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Who published Markov Analysis?

ashfordeOU (@ashfordeou) published this skill. Their other Agent Skills are listed on their SkillMD profile.

Markov Analysis by ashfordeou — Agent Skill

Name: Markov Analysis
Author: ashfordeOU

Markov Analysis

Use when you must run a quantitative Markov analysis for an aircraft system safety model per ARP4761A: compute continuous time Markov chain state probabilities from the transition rate matrix, evaluate two-state failure and repair availability with steady state limits, derive the non-repairable failure probability and the mean time to failure, sum series failure rates, and estimate redundant configuration reliability with k-out-of-n combinations. Produces the state probability vector, the availability, and the MTTF that gate the quantitative safety case for the failed state. Trigger: Markov analysis, Markov chain, state probability, transition rate, failure rate, repair rate, availability, MTTF, absorbing state, k-out-of-n.

ashfordeOU ec80625 3 files · 19.4 KB Updated Sep 17, 2026

File contents

ARP4761A Markov Analysis (systems-engineering-safety/arp4761a/markov-analysis)

Use when the task is quantitative safety or reliability modeling of a system with Markov analysis per ARP4761A: state probabilities from transition rates, two-state failure and repair availability, MTTF, and redundant configuration reliability.

Domain quick reference

A continuous time Markov chain (CTMC) models system states (for example fully operational, degraded, failed) with transitions at constant rates: failure rate lambda and repair rate mu, per hour.
The state probability vector evolves by dP/dt = P Q, where Q is the transition rate matrix with the row sums zeroed on the diagonal, so P(t) = P(0) exp(Q t) conserves total probability.
Two-state failure and repair model: P_failed(t) = lam/(lam+mu) * (1 - exp(-(lam+mu) t)); steady state unavailability is lam/(lam+mu) and availability is mu/(lam+mu).
Non-repairable model (absorbing failed state): R(t) = exp(-lam t), failure probability 1 - exp(-lam t), mean time to failure 1/lam.
Series chain total failure rate is the sum of the rates; an n-unit active redundancy without repair has MTTF (1/lam) * (1/n + 1/(n-1)
- ... + 1), so two units give 3/(2 lam).
At least k of n identical units surviving: sum over i from k to n of C(n,i) R^i (1-R)^(n-i) with R the per-unit reliability.
Markov analysis quantifies failure conditions whose probability is then compared against the severity-based requirement in the SSA.

Workflow

Define the states of the system (operational, degraded, failed) and the constant transition rates between them.
Build the transition rate matrix; diagonal entries are the negative row sums.
Compute the state probability vector at the mission time with state_probabilities.
For a two-state model, check availability with two_state_availability and the steady state limit.
For a non-repairable unit, compute the failure probability and MTTF with nonrepairable_probabilities and mttf_exponential.
Combine independent units with series_failure_rate and estimate redundant configurations with redundancy_mttf and k_of_n_reliability.
Compare the resulting failure probability against the safety requirement for the failure condition.

Pitfalls

Using the transition rate matrix with the diagonal filled instead of rebuilt as the negative row sum, which breaks probability conservation.
Treating a repairable model as non-repairable and quoting 1/lam as the MTTF when repair restores the state.
Summing failure rates for a redundant parallel pair as if the pair were in series, which understates the true reliability.
Forgetting that k-of-n reliability needs the per-unit reliability at the mission time, not the steady state availability.
Reporting a state probability vector that does not sum to 1 as evidence of a modeling error, when the chain conserves probability by construction.

Behavior contract (gate 3)

The Markov logic is exercised by the gate 3 contract test: scripts/test_markov_analysis.py against scripts/markov_analysis_logic.py (stdlib unittest, offline). Run: python3 scripts/test_markov_analysis.py

Compliance

Standards referenced, not reproduced: ARP4761A text is proprietary (SAE); summary-only per standards-map.yaml and brief 06.
Markov analysis is common reliability methodology (Annex L of ARP4761A), paraphrased; no verbatim standard text.
compliance: STANDARDS-REF, gated: false.

