# Frequency Analysis

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## Frequency Analysis

This material in this ebook describes risk quantification. It shows how to estimate how often incidents may occur. Topics covered include risk ranking, fault tree and event tree analysis, Monte Carlo simulations, Markov models, safeguards and Layer of Protection Analysis (LOPA). Also discussed are the sources and management of failure rate data.

The quotations, “What gets measured gets done” and “You become what you measure” are well known. This means that, if risk is to be properly understood and reduced to an acceptable level then quantification is needed.

The basic risk equation is:

RiskHazard = Consequence * Predicted Frequency............. (17.1)

This ebook discusses how a value for the Predicted Frequency term can be obtained (the distinctions between frequency, predicted frequency, probability and likelihood are explained in Chapter 1 of *Process Risk and Reliability Management*. Concepts covered include:

- The Pareto Principle: identifying the important few and ignoring the unimportant many;
- Fault trees;
- Event trees;
- Monte Carlo simulation;
- Markov models; and
- Layers of Protection Analysis.

## The Pareto Principle

In the late nineteenth century the Italian economist and misanthrope Vilfredo Pareto (1848 – 1923) famously noted that most of the wealth in a community was held by a small proportion of the population. From this insight he developed the 80/20 rule, or the Pareto Principle, which, in the case of community wealth, meant that about 20% of any population owns about 80% of the wealth.

His principle, which has no theoretical underpinning, is widely observed to be true in many fields of human activity (Juran 1951). It is applied here to the topic of risk analysis in the process industries.

The Pareto Principle can be expressed mathematically.

log n = c + (m * log x).................................................... (2)

where n is the number of items whose value is greater than x; c and m are constants.

Examples of the principle’s applications in an industrial context include:

- 80% of a company’s sales come from 20% of its customers.
- 80% of a company’s sales are made by 20% of the sales force.
- 20% of the workers are involved in 80% of the accidents.
- 20% of the equipment items cause 80% of the facility shutdowns.
- 20% of a company’s products will account for 80% of the total product defects.
- 80% of weight loss during a diet is achieved with the first 20% of the effort.

In each of the above examples, an ‘important few’ or the ‘vital few’ have a great impact on the business, whereas the ‘unimportant many’ are much less significant. Therefore a safety manager should direct his or her program toward that minority of incident-creating workers. Spending time on the ‘unimportant many’ is not likely to have much benefit.

One commonly-held misconception to do with the Pareto Principle is that 80% of the problems can be resolved with 20% of the resources. In fact the Principle makes no statement at all as to how much effort is needed to address the contributing factors.

## Table of Contents

Introduction

The Pareto Principle

Importance Ranking

Fault Tree Analysis

Gates

OR Gate

AND Gate

VOTING Gate

Events

Top Event

Intermediate Events

Base Events

House Event

Top-Down Development of a Fault Tree

1. Define the Top Event

2. Build the Tree

3. Identify the Cut Sets

4. Eliminate Repeat Sets

5. Eliminate Repeat Events in a Set

6. Eliminate Redundant Events

7. Quantify the Risk

Mathematics of an OR Gate

Mathematics of an AND Gate

Mathematics of a Voting Gate

Cut Set Quantification

8. Risk Rank

Event Contribution

Important Few

Unimportant Many

Power of the AND Gate

Importance Equalization

Cost-Benefit Analysis

Importance Ranking Using Cut Sets

Birnbaum Factor Method

Fussell-Vesely Method

Perturbation Method

Common Cause Events

Fukushima-Daiichi

Generic Fault Trees

Generic Safety Fault Tree

Generic Reliability Fault Tree

Discussion of the Fault Tree Method

Qualitative Fault Tree Analysis

Event Tree Analysis

Quantification of an Event Tree

Scope of Event

Combining Event Trees and Fault Trees

Short Sequence of Events

Many Events

Partial Success

Discrete Event Analysis

Non-Linearities and Complexities

Conveying Statistical Uncertainty

Monte Carlo Simulation

Random Number Generators

Seed Numbers

Speeding the Simulation

Markov Models

Top-Down / Bottom-Up Approach

Top-Down

Bottom-Up

Qualitative Insights

Limitations to Quantification

Mathematical Understanding

Value-Laden Assumptions

Lack of Exhaustivity

Human Behavior

Data Quality

Safeguards

Safeguard Level 1: Normal Operations

Safeguard Level 2: Procedural Safeguards

Safeguard Level 3: Safety Instrumented Systems

Safeguard Level 4: Mechanical Safeguards

Safeguard Level 5: Passive Safeguards

Safeguard Level 6: Emergency Response

Layer of Protection Analysis

The LOPA Process

Single Scenarios

IPLs

Specific

Independent

Dependable

Auditable

Human Response

Implementing LOPA

Team Makeup

Timing

Tools

Procedures and Inspections

Risk Criteria

Failure Rate Data

Conditional Probability / Bayes’ Theorem

Evaluation of Tests

Sequential Observations

Combining Data Sources