samples were prepared from premium-grade cow’s milk sourced from the Zher-Ana farm and divided into three experimental groups: Control (without additives), Opt1 (3% additive), and Opt2 (4% additive). A sequence of states was created for the three studied groups (Control, Opt1, and Opt2), and the transition states of the matrix were calculated. The Reliability Reserve quantifies how much time is left before the system transitions from the target state to the acidification state. For the first group, P45 was 0.200, corresponding to τ = 26.9 min. The incorporation of functional additives increased P45 to 0.250, reducing τ to 20.9 min and shortening the operator intervention window by approximately 6 min. Markov chains were constructed using 101 interpolated time points obtained from five experimental pH measurements for each group. The original pH values were recorded at 2, 4, 6, 8, and 10 h of fermentation and linearly interpolated with a step size of 0.1 h to improve temporal resolution. Model robustness was evaluated using sensitivity analysis (±0.05 pH boundary shifts) and bootstrap resampling (n = 1000, 95% confidence intervals). The concept of Reliability Reserve is a practical decision-making tool in real time. It offers an alternative to traditional reliability indicators such as MTTF and allows integration into digital twin-based control systems.

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