Reliability Allocation

Reliability Allocation (systems-engineering-safety/arp4761a/reliability-allocation)

Use when you must allocate a system reliability requirement down to the design items: flow one top-level reliability budget, expressed as a system failure rate per flight hour or as an MTBF, into per-item target rates and MTBF values by deterministic apportionment, then verify that the item budgets close against the system rate and report the item margins against predicted capability rates. This leaf implements the equal split and complexity-weighted apportionment schemes in pure Python, stdlib only, with no RNG and no base-rate tables: the weights are design-team inputs. It pairs with the rate-estimation leaf for the predicted item capability rates the margin report consumes and with the block-evaluation leaf that later consumes the per-item targets this leaf hands down.

Out of scope, owned by sibling leaves: apportioning failure-condition severity probability targets across channels and assigning item assurance levels (preliminary-system-safety-assessment); evaluating series or parallel block structures from given rates (reliability-block-diagram); assigning requirement statements to items with coverage registers (requirements-allocation in the arp4754a pack); estimating item rates from test or service data (failure-rate-estimation); repair-time statistics (maintainability- prediction).

Domain quick reference

• MTBF to failure rate: rate = 1 / MTBF. An MTBF system requirement enters the flow-down as its rate equivalent (mtbf_to_rate).
• Equal split: item target i = system_rate / n_items. For a non-redundant series chain the item rates sum to the system rate, so every item carries the same share and the same MTBF target, n_items x system MTBF.
• Complexity-weighted apportionment: item target i = system_rate x w_i / sum(w), with w the item complexity or part-count weights chosen by the design team (complexity_weighted_alloc). The normalized weighted split also sums to the system rate.
• Series-sum closure: total = sum(item rates), relative_error = (total - system_rate) / system_rate, exact when total matches the system rate within rel_tol 1e-12 (closure_check). The allocated item budgets must add back up to the system budget.
• Capability margin: margin = capability / target - 1 (margin_report). A negative margin means the predicted capability rate is better, lower, than the budget (slack); a positive margin means the predicted rate exceeds the budget (deficit, an overrun).
• Item MTBF target: MTBF_i = 1 / rate_i, the reciprocal of the item target rate, is the value recorded in the item development specification.
• Units: failure rate in failures per flight hour, MTBF in hours.
• ARP4761A frames the safety assessment process and ARP4754A the allocated-requirements practice that carries numeric targets into item development specifications; both are reference-only here.

Workflow

1. Fix the top-level reliability budget: state the system requirement as system_rate failures per flight hour. If the requirement is an MTBF, convert it with mtbf_to_rate so the budget is a rate.
2. Choose the apportionment scheme: equal_split when the items are comparable, or complexity_weighted_alloc with the design-team complexity weights when part counts differ. The weights are item complexity or part-count estimates, design data, not table lookups.
3. Produce the per-item failure-rate and MTBF targets: take the item target rates from the chosen split and convert each to its MTBF target with mtbf_to_rate.
4. Verify the series-sum closure: run closure_check on the allocated item rates against the system rate and confirm the relative error sits at float noise with the exact flag True.
5. Report the capability margins: run margin_report with the item target rates and the predicted item capability rates, then flag the items with positive margin (budget overrun) and negative margin (slack) and weigh the rebalancing trade between them.
6. Record the per-item targets in the item development specifications: one failure-rate target and one MTBF target per item, per ARP4754A allocated-requirements practice.
7. Confirm the deterministic checks with the contract test scripts/test_reliability_allocation.py.

Worked example

Wing flap actuation system. Top-level reliability requirement: the system shall fail at no more than 1e-4 per flight hour, the equivalent of a 10000 h MTBF target, flowed down across six items: pump, valve, sensor, computer, actuator, wiring, with design-team complexity weights [1, 2, 1, 3, 2, 1] (sum 10). The dual-redundant computer is the most complex item, the wiring run and the sensor the least.

• Equal split, equal_split(1e-4, 6): every item budget is 1.6666666666666667e-05 per flight hour, an MTBF target of 60000 h. closure_check reports total 0.00010000000000000002 against system rate 0.0001, relative error 1.36e-16, exact True: the budgets sum to the system rate. Equal split asks the simple wiring run and the complex computer to meet the same 60000 h MTBF, ignoring their different part counts.
• Complexity-weighted split, complexity_weighted_alloc(1e-4, [1, 2, 1, 3, 2, 1]): pump 1e-05 per flight hour (100000 h MTBF), valve 2e-05 (50000 h), sensor 1e-05 (100000 h), computer 3e-05 (33333.3 h), actuator 2e-05 (50000 h), wiring 1e-05 (100000 h). closure_check reports relative error 0 and exact True: the normalized weighted split sums exactly to the target. The computer carries the largest budget, 3e-05, and the light items the smallest, 1e-05, matching the weight ratio.
• Margin report against predicted capability rates [9e-6, 1.6e-5, 1.2e-5, 2.8e-5, 1.8e-5, 1.1e-5] per flight hour: margins -0.1000 (pump), -0.2000 (valve), +0.2000 (sensor), -0.0667 (computer), -0.1000 (actuator), +0.1000 (wiring). The sensor and wiring predictions overrun their budgets while the valve and actuator hold slack. The capability rates sum to 9.4e-05 against the 1e-4 system target, so the item-level prediction still closes at system level with headroom, and the report flags the rebalancing trade: tighten the valve budget or relax the sensor budget and the system requirement stays met. A capability sum above the system rate would mean the item budgets as predicted cannot meet the top-level requirement and the design or the requirement must be revisited.

