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arXiv · 2608.19471

Exact Multistate Reliability and Upgrade Design for Heterogeneous HBM Systems via Threshold-Pruned BAT

Wei-Chang Yeh
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Abstract

High Bandwidth Memory (HBM) systems can exhibit partial service rather than only full service or complete isolation: a controller-visible service unit may deliver full, reduced, or zero bandwidth because of sub-channel isolation, lane remapping, or protection overhead. This paper develops an exact multistate reliability framework in which each service unit carries an arbitrary finite set of bandwidth states and reliability is the probability that aggregate delivered bandwidth meets a demand. The binary k-out-of-n model is recovered as a special case, while closed-form binary-mapping error relations quantify mean-bandwidth distortion and provide a screening test for the simpler abstraction. For exact single-threshold evaluation a threshold-pruned multistate Binary-Addition-Tree (TP-mBAT) algorithm is proposed. It is deliberately regime-specific: fixed-grid dynamic programming is preferable on a compact common grid, where a 16-unit commensurate control required 273 pruned-DP updates versus 362,506 TP-mBAT node visits. On a reproducible 14-unit incommensurate benchmark, TP-mBAT is compared against a dynamic program carrying the same threshold rules, so that no baseline is weakened. Both expand the same state space, 6,862 nodes against 6,861 updates, and the separation lies entirely in retained state: 17 traversal entries against 412,121 probability states at central demand, reducing measured peak storage from 25.02 MB to 1,152 B. An exact probability-transfer sensitivity identifies when moving mass from a degraded state to a higher-bandwidth state changes system success, and a reserved-unit floor model admits a third exact pruning rule that is vacuous without such floors. A latent package-state mixture captures shared stress, where ignoring dependence overstates reliability by 8.73 percentage points.

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Wei-Chang Yeh. 2026-08-19. Exact Multistate Reliability and Upgrade Design for Heterogeneous HBM Systems via Threshold-Pruned BAT. https://arxiv.org/abs/2608.19471

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