Joint Chance-Constrained Max–Min Service Guarantees on Directed Irrigation Trees with Conveyance Losses: Certified Evaluation and Bottleneck Diagnostics

On lossy irrigation trees, the nominal max–min guarantee has a closed form for deterministic data. With random demands and source capacity, the joint chance-constrained guarantee is the largest service level all users and periods meet jointly with a prescribed probability. A recursion on a loss-rescaled lattice brackets the probability that all capacities suffice; the one-period bracket on the guarantee has a leading-order relative width at most twice the binding-probability-weighted relative lattice step. In one-factor models, certified integration brackets the guarantee with bounded rounding errors and, under stated conditions, converges; in a generalized factor model, the guarantee does not decrease as period-factor correlations increase. Under stated conditions, its capacity elasticities equal binding probabilities. In the Gone Abat Jap network (rule-based capacities, assumed source parameters), all brackets at probability 0.95 lie below the nominal guarantee 0.85. If dry years also lower the source, the guarantee lies in [0.52755, 0.53282], whose lower end is 6.5–6.7% above estimated safe individual-constraint designs; the naive design misses the target. The source, the nominal bottleneck, binds at the guarantee with estimated conditional probability 0.926 (0.040 if deterministic).