Maintainability and Availability: Method

Maintainability and Availability

What the tool answers

Reliability says how often a system fails; maintainability says how quickly it is put back; availability is the fraction of the time it is ready. The tool starts from the maintenance tasks of a system (one row per replaceable item or repair action) and gives the three groups of numbers a maintainability engineer reports: the predicted mean, median and maximum corrective maintenance times; the inherent, achieved and operational availabilities with the maintenance man-hours per operating hour; and, for a redundant group, the availability and mean time to failure that repair buys. Two support questions follow: how many spares the lead time needs, and how large a demonstration test must be to prove the MTTR. The methods are those of MIL-HDBK-470A (1997) and MIL-HDBK-338B (1998), whose worked examples the tool reproduces (test-scripts/test_maintainability_availability.php, 256 checks).

Repair-time distributions

Each task has a failure rate λ_i (or MTBF_i = 1/λ_i) and a repair-time distribution. MIL-HDBK-470A §4.4.3 uses three: the lognormal, which fits most repair actions made of several sub-tasks of unequal length (ln t normal with mean θ and standard deviation σ; mean e^{θ + σ²/2}, median e^θ, standard deviation mean·√(e^σ² − 1)); the normal, for remove-and-replace tasks that always take about the same time; and the exponential, for tasks whose completion has no memory (substitution fault isolation), with median 0.693 M̄ and standard deviation equal to the mean. The repair time of a task may be entered as its mean or its median; for the lognormal they differ by e^σ²/2. When no repair data exist MIL-HDBK-338B §5.6.2.4 quotes σ = 0.55 as the value prediction procedures have suggested, which is the default of the table; the handbook's own ground-electronic example (Table 5.6-3) has σ = 1.11.

MTTR, median and maximum repair time

The mean time to repair of the system is the failure-rate-weighted mean of the task means, MTTR = Σλ_i M̄_i / Σλ_i (MIL-HDBK-338B eq. 5.75, MIL-HDBK-470A eq. 9): a task that fails often counts for more. The maintainability function M(t), the probability that a corrective action is finished within t, is the same mix of the task functions, M(t) = Σ w_i M_i(t), w_i = λ_i/Σλ_i. The median and the maximum times M_max (the 90 % and 95 % points, Table 5.6-2) are exact for each task (e^{θ + zσ}, M̄ + z s, −M̄ ln(1 − p)) and, for the mixture, found by numerical inversion of M(t) by bisection; the mixture's 95 % point is not the weighted mean of the task points. The maintenance man-hours per repair are Σ w_i crew_i M̄_i, and per operating hour λ times that plus the preventive-maintenance man-hours f M_pt crew_pm (the maintenance ratio MMH/OH).

The availability family

Three definitions are in use, and they answer different questions (MIL-HDBK-470A §2.2.1, MIL-HDBK-338B §10.4.1):

Inherent availability A_i = MTBF/(MTBF + MTTR) counts failures and active repair only: what the design delivers with an ideal support system. With the maintenance time ratio α = MTTR/MTBF it is 1/(1 + α), so any MTBF / MTTR pair with the same ratio gives the same A_i, the trade-off of MIL-HDBK-338B §5.8. For a series system the handbook uses the system MTBF 1/Σλ and the weighted MTTR; the page also prints the product of the item availabilities Π MTBF_i/(MTBF_i + M̄_i), the value when every item keeps failing and being repaired independently while the system is down (slightly lower, 0.735 against 0.752 in the default case).

Achieved availability A_a = MTBM/(MTBM + M̄) adds scheduled maintenance: MTBM = 1/(λ + f) is the mean time between any maintenance action (f = 1/T_pm the preventive rate) and M̄ = (λ MTTR + f M_pt)/(λ + f) the mean active maintenance time of corrective and preventive actions together (eq. 10.21).

Operational availability A_o = (MTBM + RT)/(MTBM + RT + MDT) adds the delays: MDT = M̄ + mean logistics delay + mean administrative delay is the mean downtime per event, and RT the ready time of the cycle (eq. 10.22). With RT = 0 it is A_o = MTBM/(MTBM + MDT), the form of MIL-HDBK-470A eq. 2 and of the DoD RAM-C Rationale Report Manual (2009), whose materiel availability A_M = uptime/(uptime + downtime) is the same quotient over a fleet. The task-table delays apply to corrective events, the system-level delay to every event.

A_i ≥ A_a ≥ A_o always. MIL-HDBK-338B §10.4 Example 3 (MTBF 100 h, MTTR 0.5 h, M_pt 0.25 h, delays 0.3 + 0.4 h, MTBM 75 h, RT 20 h) gives 0.995 / 0.99 / 0.985, which the tool's test reproduces.

