Introduction to Reliability and Maintainability

Drag the shape (β) and scale (η) sliders, or click anywhere on a chart to move the marker x. All four curves — PDF, CDF, Reliability, Hazard — update live from the same Weibull model, so you can see how differentiation/integration link them in real time.

Lesson

Introduction to Reliability & Maintainability — Contents

What Is Reliability? What Is Maintainability? — plain-language start
Repairable vs. Non-Repairable Systems
Core Calculus Relationship — interactive PDF/CDF/R/h visualizer
Core Calculus Relationship — written derivation & summary chain
The Bathtub Curve — hazard rate over a product's life
Failure Rate vs. ROCOF — two different "rates"
Maintainability — the M(t) function & MTTR
Availability — combining reliability and maintainability
Statistics Implementation — Excel, Minitab, and exam tips
Index Summary — non-repairable vs. repairable metrics

What Is Reliability? What Is Maintainability?

Start here — plain-language definitions, no formulas yet. Everything later in this lesson is just these two ideas made precise.

Reliability

Reliability is the probability that something keeps working, without failing, for a stated length of time, under stated conditions.

Say a batch of light bulbs is rated "90% reliable at 1,000 hours." That means if you ran a lot of them for 1,000 hours, about 90% would still be working and 10% would have burned out. Reliability always comes with a time attached — a bulb that's 90% reliable at 1,000 hours will be less reliable at 2,000 hours, because more time means more chances to fail.

Reliability answers: "What's the chance this still works after time t?"

Maintainability

Maintainability is the probability that something broken gets fixed and working again within a stated length of time, once repair begins.

Say a machine is "80% maintainable within 4 hours." That means if it breaks down, there's an 80% chance the repair crew has it running again within 4 hours of starting the fix. The other 20% of the time, the repair drags on longer — parts aren't on hand, the fault is hard to find, and so on.

Maintainability answers: "What's the chance it's fixed within time t?"

Notice these two ideas are mirror images of each other. Reliability is about staying up; maintainability is about getting back up. Put them together and you get availability — the fraction of time something is actually usable, which depends on both how rarely it breaks and how quickly it gets fixed. A machine that almost never breaks but takes a month to repair when it does can end up less available than one that breaks often but is fixed in ten minutes. This lesson builds up to that idea at the end.

Why calculus shows up at all: "probability of still working by time t" is a curve that changes as t grows — it starts near 100% and drifts down toward 0% as more things get a chance to fail. Calculus is just the tool for describing curves and the rates at which they change. You don't need to be a mathematician to follow this — every formula below is introduced with the plain-language idea first, and the interactive charts let you watch the curves move instead of only reading the equations.

The vocabulary, defined before it's used

These five terms appear constantly in reliability work. Skim them now — they'll be defined again, more precisely, as each one comes up later in the lesson.

Term	Plain-language meaning
TTF	Time To Failure — how long a single item ran before it broke. Only makes sense for something that gets scrapped, not repaired.
MTTF	Mean Time To Failure — the average TTF across many identical items. "On average, how long do these last?"
MTBF	Mean Time Between Failures — for something that gets repaired and put back in service, the average time from one failure to the next.
MTTR	Mean Time To Repair — the average time it takes to fix something once repair starts.
Hazard rate	How likely something is to fail right now, given that it has survived up to this point. Not the same as "how likely to fail overall" — it's a conditional, moment-by-moment risk.

One more distinction worth having in mind from the very start, because it decides which of these terms applies:

Non-repairable item

Fails once, gets thrown away or replaced — never fixed. Think: a light bulb, a fuse, a weld test coupon. Described using TTF / MTTF.

Repairable item

Fails, gets fixed, goes back into service — this can happen many times over its life. Think: a pump, a rolling mill stand, a furnace. Described using MTBF / MTTR.

The next section digs into this split in more detail with a timeline diagram. After that, the lesson works through the calculus of reliability for a single non-repairable item first (it's the simpler case — one item, one failure, done), then returns to repairable systems to cover MTBF, repair-time behavior (maintainability), and finally availability.

