A Phase-Tagged Matrix Functional for Continuous-Review (s, S) Inventory Systems with Phase-Type Demand Inter-Arrival Times

The continuous-review (s,S) inventory system poses a fundamental analytical challenge: the cycle time and reorder undershoot are not independent when demand inter-arrival times are non-exponential, and scalar transform methods cannot resolve their joint distribution. We introduce a phase-tagged matrix functional Ψ(s,S;ξ,θ,z), an m×m matrix that encodes, in a single closed-form object, the joint distribution of the reorder index, the cycle time, the reorder undershoot, and the operational phase of the inter-arrival process at the reorder epoch for any phase-type PH(α,S) inter-arrival distribution. The functional is derived via a finite discounted-visit recurrence with O(Δ2) computational cost, where Δ=S−s is the inventory cushion. From it we extract, in closed form, the expected demands per cycle, the mean cycle length, the undershoot distribution, and the long-run average cost rate. The classical scalar results of Sahin’s work in 1983 are recovered as the projection αΨ(s,S)e. We prove a phase-coupling impossibility theorem: under PH renewal inter-arrivals, the phase at reorder is independent of any subsequent lead-time demand, delineating precisely where Markovian arrival process (MAP) inter-arrivals are needed. All closed forms are verified against N=500,000 independent Monte Carlo replenishment cycles on two instances with structurally different PH generators, with all relative errors below 0.5%.