We develop a closed-form analysis of the joint distribution for cumulative shock reliability systems with phase-type inter-shock times. The analytical literature on shock-driven reliability has hitherto been split into two largely separate traditions: scalar fluctuation theory, which delivers closed-form joint distributions of pre-failure and failure-time observables but cannot accommodate matrix phase structure; and matrix-analytic methods, which handle phase-type dynamics naturally but focus on stationary indicators rather than first-passage distributions. We bridge these traditions by introducing a matrix-valued reliability functional Φν(ξ,u,v,ϑ,θ) that encodes the joint distribution of the failure index, pre-failure damage and time, failure-time damage and time, and the operational phase at the moment of failure. We derive Φν in closed form via Sherman–Morrison reduction of the matrix Laplace–Stieltjes transform together with the Dshalalow error-operator, and establish a span-reduction theorem showing that Φν lies in a three-dimensional matrix subspace generated by the identity and two matrix LSTs. The functional simultaneously generalizes the scalar fluctuation functional of Dshalalow and White and the phase-tagged first excess functional of Tadj, recovering both as projections. We extract twelve closed-form reliability indices, including the reliability function, mean time to failure, mean overshoot, joint pre-failure and failure transforms, and, new to the cumulative shock literature, the phase distribution at failure and the phase-resolved failure-time distribution. Two structural identities of Wald type emerge as corollaries. The framework reduces to elementary arithmetic for rational model primitives and is verified against 2×105 Monte Carlo trajectories in a worked example.
Lotfi Tadj (Mon,) studied this question.