Steady-State Detection

The primary purpose of steady-state detection is to identify and truncate the transient phase of a simulation. During this initial period, output data is heavily influenced by the arbitrary starting conditions (e.g., empty queues, zero inventory) rather than the system's true long-run behavior. Failing to remove this initialization bias leads to systematically skewed performance metrics, such as underestimating average wait times or overestimating throughput. The detection algorithm pinpoints the truncation point—the observation index after which the process is considered stationary—ensuring only representative data enters the final statistical analysis.

Detection algorithms rely on formal hypothesis tests for stationarity to mathematically determine when equilibrium is reached. Common approaches include:

• Schruben's Test: Applies a Brownian bridge process to detect variance shifts across the time series.
• KPSS Test: Tests the null hypothesis that the series is trend-stationary against a unit root alternative.
• CUSUM Charts: Cumulative sum control charts that visually and statistically flag when the mean process deviates from a target value. These methods operate on batch means—aggregated, non-overlapping groups of observations—to smooth high-frequency noise and reveal underlying trends.

To achieve reliable detection, raw simulation output is transformed using batching techniques. Observations are partitioned into sequential batches of equal size, and the mean of each batch is treated as a single, approximately independent data point. This addresses the autocorrelation inherent in simulation data. Two dominant strategies exist:

• Single Long Run: One extended simulation run is analyzed post-hoc, dividing the entire trace into batches to locate the truncation point.
• Multiple Replications: Several independent runs with different random seeds are executed. Steady-state is assessed across replications, providing cross-validation of the detected equilibrium point and enabling confidence interval construction.

A practical heuristic for steady-state detection involves monitoring the marginal confidence interval width of the output mean. The algorithm sequentially adds observations and recalculates the standard error. The system is declared stable when the relative precision—the ratio of the confidence interval half-width to the cumulative mean—falls below a predefined threshold (e.g., 0.05). This precision-based stopping rule ensures that data collection continues only until the estimate achieves the required statistical accuracy, optimizing computational effort in large-scale digital twin simulations.

Before applying formal statistical tests, analysts use visual time-series plots to qualitatively assess convergence. Key diagnostic tools include:

• Running Mean Plots: A cumulative average plotted over time; stabilization into a flat line suggests equilibrium.
• Welch's Graphical Procedure: Overlaying multiple independent replication traces with a moving average to visually identify where the ensemble variance stabilizes.
• Autocorrelation Function (ACF) Plots: Confirming that lagged correlations decay rapidly, indicating that the process has forgotten its initial state. These heuristics provide an essential sanity check and guide the parameterization of automated detection algorithms.

For systems exhibiting regenerative structure, steady-state analysis can bypass the warm-up problem entirely. A regenerative process probabilistically restarts from a fixed state at random regeneration points (e.g., an empty-and-idle state in a queue). By collecting data only within complete regeneration cycles, the output forms independent and identically distributed blocks. The steady-state mean is then estimated as the ratio of expected accumulation per cycle to expected cycle length, eliminating initialization bias without requiring a truncation point. This method is particularly powerful for Markovian systems.

The initial simulation interval during which the system transitions from its empty or arbitrary starting conditions to a representative operating state. Data collected during this phase is statistically biased and must be discarded before analysis.

• Initialization Bias: The primary source of error that steady-state detection eliminates.
• Truncation Point: The specific observation index where the warm-up ends and steady-state begins.
• Rule of Thumb: Common heuristics include discarding the first 10-30% of observations, but formal detection algorithms are far more reliable.

A formal statistical hypothesis test for detecting initialization bias. It partitions a single long simulation run into batches and tests for a significant trend in the batch means, indicating the system has not yet reached equilibrium.

• Null Hypothesis: The batch means exhibit no significant upward or downward trend.
• Rejection: If the null is rejected, the initial batches are discarded, and the test is repeated iteratively.
• Advantage: Fully automated and objective, unlike graphical methods.

A foundational output analysis technique where a single long simulation run is divided into contiguous, non-overlapping batches of observations. The batch means are treated as approximately independent and identically distributed (i.i.d.) samples.

• Purpose: Estimates the steady-state mean and constructs valid confidence intervals without requiring multiple replications.
• Batch Size Trade-off: Larger batches reduce autocorrelation between means but yield fewer degrees of freedom.
• Non-Overlapping: A strict requirement to ensure statistical independence between batches.

A probabilistic approach that identifies regeneration points—specific system states where the future evolution becomes probabilistically independent of the past (e.g., an empty-and-idle state in a queue).

• Regenerative Cycles: The simulation is decomposed into i.i.d. cycles between successive regeneration points.
• Classic Ratio Estimator: The steady-state mean is estimated as the ratio of expected cycle cost to expected cycle length.
• Limitation: Requires identifiable regeneration points, which may be rare or non-existent in complex supply chain models.

An advanced method for estimating the variance of the sample mean in steady-state simulations by analyzing the time series in the frequency domain. It directly models the autocorrelation structure without requiring batch partitioning.

• Power Spectrum at Zero Frequency: The key quantity estimated, which is proportional to the asymptotic variance.
• Tukey-Hanning Window: A common smoothing kernel applied to reduce spectral leakage.
• Advantage: Often yields more precise confidence intervals than batch means for highly autocorrelated processes.