Optimal Inspection Scheduling for Multi-State Pure-Birth Degradation Models: Spectral Theory, Statistical Estimation, and a Corrected Cost Functional

Abstract

We revisit the pure-birth continuous-time Markov chain framework for infrastructure degradation modeling, generalizing prior fixed-state (n = 3, 4) treatments to arbitrary n by identifying the absorption time with the classical hypoexponential distribution. This identification unifies and correctly generalizes closed-form expressions for reliability, hazard, and mean residual life that had previously been rederived case by case. It exposes a genuine internal-consistency error in at least one prior single-instance analysis of this model class, which we identify and correct. Building on this foundation, we identify a previously unexamined structural degeneracy in the periodic-inspection cost-rate functional used throughout this modelling tradition: the naive cost rate tends to zero as the inspection interval grows without bound, trivially favoring indefinitely infrequent inspection regardless of parameters. We derive a corrected downtime-accrual cost functional together with a complete existence-and-uniqueness theory for the resulting optimal inspection interval via an auxiliary monotone function that requires no increasing-failure-rate assumption and yields an explicit, previously unstated existence threshold. We further derive a maximum-likelihood estimation theory for the degradation rates under both full and interval-censored observation and, by combining these two results for the first time, establish an asymptotic O(1/N) regret bound quantifying the excess long-run cost incurred by using estimated rather than true rates when computing the optimal inspection interval. Numerical experiments validate every theoretical rate directly, including one practically important finding: a naive (unsafeguarded) Newton iteration for the optimal interval can diverge, and a bisection safeguard, justified by the same monotonicity property used in the existence proof, is required for a reliable algorithm.