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Repairable vs. Non-Repairable Systems

Before any formulas, the single most important fork in reliability engineering: does this item get thrown away when it fails, or fixed and put back into service? The answer changes which index you track for the rest of this lesson.

Non-repairable

Once it fails, it is discarded or replaced — never restored to service. There is exactly one Time-to-Failure (TTF) per unit, so a population of units gives you a distribution of TTFs. This is precisely the f(x)/F(x)/R(x)/h(x) machinery above.

Examples: a light bulb, a weld coupon, a fuse, a single bearing (if scrapped rather than reworked), a tensile test specimen.

Governing index: MTTF — Mean Time To Failure, \(E[T]=\int_0^\infty R(t)\,dt\).

Repairable

Once it fails, it is restored to an operating state (repaired, refurbished) and returned to service. A single unit now generates a sequence of up-times and down-times over its life — a point process of failures, not one number.

Examples: a rolling mill stand, a reheat furnace, a pump, a CNC line — almost all production equipment.

Governing indices: MTBF (Mean Time Between Failures), MTTR (Mean Time To Repair), and Availability.

Life of a repairable asset: alternating Up and Down states. Each up interval is one Time Between Failures (TBF); each down interval includes active repair (MTTR) plus any logistics/admin delay. A non-repairable item would only ever show the first green segment, ending permanently at the first red mark.

The down interval is rarely just "repair" — it usually splits into MTTD (Mean Time To Diagnose: find the fault) and MTTR (Mean Time To Repair: fix it once found), sitting between First Failure and End Repair. MTBF spans the whole cycle — diagnose + repair + the following correct-behavior interval — from one failure to the next. Driving MTTD down (better instrumentation, clearer alarms, condition data) is often cheaper than driving MTTR down, and both shrink the down interval that Availability above is built from.

Because repair resets the clock, reliability engineers distinguish "as good as new" repair (full renewal — the classic renewal-process assumption, appropriate for a full bearing/roll change) from "as good as old" minimal repair (fix the immediate fault only, wear state otherwise unchanged — often the realistic case for a quick swap of a sensor or seal on an otherwise worn machine). This distinction is exactly why repairable systems eventually need a different rate function than h(x) (see ROCOF later in this lesson) — but first, the next two sections build the core calculus (f, F, R, h) that describes a single non-repairable item's life. That math is the foundation everything else in this lesson builds on, so it's worth working through carefully before moving on to repairable systems.

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Now the math

With reliability, maintainability, and the repairable/non-repairable split defined above, here's the calculus that makes "probability of still working" precise for a single non-repairable item — drag the sliders and watch it move.

What's actually being plotted: the charts below use the Weibull distribution, the standard model for time-to-failure in reliability engineering, because two knobs let it produce all three bathtub-curve phases (decreasing, flat, or increasing risk over time):

\[ f(x) = \frac{\beta}{\eta}\left(\frac{x}{\eta}\right)^{\beta-1} e^{-(x/\eta)^{\beta}} \]

η (eta) — the scale parameter: roughly, the "typical" lifetime. Bigger η stretches everything further out in time; it doesn't change the shape of the risk, just how far along the time axis it happens.
β (beta) — the shape parameter: controls whether risk falls, stays flat, or rises with age — this is the exact same β that defines the three zones of the bathtub curve later in this lesson.

You don't need to memorize this formula or plug numbers into it by hand — treat β and η as knobs, move the sliders, and watch how the four curves respond. The formula is here so the curves aren't a black box.

Model Controls

Shape β (beta)2.00

Scale η (eta)100

Marker position x80

Tip: click anywhere on any chart to the right to jump the dashed marker to that x — no need to use the "Marker position" slider.

Live values at x = 80

f(x) — PDF

F(x) — CDF

R(x) — Reliability

h(x) — Hazard rate

H(x) — Cum. hazard

Check: f(x)/R(x) should equal h(x), and e^−H(x) should equal R(x).

f(x) — PDF

slope of F(x) at x · f(x) = dF/dx

In plain terms: how densely failures are clustered right around this exact moment.

