Reliability

Motivation

What is reliability?

System reliability with respect to component reliability

System design with respect to system reliability

As the number of components is increased, the reliability of the serial system decreases.

Consider airplane with 1000000 components. If each component has a reliability of 0.999999 (that is a failure only one time in a million chances), the reliability of the plane would be 0.9999991000000=0.0368. Hardly what we would call a reliable system. Would you fly on a plane when the chance of a safe flight are scarcely better than 1 in 3?

Application

Safety critical application:

  • aircraft control and navigation
  • railroad control
  • spacecraft
  • military application
  • ...

Reliability theory:

  1. analysis of system reliability
  2. design of system with defined reliability

Analysis of system reliability

Generally you don’t know the parameters of reliability of all components in the system. Therefore you cannot express exactly the reliability of the whole system.

The reliability of the system is measured in practice. This is very expensive and long-lasting. (If you want to prove that your system has one failure in 10 years, you need to watch this system at least 10 years)

The reliability is set from the long set of parameters of the working system. You need long time measuring system and statistical analysis.

Design of the system with defined reliability

there are some criterion to define the design of the system:

  • defined reliability for defined system’s lifetime (for example car, you accepts 5 failures for 20 years of car lifetime)
  • optimalization of system reliability and system cost (for example production line, you can accept one bad products per hour with defined cost of the production line)
  • elimination of failure (for example control system of aircraft)

Methods how to control the system reliability:

  • structure of the system
  • increase reliability of system components

Definition of reliability

What is reliability?

We will define reliability as a :

  1. probability that a system will perform an a given trial
  2. probability that an item will last for a given length of time

The reliability strongly depends on definition of the failure

It is important to define what is considered to be failure.

For example in a sound recording on magnetic tape, there can be slight degradation with use. What constitutes a failure? When we lose one decibel? The typical human ear cannot hear this degradation, but it can easily be measured. Is this a failure? If not, how about two decibels?

At what point does the taste of milkshake shift from acceptable (success) to unacceptable (failure)?

There some types of failures

  • sudden or progressive
  • partially or complete
  • permanent or temporally
  • dependent or independent
  • early or end-of-life

In this course we will consider only permanent and random independent failures

The reliability depends on operating conditions

There are light bulbs designed for exterior use and those that are designed for interion use. The useful life (reliability) of an interior bulb is significantly decreased when used outside because it is not designed for the temperature fluctuations and moisture levels of the outdoor environment. It is being utilized outside its prescribed operating conditions.

reliabilitythe probability is 0.02 that this light bulb

definition of failurewill not provide light for at least 1000 hours

prescribed operatingwhen operated in a normal household environment

conditions

Characterization of reliability by:

  • failure density
  • failure rate
  • mean time between failure (MTBF), for non-repairable product mean time to failure (MTTF)

Usually it depends on time, sometimes on some operational parameter.

It can be described by:

  • continuous independent variable (time)
  • discrete independent variable (time amounts)
  • dependent variables, two states: failure, working

Continuous system

We are investigating product in timewith the same operating conditions. At the time 0 the product satisfies functionality and in random time the product fails. The value  is continuous random value.

The reliability it means the probability that the product would work without failure from time 0 to time t is marked as R(t) and is defined as

Probability of failure in time interval <0, t> is Q(t) an is defined as .

As the probability of failure and probability of working without failure are exclusive, so

Suppose that exist derivation of function F(t):

The function f is density of random value , called density of failure.

Failure rate value derivates from failure density and probability of working without failure and is defined as

Each function R(t), Q(t), f(t),describes fully the reliability of the system and other three function can be derived from this function. For example if you have function for your system, you can express R(t), Q(t) and f(t) from this function.

Mean time to failure (MTTF) TS is defined as mean value of random value :

Dispersion D of random value :

or

Guaranteed time without failure T is such time that R(T) = 

Typical progress of functions R(t), Q(t), f(t) and

Can be demonstrated on function , that is usually divided into three parts

  1. Early stage failure (failures in design, development and production)
  2. normal life (constant failure rate)
  3. End-of-life failure (the failure rate is increasing depending on depreciation)

Discrete systems

System reliability with discrete operation depends on number of performed operations.

Discrete random variable X that can have value xi, i = 1,2 .. n. Discrete probability density f(x) is probability and it satisfies and

Probability of failure is defined by distribution function F(X) and satisfies

Probability density f(x) can be computed from probability distribution F(x) as , where x- is predecessor of x.

Mean time to failure is defined as mean value

Dispersion of discrete random value or as

Suppose we have n independent random variables X1, .. Xn with the same probability distribution. The arithmetic mean is . The mean value of sum can be expressed as and

Dispersion satisfies and

Meaning of characterization of reliability

Suppose we are testing NO number of same products and we remember the times of failure of products. Suppose N(t) is the number of correctly working products in time t.

From definition of probability working without failure is N(t)=NOR(t). From this we can express and probability of failure is

Suppose N is number of system that has failure in time interval <t, t+t>, where t0. So

. From this we can express the probability density

For one amount of time there is in average failure for number of products. It means that failure density is equal mean number of failures in one amount of time in time t with respect to initial number of working systems NO in time t = 0.

The failure rate satisfies . It means that failure rate is equal mean number of number of failures in one time amount in time t with respect to number of product working without failure in time t.

Failure rate can be define with conditional probability. Let’s mark Q’ conditional probability that product has failure in small time amount <t, t+t> with respect that in time t it worked without failure. It’s true that Q = R(t) Q’. The Q represent unconditional probability of failure in time interval <t, t+t>. With previous equation we can receive following . From this , failure rate is equal conditional probability of failure in small amount of time from time t with respect working without failure until time t.