Markov Availability Models

State-based reliability modeling that captures degraded modes and repair dynamics an RBD cannot, including graceful-degradation rungs.

When structure is not enough

Reliability block diagrams assume components are simply up or down. Real plants have in-between states: degraded operation, awaiting-repair, in-repair, redundancy-exhausted. Markov models represent these as states with transition rates (failures and repairs), letting us compute the long-run fraction of time in each - including the degraded-but-serving states that graceful degradation creates.

A minimal model

Consider a unit with states UP, DERATE, DOWN. Failures move it down the ladder at rates driven by MTBF; repairs move it back up at rates driven by MTTR. The stationary distribution gives availability as time in UP plus a partial credit for time in DERATE.

import numpy as np
def stationary(Q):
    # solve pi Q = 0, sum(pi)=1  (Q rows sum to zero)
    n = Q.shape[0]
    A = np.vstack([Q.T, np.ones(n)])
    b = np.zeros(n + 1); b[-1] = 1.0
    pi, *_ = np.linalg.lstsq(A, b, rcond=None)
    return pi

def availability(pi, credit=(1.0, 0.5, 0.0)):
    return float(sum(p * c for p, c in zip(pi, credit)))  # DERATE gets 0.5

Why it matters for the gate

Because graceful degradation keeps the plant partly serving, a pure up/down model understates true availability while a naive model can overstate it. The Markov model credits derated service honestly, which is essential when arguing how close a fleet gets to the 0.99982 Tier III target. It still cannot make a single stressed-plug burner reach that target; it shows exactly how far short and why.

These transition rates come from MTBF/MTTR and the derate rungs from graceful degradation; the results feed the plant Monte-Carlo and fleet model.