Verification

• Confirm equal_split(1e-4, 6) returns six budgets each 1.6666666666666667e-05 per flight hour and that 1.0 / budget is 60000 h within 1e-9.
• Confirm complexity_weighted_alloc(1e-4, [1, 2, 1, 3, 2, 1]) returns [1e-05, 2e-05, 1e-05, 3e-05, 2e-05, 1e-05] and that the computer share, 3e-05, is the largest budget.
• Confirm the closure identities: every equal and weighted split of a positive system rate closes with relative error at float noise and the exact flag True; mtbf_to_rate round-trips (rate x MTBF = 1.0).
• Confirm margin_report margins equal capability / target - 1.0 with input order preserved and that a capability sum below the system rate leaves headroom while a sum above it flags a requirement that cannot be met.
• Confirm every non-positive rate, weight, MTBF or capability, empty input list, item count outside 1..ITEMS_MAX, and length mismatch raises ValueError.
• Run the contract test offline: python3 skills/systems-engineering-safety/arp4761a/reliability-allocation/ scripts/test_reliability_allocation.py (33 tests, deterministic).

Related leaves

• systems-engineering-safety/arp4761a/reliability-block-diagram: the downstream consumer that evaluates a series chain of blocks from the per-item target and capability rates this leaf allocates.
• systems-engineering-safety/arp4761a/preliminary-system-safety- assessment: the sibling that apportions failure-condition severity probability targets and assigns item assurance levels, the per-condition side of the safety requirement flow.
• systems-engineering-safety/arp4761a/failure-rate-estimation: demonstrates or estimates item rates from test and service data, the source of the predicted capability rates the margin report needs.
• systems-engineering-safety/arp4754a/requirements-allocation: the qualitative register that assigns requirement statements to design items, complementing the numeric targets this leaf hands to items.

Pitfalls

• Equal splitting ignores part counts: the simple wiring run and the complex dual computer receive the same 60000 h MTBF budget, so use complexity_weighted_alloc whenever the item complexities differ.
• Reading the margin sign backwards: a negative margin is slack (the predicted capability is better than the budget) and a positive margin is a deficit (the predicted rate overruns the budget), the inverse of the intuitive reading.
• Treating the closure check as a design endorsement: closure only confirms the arithmetic split sums to the system rate; the design still fails the requirement if the predicted capability sum exceeds the system rate, and then the design or the requirement must change.
• Confusing the MTBF target with a demonstrated MTBF: the targets are budgets flowed down for the item development specifications; demonstrating rates from test or service data is the failure-rate-estimation job.
• Inventing weights on the fly: the apportionment is only as good as the design-team complexity weights, which should reflect part count or complexity estimates agreed with the design team, not analyst guesses.
• Applying the series-sum identity to redundant chains: the identity holds for a non-redundant series item chain; evaluating redundant or parallel structures from given rates is the reliability-block-diagram scope.

Contract test

Run the deterministic contract test (stdlib unittest, offline, <20 s):

python3 skills/systems-engineering-safety/arp4761a/
reliability-allocation/scripts/test_reliability_allocation.py

The test covers the flap actuation worked example (six-item equal and weighted target rates, closure relative error 1.36e-16 and exact True, item MTBF targets 60000 h and 33333.3 h, margin report values), the MTBF rate round trip, the share identity rate_i = system_rate x w_i / sum(w), the largest-complexity-item largest-budget check, the closure identity over arbitrary positive weights, the capability-sum headroom check, ITEMS_MAX and single-item boundaries, and ValueError rejection of every non-physical input.

Behavior contract (gate 3)

Run the deterministic contract test (stdlib unittest, offline):

python3 skills/systems-engineering-safety/arp4761a/
reliability-allocation/scripts/test_reliability_allocation.py

The test covers the 1e-4 per flight hour six-item flow-down contract (equal and complexity-weighted target rates, MTBF targets, series-sum closure, capability margins), the MTBF rate round trip, the weighted share identity, deterministic repeatability, and ValueError rejection of non-positive rates, weights, MTBF values and capabilities, invalid item counts, empty lists and length mismatches. All numeric asserts are tolerance-based (assertAlmostEqual / math.isclose), never exact equality on computed sums.

Compliance

• Standards referenced, not reproduced: ARP4761A and ARP4754A are proprietary SAE documents (sae.org/standards); the apportionment relations above are standard engineering methodology, summary-only per standards-map.yaml, with both ids marked reference-only.
• compliance: STANDARDS-REF, gated: false.