F(x) — CDF

area under f(x) up to x · F(x) = ∫f dt

In plain terms: what fraction of units have failed by this point.

R(x) — Reliability

mirror of CDF · R(x) = 1 − F(x)

In plain terms: what fraction of units are still working at this point.

h(x) — Hazard rate

conditional rate · h(x) = f(x)/R(x) = −d(ln R)/dx

In plain terms: the risk of failing right now, for units that have made it this far without failing.

R(x) vs F(x) — Overlap

R(x) + F(x) = 1 at every x · crossover marks the median life (F=R=0.5)

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Core Calculus Relationship

The written derivation behind the interactive panels above.

How the PDF and CDF are connected

The probability density function (f(x)) and cumulative distribution function (F(x)) are two views of the same continuous probability model. The PDF shows where values are concentrated; the CDF keeps a running total of probability from the far left up to x.

\[ F(x)=\int_{-\infty}^{x} f(u)\,du \qquad\text{and}\qquad f(x)=\frac{dF(x)}{dx} \]

PDF ↔ CDF Explorer

Move x₁ and x₂ with the sliders, or drag either labelled marker directly on either live plot. The shaded PDF area, CDF increase, and live values stay synchronized.

Mean μ0.0

Standard deviation σ1.0

Lower bound x₁−1.0

Upper bound x₂1.0

P(x₁ < X ≤ x₂)

68.27%

F(x₂)

0.8413

f(x₂) = CDF slope

0.2420

Probability density · f(x)

The shaded area between x₁ and x₂ is the interval probability.

Cumulative probability · F(x)

The vertical increase ΔF equals the shaded PDF area; the tangent slope at x₂ equals f(x₂).

For μ = 0.0 and σ = 1.0, the probability between x₁ = −1.0 and x₂ = 1.0 is 68.27%.

What each graph means

The bell-shaped curve is the PDF, (f(x)). Its height describes the concentration or density of possible values around x.
The S-shaped curve is the CDF, (F(x)). Its height is the total probability accumulated up to x.
The slope of the CDF at any point equals the height of the PDF at the same point.

Therefore, when (f(x)) is small, (F(x)) is nearly flat. As (f(x)) increases, (F(x)) becomes steeper. Where (f(x)) reaches its peak, (F(x)) has its steepest slope. As (f(x)) falls, (F(x)) continues increasing, but more slowly. The PDF eventually approaches zero while the CDF approaches 1.

Probability between x₁ and x₂

The probability of falling between two values is the area under the PDF over that interval, which is also the change in CDF height:

\[ P(x_1<X\le x_2)=\int_{x_1}^{x_2}f(x)\,dx=F(x_2)-F(x_1) \]

Important diagram correction: a probability interval is the shaded area under the PDF, not the area under the CDF. An area under the CDF would represent (int F(x)\,dx), which is not (P(x_1<X\le x_2)).

Quantity at x₂	Meaning
f(x₂)	PDF height and CDF slope at x₂
F(x₂)	Total probability accumulated up to x₂
F(x₂) − F(x₁)	Probability between x₁ and x₂

Reliability interpretation

When (T) is time to failure, (F(t)=P(T\le t)) is the probability that the unit has failed by time t, while (R(t)=1-F(t)=P(T>t)) is the probability that it is still operating. The failure density connects both curves:

\[ f(t)=\frac{dF(t)}{dt}=-\frac{dR(t)}{dt} \]

Bottom line: (F(x_2)) is the accumulated probability up to x₂, (f(x_2)) is the CDF's slope at x₂, and (F(x_2)-F(x_1)) is the probability between the two draggable bounds. The same calculus relationship applies to the Weibull failure-time model used elsewhere in this lesson.

CDF → PDF via differentiation

\[ f(x) = \frac{d}{dx}F(x) \]

The PDF is the instantaneous rate of change of the CDF — it tells you how fast probability is accumulating at any point x.

