1. Introduction to Reliability Engineering

Reliability engineering is the discipline of ensuring that systems and components perform their intended functions over a specified period under stated conditions. It integrates probability theory, statistics, and engineering to predict, analyze, and improve product performance over time.

Key applications include:

2. Basic Reliability Concepts

Definitions

Reliability: The probability that a system will perform its intended function for a specified time period under stated conditions.

Failure: The termination of the ability of an item to perform its required function.

Life Data: Observations of times-to-failure or times-to-event.

Key Metrics

Mean Time To Failure (MTTF)
MTTF = E[T] = ∫0∞ R(t) dt
For non-repairable items. Represents the average time until failure.
Mean Time Between Failures (MTBF)
MTBF = Total Operating Time / Number of Failures
For repairable systems. Includes repair time in the cycle.

3. Reliability Functions

Reliability Function R(t)

R(t) = P(T > t) = 1 - F(t) = ∫t∞ f(u) du
Probability that the item survives beyond time t. Also called the survival function S(t).

Cumulative Distribution Function F(t)

F(t) = P(T ≤ t) = 1 - R(t)
Probability of failure by time t (unreliability).

Probability Density Function f(t)

f(t) = dF(t)/dt = -dR(t)/dt

Hazard Function h(t)

Hazard Rate (Failure Rate)
h(t) = f(t) / R(t) = -d[ln R(t)] / dt
Instantaneous failure rate at time t, given survival to time t. Units: failures per unit time.

Cumulative Hazard Function

H(t) = ∫0t h(u) du = -ln R(t)
R(t) = exp[-H(t)]

4. Exponential Distribution

The exponential distribution is the simplest lifetime distribution, characterized by a constant failure rate. It represents the "memoryless" property: the probability of failure in the next interval is independent of age.

Exponential Distribution Functions
f(t) = λ e-λt

R(t) = e-λt

h(t) = λ (constant)

MTTF = 1/λ
Where λ is the failure rate parameter.
When to Use Exponential
  • Electronic components during useful life
  • Complex systems with many components
  • Situations with random failures (no wear-out)

5. Weibull Distribution

The Weibull distribution is the most widely used distribution in reliability engineering due to its flexibility in modeling various failure patterns.

Two-Parameter Weibull
f(t) = (β/η)(t/η)β-1 exp[-(t/η)β]

R(t) = exp[-(t/η)β]

h(t) = (β/η)(t/η)β-1
β = shape parameter (determines failure pattern)
η = scale parameter (characteristic life, R(η) = 0.368)

Interpretation of Shape Parameter β

β Value Hazard Pattern Failure Mode
β < 1DecreasingEarly-life failures ("infant mortality")
β = 1ConstantRandom failures (exponential)
1 < β < 4IncreasingWear-out failures
β ≈ 3.5IncreasingApproximates normal distribution
β > 4Rapidly increasingRapid wear-out

Three-Parameter Weibull

R(t) = exp[-((t-γ)/η)β] for t ≥ γ
γ = location parameter (minimum life or failure-free period)
Example: Bearing Failure Analysis

Analysis of 50 bearing failures yields Weibull parameters:

  • β = 2.1 (increasing hazard, wear-out)
  • η = 15,000 hours

Calculate reliability at 10,000 hours:

R(10,000) = exp[-(10,000/15,000)2.1] = exp[-(0.667)2.1] = exp[-0.417] = 0.659

65.9% of bearings survive to 10,000 hours.

6. Lognormal Distribution

The lognormal distribution is appropriate when the logarithm of failure times is normally distributed. Common for failure modes involving fatigue, corrosion, and material degradation.

Lognormal Distribution
f(t) = (1 / tσ√(2π)) exp[-(ln(t) - μ)2 / (2σ2)]
μ = mean of ln(T)
σ = standard deviation of ln(T)
Median = eμ

7. Parameter Estimation

Maximum Likelihood Estimation (MLE)

MLE finds parameters that maximize the probability of observing the collected data.

Log-Likelihood for Weibull
ln L = r ln(β) - rβ ln(η) + (β-1) ∑ ln(ti) - (1/ηβ) ∑ tiβ
Where r = number of failures, and the sums include all failure times.

Probability Plotting

Graphical method using linearized CDF. For Weibull:

ln[-ln(1-F)] = β ln(t) - β ln(η)
Plot ln[-ln(1-F̂)] vs ln(t). Slope = β, intercept = -β ln(η)

Median Ranks

For plotting positions with n items:

F̂i ≈ (i - 0.3) / (n + 0.4)
Bernard's approximation for median ranks.

8. Handling Censored Data

Censored observations occur when the exact failure time is unknown. Types include:

MLE with Right Censoring

L = ∏failures f(ti) × ∏censored R(tj)
Failures contribute density; censored observations contribute survival probability.

9. Kaplan-Meier Estimator

The Kaplan-Meier (product-limit) estimator provides a non-parametric estimate of the survival function, handling censored data.

Kaplan-Meier Estimator
R̂(t) = ∏ti≤t (1 - di/ni)
ti = ordered failure times
di = number of failures at ti
ni = number at risk just before ti

Confidence Intervals

Greenwood's formula for variance:

Var[R̂(t)] = R̂(t)2 ∑ti≤t di / [ni(ni - di)]

10. System Reliability

Series Systems

System fails if any component fails:

Rsystem = ∏i=1n Ri

Parallel Systems

System fails only if all components fail:

Rsystem = 1 - ∏i=1n (1 - Ri)

k-out-of-n Systems

System operates if at least k of n components work:

Rsystem = ∑j=kn C(n,j) Rj (1-R)n-j
For identical components with reliability R.

11. Reliability Testing

Test Types

Sample Size Determination

For demonstration testing with zero failures:

n ≥ ln(1-C) / ln(R0)
C = confidence level
R0 = reliability to demonstrate
Example: To demonstrate R=0.90 at 95% confidence: n ≥ ln(0.05)/ln(0.90) = 28.4 ≈ 29

References and Further Reading

  1. Meeker, W.Q., Escobar, L.A., & Pascual, F.G. (2022). Statistical Methods for Reliability Data, 2nd Edition. Wiley.
  2. Nelson, W. (2004). Accelerated Testing: Statistical Models, Test Plans, and Data Analysis. Wiley.
  3. Abernethy, R.B. (2006). The New Weibull Handbook, 5th Edition.
  4. O'Connor, P. & Kleyner, A. (2012). Practical Reliability Engineering, 5th Edition. Wiley.
  5. Kaplan, E.L. & Meier, P. (1958). "Nonparametric Estimation from Incomplete Observations." JASA, 53, 457-481.
← Previous: Statistics Back to Resources →