---name: markov-analysisdescription: Use when you must run a quantitative Markov analysis for an aircraft system safety model per ARP4761A: compute continuous time Markov chain state probabilities from the transition rate matrix, evaluate two-state failure and repair availability with steady state limits, derive the non-repairable failure probability and the mean time to failure, sum series failure rates, and estimate redundant configuration reliability with k-out-of-n combinations. Produces the state probability vector, the availability, and the MTTF that gate the quantitative safety case for the failed state. Trigger: Markov analysis, Markov chain, state probability, transition rate, failure rate, repair rate, availability, MTTF, absorbing state, k-out-of-n.license: Apache-2.0---# ARP4761A Markov Analysis (systems-engineering-safety/arp4761a/markov-analysis)Use when the task is quantitative safety or reliability modeling of asystem with Markov analysis per ARP4761A: state probabilities fromtransition rates, two-state failure and repair availability, MTTF,and redundant configuration reliability.## Domain quick reference- A continuous time Markov chain (CTMC) models system states (for example fully operational, degraded, failed) with transitions at constant rates: failure rate lambda and repair rate mu, per hour.- The state probability vector evolves by dP/dt = P Q, where Q is the transition rate matrix with the row sums zeroed on the diagonal, so P(t) = P(0) exp(Q t) conserves total probability.- Two-state failure and repair model: P_failed(t) = lam/(lam+mu) * (1 - exp(-(lam+mu) t)); steady state unavailability is lam/(lam+mu) and availability is mu/(lam+mu).- Non-repairable model (absorbing failed state): R(t) = exp(-lam t), failure probability 1 - exp(-lam t), mean time to failure 1/lam.- Series chain total failure rate is the sum of the rates; an n-unit active redundancy without repair has MTTF (1/lam) * (1/n + 1/(n-1) + ... + 1), so two units give 3/(2 lam).- At least k of n identical units surviving: sum over i from k to n of C(n,i) R^i (1-R)^(n-i) with R the per-unit reliability.- Markov analysis quantifies failure conditions whose probability is then compared against the severity-based requirement in the SSA.## Workflow1. Define the states of the system (operational, degraded, failed) and the constant transition rates between them.2. Build the transition rate matrix; diagonal entries are the negative row sums.3. Compute the state probability vector at the mission time with state_probabilities.4. For a two-state model, check availability with two_state_availability and the steady state limit.5. For a non-repairable unit, compute the failure probability and MTTF with nonrepairable_probabilities and mttf_exponential.6. Combine independent units with series_failure_rate and estimate redundant configurations with redundancy_mttf and k_of_n_reliability.7. Compare the resulting failure probability against the safety requirement for the failure condition.## Pitfalls- Using the transition rate matrix with the diagonal filled instead of rebuilt as the negative row sum, which breaks probability conservation.- Treating a repairable model as non-repairable and quoting 1/lam as the MTTF when repair restores the state.- Summing failure rates for a redundant parallel pair as if the pair were in series, which understates the true reliability.- Forgetting that k-of-n reliability needs the per-unit reliability at the mission time, not the steady state availability.- Reporting a state probability vector that does not sum to 1 as evidence of a modeling error, when the chain conserves probability by construction.## Behavior contract (gate 3)The Markov logic is exercised by the gate 3 contract test:scripts/test_markov_analysis.py againstscripts/markov_analysis_logic.py (stdlib unittest, offline). Run:python3 scripts/test_markov_analysis.py## Compliance- Standards referenced, not reproduced: ARP4761A text is proprietary (SAE); summary-only per standards-map.yaml and brief 06.- Markov analysis is common reliability methodology (Annex L of ARP4761A), paraphrased; no verbatim standard text.- compliance: STANDARDS-REF, gated: false.

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