PDF → CDF via integration

\[ F(x) = \int_{-\infty}^{x} f(t)\,dt \]

The CDF is the accumulated area under the PDF up to point x — i.e., \(F(x) = P(X \le x)\).

So differentiation and integration are inverse operations linking the two: F(x) is the "running total," f(x) is the "instantaneous slope" of that running total.

Extending to Reliability

Reliability function R(x) — probability a unit survives past time x:

\[ R(x) = 1 - F(x) = \int_x^{\infty} f(t)\,dt \]

It's just the complement/mirror of the CDF — the unaccumulated area (survivors), rather than the accumulated area (failures).

Since R(x) = 1 − F(x):

\[ f(x) = -\frac{d}{dx}R(x) \]

The PDF is also the negative derivative of the reliability function — the instantaneous rate of failure (in absolute terms, unconditioned on survival).

Hazard Rate — the conditional rate

This is the piece that ties it all together. The hazard rate h(x) is the instantaneous failure rate given the unit has survived to time x — a conditional density, not an unconditional one:

\[ h(x) = \frac{f(x)}{R(x)} = \frac{f(x)}{1-F(x)} \]

Why divide by R(x)? f(x) tells you the rate of failure across the whole original population. But by time x, some units have already failed and left the population — only R(x) fraction remain "at risk." Dividing by R(x) rescales the rate to apply only to survivors — same logic as conditional probability.

Key identity (since \(f(x) = -R'(x)\)):

\[ h(x) = \frac{-R'(x)}{R(x)} = -\frac{d}{dx}\ln R(x) \]

This means the hazard rate is the negative logarithmic derivative of the reliability function — a very clean link back to differentiation.

Tying it back together (integrate to recover R)

Integrating h(x) recovers the cumulative hazard function H(x):

\[ H(x) = \int_0^x h(t)\,dt = -\ln R(x) \]

\[ R(x) = e^{-H(x)} = \exp\left(-\int_0^x h(t)\,dt\right) \]

Summary Chain

Quantity	Relationship	Interpretation
f(x)	dF/dx	instantaneous rate of failure (unconditional)
F(x)	∫f dt	cumulative probability of failure
R(x)	1 − F(x)	probability of survival
h(x)	f(x)/R(x) = −d(ln R)/dx	instantaneous failure rate given survival so far
H(x)	∫h dt = −ln R(x)	cumulative hazard
R(x)	e^(−H(x))	recovers reliability from cumulative hazard

Practically, for Weibull (common in rolling-mill/pipe reliability work): with shape β and scale η,

\[ h(x) = \frac{\beta}{\eta}\left(\frac{x}{\eta}\right)^{\beta-1} \]

β < 1 gives a decreasing hazard (infant mortality/early defects), β = 1 is constant hazard (random failures, exponential), β > 1 is increasing hazard (wear-out). This is the same f/F/R/h chain above, just with the Weibull forms plugged in — useful for failure analysis on tube/pipe test data or bearing life in mill equipment.

Worked example

Take β = 2, η = 100 (a fairly typical wear-related shape) and evaluate everything at x = 50 — halfway to the scale parameter. This matches the default sliders in the interactive panel above, so you can set the marker to x = 50 and check these numbers against the live readout.

\[ \begin{aligned} f(50) &= \frac{2}{100}\left(\frac{50}{100}\right)^{1} e^{-(50/100)^2} \approx 0.0078 \\ F(50) &= 1 - e^{-(50/100)^2} \approx 0.2212 \\ R(50) &= e^{-(50/100)^2} \approx 0.7788 \\ h(50) &= f(50)/R(50) \approx 0.0100 \\ H(50) &= (50/100)^2 = 0.25 \end{aligned} \]

Sanity checks worth doing by hand once: F(50) + R(50) should equal 1 (0.2212 + 0.7788 = 1.0 ✓); \(e^{-H(50)}\) should equal R(50) (\(e^{-0.25}\approx 0.7788\) ✓). Those two identities are the whole point of this section — if they don't hold, one of the numbers is wrong.

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The Bathtub Curve

The classic qualitative shape of h(x) over a product's life — three failure mechanisms, three slopes.

