 # Reliability Basics

## Introduction

The history of the reliability field may be traced back to the early years of 1930s, when probability concepts were applied to problems associated with electric power generation [1-3]. However, the real beginning of the reliability field is generally regarded as World War II, when German scientists applied basic reliability concepts for improving the performance of their V1 and V2 rockets. Today, the reliability field has become a well-developed discipline and has branched out into many specialized areas, including power system reliability, human reliability and error, mechanical reliability, and software reliability [3-5]. There are many reliability basics.

This chapter presents various reliability basics considered useful to understand subsequent chapters of this book.

## Bathtub Hazard Rate Curve

The bathtub hazard rate curve shown in Figure 3.1 is normally used to describe engineering systems’ failure rate. As shown in figure the curve is divided into three parts: burn-in period, useful-life period, and wear-out period.

During the burn-in period, the engineering system/item hazard rate (i.e., time- dependent failure rate) decreases with time t, and some of the reasons for to the occurrence of failures during this period are substandard materials and workmanship, poor quality control, poor processes, poor manufacturing methods, human error, and inadequate debugging [6,7]. The other terms used for this region are infant mortality region, break-in region, and debugging region.

During the useful-life period, the hazard rate remains constant and some of the causes for the occurrence of failures in this region are higher random stress than expected, low safety factors, natural failures, abuse, human errors, and undetectable defects. Finally, during the wear-out period, the hazard rate increases with time t and some of the reasons for the occurrence of failures in this region are wear due to aging; wrong overhaul practices; wear due to friction, creep, and corrosion; short designed- in life of the item under consideration; and poor maintenance. FIGURE 3.1 Bathtub hazard rate curve.

To represent the bathtub hazard rate curve mathematically, shown in Figure 3.1, Equation (3.1) can be used : where

A (7) is the hazard rate (time-dependent failure rate). a is the scale parameter. p is the shape parameter. t is time.

At p = 0.5, Equation (3.1) gives the shape of the bathtub hazard rate curve shown in Figure 3.1.

## General Reliability-Related Formulas

There are a number of formulas used to perform various types of reliability-related analysis. Four of these formulas are presented in the next four sections.

### Failure (or Probability) Density Function

The failure (or probability) density function is expressed as shown in Equation (3.2) where

t is the time.

fit) is the failure (or probability) density function.

R(t) is the item/system reliability at time t.

Example 3.1

Assume that the reliability of a system is expressed by the following function: where

t is the time.

(r) is the system reliability at time t. ks is the system constant failure rate.

Obtain an expression for the system’s failure density function by using Equation (3.2).

By substituting Equation (3.3) into Equation (3.2), we obtain Thus, Equation (3.4) is the expression for the system’s failure density function.

### Hazard Rate Function

This is expressed by where

A(/) is the item/system hazard rate (i.e., time-dependent failure rate).

By substituting Equation (3.2) into Equation (3.5) we obtain Example 3.2

Obtain an expression for the system’s hazard rate using Equations (3.3) and

(3.6) and comment on the end result.

By inserting Equation (3.3) into Equation (3.6), we get Thus, the system’s hazard rate is given by Equation (3.7), and the right-hand side of this equation is not a function of time t (i.e., independent of time t). Needless to say, L is normally referred to as the constant failure rate of an item (in this case, of the system) because it does not depend on time t.

### General Reliability Function

The general reliability function can be obtained using Equation (3.6). Thus, we have By integrating both sides of Equation (3.8) over the time interval [(),?], we obtain Since, at t = 0, R(t) = 1.

By evaluating the right-hand side of Equation (3.9) and rearranging, we obtain Thus, from Equation (3.10), we obtain Equation (3.11) is the general expression for the reliability function. Thus, it can be used for obtaining the reliability of an item/system when its times to failure follow any time-continuous probability distribution (e.g., exponential, Weibull, and Raleigh).

Example 3.3

Assume that a system’s times to failure are exponentially distributed and the constant failure rate is 0.002 failures per hour. Calculate the system’s reliability for a 50-hour mission.

By substituting the specified data values into Equation (3.11), we get Thus, the system’s reliability is 0.9048. In other words, there is a 90.48% chance that the system will not malfunction during the stated time period.

### Mean Time to Failure

The mean time to failure of a system/item can be obtained using any of the following three formulas [7,9]: or or where

MTTF is the mean time to failure.

E(t) is the expected value.

5 is the Laplace transform variable.

R(s) is the Laplace transform of the reliability function R(t).

Example 3.4

Prove by using Equation (3.3) that Equations (3.12) and (3.14) yield the same result for the system’s mean time to failure.

By substituting Equation (3.3) into Equation (3.12), we get where

MTTFS is the system’s mean time to failure.

By taking the Laplace transform of Equation (3.3), we obtain where

Rs (.s) is the Laplace transform of the system reliability function Rs (/).

By inserting Equation (3.16) into Equation (3.14), we get Equations (3.15) and (3.17) are identical, w'hich proves that Equations (3.12) and (3.14) yield same result for the system’s mean time to failure.