Demystifying Failure Rate Math for Reliability Success

Failure rate analysis is a crucial engineering skill for ensuring reliable products and systems. This guide will explore the mathematical foundations behind failure rate, how it is calculated from data, typical distributions used, and practical applications across industries. By the end, you‘ll have an expert grasp of this vital reliability metric.

Defining Failure Rate

The failure rate function λ(t) gives the instantaneous rate of failure at time t for units that have survived up to that point. Mathematically, it is defined as:

λ(t) = lim dt→0 (Probability of failure in interval [t, t + dt]) / dt

Note λ(t) is a conditional probability that accounts for surviving until time t. It allows modeling the chance of failure per unit time for the survivor population.

Another interpretation is that λ(t)dt gives the approximate fraction of units expected to fail in the interval [t, t + dt], given survival to time t. This makes intuitive sense – it is the failure probability per unit time.

Estimating Failure Rates from Data

In practice, the failure rate function is rarely known a priori. Instead, it must be estimated from lifetime or failure data using statistical techniques. Two common estimators are:

• Maximum Likelihood Estimation – Used to fit parameters of standard distributions like Weibull and Exponential.

• Kaplan-Meier Estimator – Non-parametric empirical estimator from failure times.

Consider the following example lifetimes for a sample of 12 units:

[450, 1350, 1050, 1950, 2250, 450, 750, 1450, 1550, 1950, 750, 750]

The Kaplan-Meier plot and empirical failure rate estimate for this data is shown below:

These estimators allow extracting failure rate distributions from field data.

Modeling Failure Rates over Life Cycle

Exponential and Weibull distributions are commonly used to model failure rates. But the most insightful is the bathtub curve, which captures how failure rate evolves over a unit‘s lifecycle:

The bathtub curve has three phases:

1. Infant mortality period – High initial failure rate due to manufacturing defects.

2. Useful life period – Low constant failure rate dominated by random failures.

3. Wear-out period – Increasing failure rate as units degrade from use.

Understanding these phases facilitates appropriate maintenance and design strategies. The bathtub curve demonstrates the need for burn-in screening to eliminate infant mortality failures. Identifying wear-out allows condition-based maintenance to prevent failure.

Industry Standards for Failure Rates

Reliability prediction standards are commonly used to estimate failure rates in product design stages or when field data is limited. Sources like MIL-HDBK-217 and Telcordia provide empirically derived component failure rates for different use environments.

For example, MIL-HDBK-217 lists a base failure rate of 0.0026 failures/million hours for a fixed carbon film resistor at 60% confidence. This empirical data forms the basis of predictive reliability models.

Physics of Failure for Proactive Reliability

The physics of failure approach focuses on understanding the underlying failure mechanisms through stress-strength interference and then predicting when failures will occur. It utilizes techniques like:

• Accelerated life testing – Exposing units to stresses like temperature, voltage, pressure higher than normal to uncover failure modes and quantify acceleration factors.

• Statistical distributions – Capture failure mechanism like metal fatigue via distributions like Weibull.

• Computer simulation – Physics-based models evaluate damage accumulation.

Physics of failure allows reliability prediction based on knowledge of failure mechanisms rather than just historical data. This facilitates design-stage analysis to reduce failures proactively.

Survival Analysis Methods

More advanced statistical techniques used in failure rate analysis and reliability engineering include:

• Competing risks modeling – Accommodate multiple failure modes acting simultaneously.

• Proportional hazards models – Assess effect of covariates like temperature or pressure on failure rates.

• Markov chains – Model sequence of events and probabilities over time.

These techniques provide deeper insight into the underlying failure dynamics beyond just a failure rate curve.

System Reliability and Failure Rates

For systems, component failure rates are aggregated based on reliability models that capture the architecture and failure dependencies. Common techniques include:

• Reliability block diagrams – Model dependencies between components with series/parallel configurations.

• Fault tree analysis – Deductive top-down model of system failure mechanisms.

• Failure modes and effects analysis (FMEA) – Methodically analyze component failure modes and their effects on the system.

This analysis facilitates prediction of system reliability from component data. Maintainability and availability analysis also leverage component failure rate inputs.

Applications Across Industries

Failure rate data enables quantitative reliability and availability targets across industries like:

• Aerospace – Aircraft engine components have stringent failure rate requirements. Analysis guides redundancy needs.

• Autos – Warranty predictions rely on expected field failure rates during service life.

• Electronics – Failure rate drives allowable defects per million and burn-in testing needs.

• Telecom – Redundancy, spare parts and maintenance time of network systems depends on component failure rates.

• Medical – Failure rates for implants like artificial joints help assess probability of re-operation.

The examples are endless – any engineered system or product benefits from knowing expected failure rates early in design and the field.

Using Failure Rates in Maintenance Optimization

Failure rates also facilitate optimizing maintenance strategies in operations:

• Failure rates help identify age where wear-out preventive maintenance is viable vs. just running equipment to failure.

• Condition monitoring systems like vibration analysis leverage failure rate data to prioritize testing frequency.

• Predictive maintenance forecasting relies on failure rate projections – the higher the failure rate, the more imminent failure is.

• Spare parts inventory levels can be set based on expected replacement rates driven by failure rates.

• Maintenance staffing needs depend directly on the overall asset failure rates.

Thus failure rates provide the crucial equipment health input for driving maintenance decision-making.

Software Reliability and Failure Rates

While hardware failure rates are often quoted in terms of failures per million hours, software has no explicit notion of operating time. However, concepts like:

• Defect density – Defects per thousand lines of code.
• Availability – Percentage uptime.

Provide analogous reliability metrics for software. Conversion between hardware and software failure rates can enable system level modeling.

In Summary

This article has provided an overview of failure rate math and statistics fundamentals. While the theory may seem complex, having an intuitive grasp of your products‘ or assets‘ expected failure rates provides invaluable engineering insight. Lifecycle failure rate analysis should be in every reliability engineer‘s toolkit to help ensure customers‘ or operations‘ success.