Same h(x) axis as the interactive panel above — this is the qualitative composite curve traced out if a single Weibull-like β shifts from <1 to ≈1 to >1 across a unit's life. Real equipment is often a mixture of failure modes rather than one clean Weibull, which is why the observed bathtub is usually smoother and asymmetric.

For your rolling-mill/pipe context: infant mortality shows up as early-life weld or roll-surface defects caught in the first production runs after a change; useful life is the steady, largely random failure regime maintenance strategies are built around (this is where MTBF-based planning is valid); wear-out is where condition monitoring (vibration, wall-thickness loss, hardness drift) earns its keep — the goal is to catch the upturn in h(x) and intervene before it, via planned replacement, not run-to-failure.

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Failure Rate vs. ROCOF

Same units (failures per unit time), two different meanings — mixing them up is one of the most common reliability-engineering errors.

Hazard rate h(x)

Defined for a non-repairable item (or population of identical non-repairable items). It is a property of a probability distribution: \(h(x)=f(x)/R(x)\). It describes the instantaneous failure risk of one unit that has never failed before, conditioned on having survived to x.

ROCOF ρ(t)

Rate of Occurrence Of Failures — defined for a repairable system's failure/repair history. It is the intensity function of a counting process: the expected number of failures per unit time at system age t, regardless of how many times it has already failed and been repaired.

\[ \rho(t) = \lim_{\Delta t \to 0} \frac{E[N(t+\Delta t) - N(t)]}{\Delta t} \]

where N(t) counts cumulative failures of the repairable system up to age t. If repairs are minimal ("as good as old"), the process is often modeled as a Non-Homogeneous Poisson Process (NHPP) — plainly, a statistical model for random events occurring over time, where the rate is allowed to drift up or down — with a Power-Law intensity — ρ(t) increasing means the system is deteriorating (aging faster than repairs can offset, a signal to move from repair to overhaul/replace); ρ(t) roughly constant means failures are occurring at a stable rate, and the classic constant-λ / exponential MTBF math is a reasonable approximation; ρ(t) decreasing means reliability growth (common right after commissioning, as early bugs are fixed out — a repairable-system echo of infant mortality).

Rule of thumb: if you're tracking a fleet of identical, discard-on-failure parts (bearings pulled and scrapped, test coupons), use h(x)/MTTF language. If you're tracking one asset's failure history over its operating life (a specific mill stand's trip log), use ROCOF/MTBF language — and check whether ρ(t) is trending before assuming a constant MTBF is valid.

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Maintainability

The mirror image of reliability: instead of "how long until it breaks," maintainability asks "how long until it's fixed."

Maintainability M(t) is defined exactly like a CDF, but over repair time T_repair instead of time-to-failure: it is the probability that an active repair, once started, is completed within time t.

\[ M(t) = P(T_{repair} \le t) \]

Repair-time data is typically right-skewed — most repairs are quick, but a long tail of difficult ones (parts not on hand, hidden damage found once the unit is opened up) pulls the mean well above the median. The lognormal distribution is the standard model for this reason (a sum of many small multiplicative delays tends to lognormal, by the same logic the CLT gives you a normal for additive delays):

\[ M(t) = \Phi\!\left(\frac{\ln t - \mu}{\sigma}\right), \qquad \text{median repair time} = e^{\mu} \]

MTTR is the mean of this distribution, \(MTTR = e^{\mu + \sigma^2/2}\) — because of the skew, MTTR is always somewhat larger than the median repair time, and a handful of very long repairs can move MTTR a lot without moving the median much at all. This is the same relationship as f(x) and F(x) above, just applied to repair duration instead of failure time.

Maintainability Controls

Median repair time (h)4.0

σ (spread / tail heaviness)0.60

Median repair time

MTTR (mean)

M(MTTR)

Because the distribution is right-skewed, M(MTTR) is usually above 50% — most repairs finish before the mean, dragged upward by a few long ones.

