Unavailability⚠ unverified

Understanding the Parameters

• MTTR (mean time to repair) — now the numerator: unavailability is driven by how long the system stays down. Halving repair time roughly halves unavailability (for MTTR $\ll$ MTBF). Fast detection and repair are the most direct levers on downtime.

• MTBF (mean time between failures) — in the denominator: longer uptime between failures reduces the downtime fraction. For the common case MTTR $\ll$ MTBF, unavailability simplifies to $U \approx \text{MTTR}/\text{MTBF}$ — the ratio of repair time to time between failures.

• The output $U$ — the "down" fraction, small for good systems. It is more informative than availability at high reliability: reading $U = 3\times10^{-5}$ is clearer than $A = 0.99997$. Multiplied by a year it gives expected annual downtime ($U \times 8760$ h).

• Why it matters for safety. For a standby protective system (a safety valve, a backup generator, an emergency shutdown), unavailability is the probability of failure on demand — the chance the safeguard is down exactly when needed. This makes $U$ the core quantity in fault-tree and SIL (safety integrity level) analysis.

Derivation (Approaching a Proof)

Unavailability is defined as the complement of availability. From the renewal-cycle derivation of availability, the steady-state up-fraction is $A = \text{MTBF}/(\text{MTBF} + \text{MTTR})$, so the down-fraction is everything else:

$$U = 1 - A = 1 - \frac{\text{MTBF}}{\text{MTBF} + \text{MTTR}} = \frac{(\text{MTBF} + \text{MTTR}) - \text{MTBF}}{\text{MTBF} + \text{MTTR}} = \frac{\text{MTTR}}{\text{MTBF} + \text{MTTR}}. \qquad\blacksquare$$

For the usual regime of reliable, quickly-repaired systems (MTTR $\ll$ MTBF), the denominator is dominated by MTBF and

$$U \approx \frac{\text{MTTR}}{\text{MTBF}} = \lambda\,\text{MTTR},$$

using $\lambda = 1/\text{MTBF}$ (Failure Rate From Mtbf). This compact form — unavailability equals failure rate times repair time — is the workhorse of fault-tree analysis, where component unavailabilities combine to give system probability of failure on demand.

Dimensional check. MTBF and MTTR share time units, so $U$ is a dimensionless probability. $\checkmark$

History and Development

• Risk and safety analysis. Unavailability is the central quantity in probabilistic risk assessment (PRA) and fault-tree analysis, developed for nuclear, aerospace, and process industries. For safety systems the relevant number is the probability of failure on demand — precisely the unavailability.

• The $\lambda\,\text{MTTR}$ shortcut. The approximation $U \approx \lambda\,\text{MTTR}$ makes component unavailabilities easy to combine through fault trees (AND/OR gates), underpinning quantitative safety cases and SIL verification in standards like IEC 61508.

• Testing intervals. For standby systems tested periodically, average unavailability also depends on the test interval (a fault can lie hidden between tests), extending this steady-state formula — a key consideration in setting inspection schedules for safety equipment.

Related Concepts: Availability, Mtbf From Failure Rate, Failure Rate From Mtbf, Exponential Reliability, Weibull Reliability, Bathtub Curve

Notes: Registry calculator unavailability (unverified). $U = 1 - A = \text{MTTR}/(\text{MTBF}+\text{MTTR})$ — correct as shipped (computes $A$ internally). For MTTR $\ll$ MTBF, $U \approx \lambda\,\text{MTTR}$ — the fault-tree workhorse and the probability of failure on demand for standby safety systems. Registry labels MTBF/MTTR dimensionless (are times).