A Simple Reliability Growth Model

For the sake of simplicity, first look at the case when you are interested in a unit that can only succeed or fail. For example, consider the case of a wine glass designed to withstand a fall of three feet onto a level cement surface.

Figure

The success/failure result of such a drop is determined by whether or not the glass breaks. [Note]

Furthermore, assume that:

Figure

Now given the above assumptions, the question of how the glass could be in the unreliable state just before trial $n$ can be answered in two mutually exclusive ways:

The first possibility is the probability of a successful trial, $(1-p)$, where $p$ is the probability of failure in trial $n-1$, while being in the unreliable state, $P_{n-1}(0)$, before the $n-1$ trial:

 MATH (1)

Secondly, the glass could have failed the trial, with probability $p$, when in the unreliable state, $P_{n-1}(0)$, and having failed the trial, an unsuccessful attempt was made to fix, with probability

$(1-\alpha )$:

MATH (2)

Therefore, the sum of these two probabilities, or possible events, gives the probability of being unreliable just before trial $n$:

MATH

or:

MATH

By induction, since $P_{1}(0)=1$:

MATH (3)

To determine the probability of being in the reliable state just before trial $n$, Eqn. (3) is subtracted from 1, therefore:

MATH (4)

Define the reliability $R_{n}$ of the glass as the probability of not failing at trial $n$. The probability of not failing at trial $n$ is the sum of being reliable just before trial $n$, MATH, and being unreliable just before trial $n$ but not failing MATH, thus:

MATH

or:

MATH (5)

Now instead of $P_{1}(0)=1$, assume that the glass has some initial reliability or that the probability that the glass is in the unreliable state at $n=1$, $P_{1}(0)=\beta $, then:

MATH (6)

When $\beta <1$, the reliability at the $n^{th}$ trial is larger than when it was certain that the device was unreliable at trial $n=1$. A trend of reliability growth is observed in Eqn. (6). Let $A=\beta p$ and MATH, then Eqn. (6) becomes:

MATH (7)

Eqn. (7) is now a model that can be utilized to obtain the reliability (or probability that the glass will not break) after the $n^{th} $ trial. Additional models, their applications and methods of estimating their parameters are presented in the following chapters.