M(t) — Maintainability function

P(repair completed by time t) · lognormal model

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Availability

Where reliability (MTBF) and maintainability (MTTR) combine into the number operations actually cares about.

Inherent Availability is the steady-state fraction of time a repairable system is up, considering only active repair time (not logistics delays, not planned/preventive maintenance):

\[ A_i = \frac{MTBF}{MTBF + MTTR} \]

Availability type	Formula	What it adds over A_i
Inherent, A_i	MTBF / (MTBF + MTTR)	baseline — active repair time only
Achieved, A_a	MTBM / (MTBM + M̄)	+ preventive maintenance time (MTBM = mean time between all maintenance, M̄ = mean maintenance time, corrective + preventive)
Operational, A_o	MTBM / (MTBM + MDT)	+ logistics & administrative delay — MDT (Mean Down Time) ≥ MTTR; this is what the plant floor actually experiences

Availability Calculator

MTBF (hours)400

MTTR (hours)6

—

Availability

Unavailability

Downtime / year

Expected failures / yr

Up/Down cycle at this MTBF : MTTR ratio

■ up (MTBF) ■ down (MTTR) — segment widths scaled to the ratio, several cycles shown

Widening the MTTR slider (or shrinking MTBF) visibly fattens the red segments — the same lever operations pulls when it invests in spares, standard work, or faster diagnostics to push MTTR down rather than only chasing a longer MTBF.

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Statistics Implementation

Move from the concepts and live curves in this lesson to reproducible calculations in Excel and fitted reliability models in Minitab.

Excel Functions

Function	Syntax	Purpose	When to use it
WEIBULL.DIST	`=WEIBULL.DIST(t,beta,eta,TRUE)`	Returns the Weibull cumulative failure probability F(t).	Use when failure time follows a Weibull model. In Excel, the argument named alpha is the lesson's shape β and the argument named beta is the lesson's scale η.
WEIBULL.DIST	`=WEIBULL.DIST(t,beta,eta,FALSE)`	Returns the Weibull probability density f(t).	Use to compare how densely failures are expected around a selected time.
Weibull reliability	`=1-WEIBULL.DIST(t,beta,eta,TRUE)`	Returns R(t)=1−F(t).	Use for the probability that a non-repairable item survives beyond time t.
EXPON.DIST	`=EXPON.DIST(t,1/mean_time,TRUE)`	Returns an exponential CDF using rate λ=1/mean time.	Use only when a constant-rate model is defensible, such as a simple exponential repair-time approximation.
EXPON.DIST	`=EXPON.DIST(t,1/mean_time,FALSE)`	Returns the exponential probability density.	Use to inspect density under the same constant-rate assumption.
LOGNORM.DIST	`=LOGNORM.DIST(t,LN(median),sigma,TRUE)`	Returns the cumulative lognormal probability.	Use for right-skewed repair times when ln(repair time) is approximately normal, matching the maintainability visual in this lesson.
LOGNORM.DIST	`=LOGNORM.DIST(t,LN(median),sigma,FALSE)`	Returns the lognormal probability density.	Use to see where repair durations are most concentrated.
GAMMA	`=eta*GAMMA(1+1/beta)`	Calculates Weibull MTTF from scale η and shape β.	Use after a defensible two-parameter Weibull model has been fitted.

Excel Use Cases

Weibull survival at 80 hours

For β=2 and η=100 hours, enter =1-WEIBULL.DIST(80,2,100,TRUE). The result is about 0.5273, so the model predicts about 52.73% survival beyond 80 hours.

The matching MTTF is =100*GAMMA(1+1/2), or about 88.62 hours.

Repair and availability planning

If exponential repair time has MTTR=6 hours, =EXPON.DIST(4,1/6,TRUE) gives about 0.4866: a 48.66% chance of repair within four hours under that model.

For MTBF=400 hours and MTTR=6 hours, =400/(400+6) gives about 98.52% inherent availability.

Minitab Navigation

Choose a lifetime distribution: use Stat → Reliability/Survival → Distribution Analysis (Right Censoring) → Distribution ID Plot when the data contain exact failure times plus units that have not yet failed. Compare candidate distributions before committing to Weibull.