Redundant groups: the Markov model

For n identical units of which k must work, with r repair channels, the state is the number of failed units j = 0 … n. A failure occurs at rate (n − j) λ and a repair at min(j, r) μ (MIL-HDBK-338B §5.7.2: exponential times, one event at a time, failed units wait for a free channel). The steady state solves π Q = 0, Σ π_j = 1 by Gaussian elimination and is checked against the birth-death product form π_j = π₀ Π_i<j (n − i) λ / (min(i + 1, r) μ). The availability is the probability of the up states (j ≤ n − k). For a 1-of-2 pair with one crew this is the handbook's (μ² + 2λμ)/(μ² + 2λμ + 2λ²) (§5.8.1, 0.946 for λ = 0.04, μ = 0.2). Three further numbers come from the same chain: the mean time to the first group failure from all units up, by the absorbing chain on the up states ((3λ + μ)/2λ² for 1-of-2; a second crew changes nothing there because it only acts once two units are down); the steady-state failure frequency ν = Σ π_i q_ij over the up-to-down transitions; and the mean up time A/ν and mean down time (1 − A)/ν per cycle. The mean up time is shorter than the first-failure MTTF because after a repair the group restarts with units still down.

The instantaneous availability A(t) from the all-up state is integrated by fourth-order Runge-Kutta with a sub-step that keeps the stiff chain stable; for a single unit it is the handbook's A(t) = μ/(λ + μ) + λ/(λ + μ) e^{−(λ + μ)t} (eq. 5.110), whose transient is gone by about 4/(λ + μ) (eq. 5.112). The interval availability over (0, T) is the time average of A(t) (eq. 10.23). The chart also shows the reliability of the same group without repair, Σ C(n, i) pⁱ (1 − p)^{n − i} with p = e^−λt: the gap between the two curves is what repair buys.

Spares

With a constant failure rate the demand for an item over the lead time L from N systems is Poisson with mean m = N λ_i L, and a stock s covers it with probability Σ_x≤s m^x e^−m/x! (MIL-HDBK-338B eq. 5.58; the handbook's example λ = 0.001, t = 500, two spares, 0.986). The tool finds the smallest s that reaches the entered fill rate, the expected backorders EBO(s) = Σ_x>s (x − s) P(x) and the item's supply availability 1 − EBO/N, the fraction of the systems not waiting for it (Sherbrooke's approximation for one item per system); the product over the items is the fleet's supply availability. The fill rate is per item, so the probability that every line is covered is lower.

Demonstration test plan

MIL-HDBK-470A Appendix B carries the fixed-sample test methods of MIL-STD-471A. Test Method 1 tests H₀: mean = μ₀ (the specified value) against H₁: mean = μ₁ > μ₀ (the maximum tolerable value) with producer's risk α and consumer's risk β, relying on the normality of the sample mean. Plan A assumes lognormal times with a prior variance σ² of ln t, so the standard deviation of the times is μ √(e^σ² − 1), and needs n = (Z_α μ₀ + Z_β μ₁)² / (μ₁ − μ₀)² (e^σ² − 1) tasks (eq. B-4); Plan B is distribution-free with a prior standard deviation d and needs n = [(Z_α + Z_β) / ((μ₁ − μ₀)/d)]² (eq. B-5). The equipment is accepted when the sample mean X̄ ≤ μ₀ + Z_α d̂/√n. At least 30 tasks are tested; the handbook rounds n to the nearest integer (56 and 43 in its 30 / 45 minute example with σ² = 0.6 and d = 30 min), and the page shows the rounded-up value beside. The operating-characteristic curve is the probability of acceptance against the true mean: 1 − α at μ₀, β at μ₁, and in between Φ[(C − μ)√n / d(μ)], which for the handbook's example reads 0.21 at 40 minutes, the value its Figure B-2 shows.

Test Method 9 needs no prior variance: the mean is accepted when the specified μ_c is at least X̄ + φ d̂/√n (φ the deviate of the consumer's risk), and M_max when the specified value is at least exp(mean ln X + ψ s_{ln X}), ψ = 1.645 or 1.282 for the 95 % or 90 % point (valid for lognormal corrective times). The page applies both criteria to the task table as if the sample behaved like the prediction, which shows whether a specification can be demonstrated with the sample size in hand.

Structure expression

Left blank, the tasks form a series system (MTBF = 1/Σλ, every failure a system failure). A structure in the System Reliability tool's notation (Sub1 and (Sub2 or Sub3) and 2of3(Sub4, Sub5, Sub6)) evaluates the mission reliability and the inherent and operational availabilities through the block diagram with every item repaired independently; the MTTR and man-hours stay the series figures, since every item failure is a maintenance action whether or not the system goes down.

Limits

Failure rates are constant (useful-life phase); repair, delay and preventive-maintenance times enter by their means except where the distribution is used (M_max, M(t), the demonstration plan). The Markov group takes identical units and ignores switching failures, imperfect coverage and the failure of idle units at a different rate. Spares assume a Poisson demand with no cannibalization and a constant lead time. The demonstration plans use the normal approximation of the sample mean, which the handbook accepts from 30 tasks upward; the sequential plans and the percentile tests of the other methods are not built.