Delay Tolerance Thresholds and Natural Recovery in a Fork-Join Production System over Max-Plus Algebra

This study investigates server-specific delay tolerance and natural recovery in a three-server fork--join production system modeled with max-plus algebra. The system has two parallel servers synchronized by a join server and is analyzed in the canonical regime where the second-server self-loop is the unique critical circuit. Using max-plus spectral theory, critical-circuit margins, relative-delay recurrences, and computational validation, we examine a single additive event-time delay introduced after the nominal trajectory has entered the eigenvector regime. Every positive delay at the critical server propagates permanently, so its tolerance threshold is zero. At the two non-critical servers, exact finite thresholds are determined by the available margins between the critical circuit and competing non-critical paths. Delays not exceeding these thresholds are naturally absorbed without switching or control intervention, and a finite upper bound on the recovery time is established from a two-cycle contraction argument. Numerical tests across three admissible parameter sets confirm below-threshold recovery, boundary recovery, above-threshold propagation, and invariance with respect to the delay cycle and eigenvector translation. These results provide a precise event-level robustness characterization for the stated canonical fork--join regime.