Basic expressions and formulas...

Estimated failure rate $\lambda$ = $\frac{num failures}{time}$

If the failure rate is constant:

$\lambda = z(t)$, MTTF = $\frac{1}{\lambda}$

Probability that component 1 fails before component 2: $$ Pr(T_2>T_1)= \int_0^\inf Pr(T_2>t|T_1=t) \cdot f_{T_1} dt =\int_0^\inf \lambda_1 \cdot e^{-\lambda_2 t}\cdot e^{-\lambda_1 t} dt = \frac{\lambda_1}{\lambda_1 + \lambda_2} $$

Reliability

The main concept of the course is reliability. This is defined somewhat as "the probability that an item will perform a required function under stated conditions for a stated period of time.". This involves components, systems, subsystems such as hardware, software and even humans.

Quality

By quality we mean the totality of features and characteristics of a product or service that bear on its ability to satisfy stated or implied needs. Quality of a product is not only characterized by the conformity to specifications at the time it is supplied to the user, but the ability to meet these specifications during the entire lifetime.

Availability

The ability of an item to perform its required function at a stated instant of time or over a stated period of time. We can denote this as

$ A(t) = P(\mbox{item is functioning at time t}) $

The Reliability Function

The reliability function of an item is defined as: $$ R(t) = 1 - F(t) = P(T>t) \mbox{ for } t > 0 $$ or equivalently $$ R(t) = 1 - \int_0^t f(u) du = \int_t^\inf f(u) du$$

Hence $R(t)$ is the probability that an item does not fail in the time interval $(0,t]$.

Failure Rate Function

The probability that an item will fail in the interval $(t, t+ \Delta t]$ when we know that the item is functioning at t is:

$$ P(t<T \leq t + \Delta t | T > t) = \frac{P(t< T \leq t + \Delta t)}{P(T>t)} $$

Dividing by the length of the time interval gives us $z(t)$, $(FOF)$:

$$ z(t) = \lim_{\Delta t\to 0}{\frac{P(t<T \leq t + \Delta t | T > t)}{\Delta t}} $$

Giving, where $F(t) = 1 - R(t)$:

$$ z(t) = \lim_{\Delta t\to 0}{\frac{F(t + \Delta t) - F(t)}{\Delta t}} \frac{1}{R(t)} = \frac{f(t)}{R(t)} $$

From this on, we can deduce different relationships between the functions $F(t), f(t), R(t) \mbox{ and } z(t)$:

MTTF

Mean Time To Failure

if $\lambda$ is constant: MTTF = $\frac{1}{\lambda}$

else: MTTF = $\int_0^\inf R(t) dt$

Main System Functions

• When a predefined prcess demand (deviation) occurs in the EUC(Equipment Under Control), the deviation shall be detectetd by the SIS sensors, and the required actuationg items shall be activated and fulfill their intended functions
• The SIS shall not be activated spuriously, that is, without the presence of a predefined process demand in the EUC

Subsystem Components

• A sensor subsystem that shall detect a specified hazardous event or deviation (In the current case, the sensor subsystem comprises three gas detectors)
• A logic solver subsystem that interprets the finals from the sensor subsystem and sends an action signal to the final element subsystem 3
• A final element subsystem that shall take action upon signal from the logicsolver (Inthecurrent case, the final element subsystem comprises a single shutdown valve, the ESDV)

Failure Classification