Estimate reliability: use Stat → Reliability/Survival → Distribution Analysis (Right Censoring) → Parametric Distribution Analysis, select Weibull only when the fit is defensible, enter the censoring column, and request survival probabilities or percentiles. Use the arbitrary-censoring version when observations are left- or interval-censored. If no parametric distribution fits adequately, use the corresponding nonparametric analysis.

Analyze repairable equipment: use Stat → Reliability/Survival → Repairable System Analysis → Parametric Growth Curve to estimate the mean cumulative function and ROCOF with a power-law or homogeneous Poisson process. This is the repairable-system route; it is not the same as fitting a hazard function to one non-repairable lifetime.

Exam Tips

For ASQ CSSBB and CQE preparation, connect each formula to the system type and the question being asked: Excel is useful for transparent hand calculations, while the Minitab routes above show how the same decisions are made with censored field data. In real work, document the distribution-fit evidence and operating assumptions instead of selecting a model from the formula alone.

Write R(t)=1−F(t) first. It prevents the common error of reporting cumulative failures as survivors.
Remember the Weibull shape rule: β<1 means decreasing hazard, β=1 means constant hazard, and β>1 means increasing hazard.
Use MTTF for a non-repairable population; use MTBF, MTTR, ROCOF, and availability for repairable systems.
Do not confuse hazard h(t), conditional on surviving to t, with ROCOF, which counts repeated failures from a repairable system.
For the simple steady-state approximation, inherent availability is MTBF/(MTBF+MTTR). State the assumptions before applying it to operational availability.
In real analysis, retain right-censored units. Discarding units that have not failed wastes exposure information and can bias the fitted life model.

Verified software references

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Index Summary

Non-repairable vs. repairable — same underlying calculus, different vocabulary.

Concept	Non-repairable system	Repairable system
Core time metric	TTF (Time To Failure)	TBF (Time Between Failures)
Central tendency	MTTF = ∫R(t)dt	MTBF
Rate function	h(x) — hazard rate	ρ(t) — ROCOF
"Survives to x" probability	R(x) = 1 − F(x)	not generally defined — use ρ(t) trend instead
Repair-time metric	n/a — item is scrapped	MTTR, M(t)
Combined figure of merit	R(x) alone	Availability A = MTBF/(MTBF+MTTR)
Typical distribution	Weibull (β shape)	Exponential (constant ρ) or NHPP power-law (trending ρ)
Example in your world	bearing sent to scrap, weld coupon, fuse	mill stand, reheat furnace, pump, CNC line

Bottom line: the same differentiation/integration relationship that links f, F, R, and h for a single non-repairable item reappears — in a modified, process-level form — once you move to repairable equipment: M(t) is a CDF over repair time, ρ(t) is a rate function like h(x) but for a renewal/counting process rather than a static distribution, and Availability is the ratio that finally lets you compare "how often it breaks" against "how fast it gets fixed" on one scale.

That's the full lesson. ↑ Back to contents to revisit any section, or scroll back up to the interactive panels and try your own numbers.

Introduction to Reliability and Maintainability

What Is Reliability? What Is Maintainability?

Reliability

Maintainability

The vocabulary, defined before it's used

Non-repairable item

Repairable item

Repairable vs. Non-Repairable Systems

Non-repairable

Repairable

Model Controls

Core Calculus Relationship

How the PDF and CDF are connected

PDF ↔ CDF Explorer

What each graph means

Probability between x₁ and x₂

Reliability interpretation

CDF → PDF via differentiation

PDF → CDF via integration

Extending to Reliability

Hazard Rate — the conditional rate

Tying it back together (integrate to recover R)

Summary Chain

Worked example

The Bathtub Curve

Failure Rate vs. ROCOF

Hazard rate h(x)

ROCOF ρ(t)

Maintainability

Maintainability Controls

Availability

Availability Calculator

Statistics Implementation

Excel Functions

Excel Use Cases

Weibull survival at 80 hours

Repair and availability planning

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