2. reliability function


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2. reliability function

  1. 1. 01-02-2012 Reliability Reliability Function• A relative frequency definition of reliability can • Let us put some N items under life test under the be stated as follows: same environmental conditions and observe the If a large number of independent systems/ number of failures at a predetermined interval of subsystems/components are operated at the same y p p time, regularly. Obviously the number of failures time, the R(t) can be estimated to be the ratio of would go on i ld increasing until all th N it i til ll the items putt systems/subsystems/ components still running under test have failed. The test is terminated only over time t and the initial number of when all the items have failed. Let us consider that systems/subsystems/components put to operation. we are considering catastrophic failures only. Let us assume that the number of failures over a time t are: nf (t). Naturally, the survivals shall be: Systems/subsystems/components can be called, in ns(t) = N- nf (t). general, as items. 1 2 Reliability Function Reliability Function• From the definition of probability, we can • The item unreliability likewise can be define reliability as: defined as:• Obviously this definition requires that the • In fact Q(t) shall also be the cumulative test is conducted for a large value of N distribution (cdf) or F(t) if the random under identical conditions of tests and the variable, t , is taken as time-to-failure for tests are conducted independently. the observation, in this test. 3 4 1
  2. 2. 01-02-2012 Reliability and Unreliability Reliability Function• Cumulative distribution function can also be defined as: • The cumulative distribution function increases from 0 to 1 as the random variable, t increases from zero to its highest , g value towards infinity. Since at t = 0, the item was operating, Q(t)=F(t)=0, but as• Reliability (Survival Function) is given by: t, F(t) 1. Also, at t = 0, R(t)= 1 and R(t) 0 as t. We also know that theDensity Function would be defined by: derivative of cdf of a continuous rv provides probability density function or f(t). 5 6 Reliability Function Reliability Function • By dividing both sides of equation (1), we can define what is known as hazard rate, viz., • Realizing that, we obtain an important relationship: 7 8 2
  3. 3. 01-02-2012 Hazard Rate Reliability Function In other words, the hazard rate is the• It is important to highlight at this stage the difference between rate of failing and hazard conditional probability of an rate. At two different points of time, say at t1 and p , y instantaneous failure given that the f g t2, we can have the same rate of failing but as the component is surviving upto time t, number of survivals would go on reducing the hazard rate would not remain the same even if the divided by the length of the short time rate of failing is the same. Hazard rate is a better interval involved. indicator of how hazardous a situation is at a given point of time. 9 10 Reliability Function Reliability Function• It is obvious from the foregoing expressions that • Now reverting to an expression derived earlier, we f(t) is the rate of failing normalized to the have: original population put to test whereas h(t) is the rate of failing normalized to the number of survivals. The difference forms of f(t) and h(t) • We can solve this first order differential equation that can be used to obtain them from the histogram by separable variable technique with initial of life-test data, is given by: condition that at t = 0, R(0) = 1. We then obtain a very important relationship in reliability studies: 11 12 3
  4. 4. 01-02-2012 Reliability Function Characterizations Reliability Characteristics of a Unit A probability distribution is completely characterized by each of the following entities: Hazard Rate • Density function (f(t))f(t), h(t) h(t)=f(t)/R(t) f(t1) • Cumulative distribution function (F(t)) • Reliability (R(t)) • Hazard rate (h(t)) R(t1) Area Q(t1) or F(t1) Area In other words: if one of these entities is known, we t1 can compute the other entities from it. 13 14 Fig. 3.1: Reliability Characteristic of a Unit Case I (Density Function is known) Case II (Distribution Function is known) t f (t )  F (t ) F (t )   f ( x )dx 0 R(t )  1  F ( t )  R(t )   f ( x )dx F (t ) t h( t )  f (t ) 1  F (t ) h(t )  R( t ) 15 16 4
  5. 5. 01-02-2012 Case III (Reliability is known) Reliability and MTTF t   h(u)du f ( t )   R ( t ) R( t )  e 0 t F ( t )  1  R( t ) Th function The f i (t )  cumulative hazard rate.  h( u)du 0 is called the i ll d h  R ( t ) The expected lifetime is often called Mean Time To h(t )  Failure (MTTF). R( t ) It is dangerous to make decisions based on MTTF only; never cross the river based on average depth. Consider the variance as well! 17 18 MTTF Reliability ExpressionsThe mean-time-to-failure in case of a continuous • Using the expression:random variable such as time- to- failure, t ,is given • We can obtain the reliability of a unit, whichby: follows a given failure distribution or has a   t    i h d F given hazard rate. For example:l  MTTF   tf ( t )dt         du f ( t )dt   f (t )dtdu  R(u)du 1. For h(t) = , we have : 0 00  0u 0 2. For linearly increasing hazard rate, h(t) =This applies equally well to component MTTF, a+bt , we have:subsystem MTTF or even to system MTTF. Wemust, however, take appropriate reliability of the 19 20entity. 5
  6. 6. 01-02-2012 Reliability Expressions Reliability Expressions3. If the hazard rate is increasing non-linearly, i.e., • Therefore, if we know the variation of the failure distribution is Weibull or so, with hazard rate with time,we can obtain the hazard rate given by: g y expression for component reliability using the very versatile general expression: The reliability expression would be given by: Where T is the mission time. 21 22 Phases of Life Typical Bathtub Curve• There are three phases of life of any unit. These are: Bathtub curve describes the variation of hazard rate with time, which is generally taken as life of a unit.• Early Life or Infancy Period• Useful Life or Prime of Life• W Wearout Phase or P i d t Ph Period• Each phase has a particular type of failure dominant and has respectively over these three phases of life either decreasing, constant or increasing hazard rate characteristic. These failures result in an overall characteristic over the life time, which apparently looks like a bath tub . Hence the name. 23 24 6
  7. 7. 01-02-2012 Early Life Useful Life• Early life has predominantly Quality failures, which • During this period, the hazard rate is very often small show up in early life of a unit and can be traced mainly and approximately constant. It is during this period to the manufacturer’s carelessness and can be that a unit is put to effective use and usefully employed attributed to defective designs, use of substandard during the entire life time. During this period, early or material, poor workmanship or poor quality control. quality failures as well as wear out failures are These failures result in a very high hazard rate in the negligible. O l sudden or catastrophic f il li ibl Only dd hi failures can beginning and keeps decreasing as the time passes. occur, which are primarily caused by sudden and step Early failures can be eliminated through the use of increase in the stress level beyond the design strength. debugging process which consists of operating a unit These failures occur randomly and unexpectedly. under conditions of use for a period of time However their frequency over a long period of time is corresponding to the preponderance of early failures. constant. One cannot eliminate these failures but their The length of debugging period is decided by observing probability can be reduced by improving reliability at the failure distribution and by following a specific the design stage. debugging procedure. This is also known as burn in period. 25 26 Wearout Life Other Hazard Models• As the unit reaches the end of its life, parts begin to • Next slide provides a list of some of the distributions wear out and the hazard rate of the unit begins to rise that are extensively used in reliability studies. But one rapidly. Early or quality failures are very rare during must lose sight of the practicability and not fit a this period and stress related failures occur with the complicated model where it is not actually necessary. same frequency as they occur in other phases of life. The failures that occur d i Th f il h during this period are aptly hi i d l called as gradual or wearout failures and are dominant in old age or towards the end of the life time. These failures keep increasing slowly over the life as the deterioration increases with age and the age at which these become predominant depends on the environment, a unit is operated. It is advisable that the replacement of a unit should be done about the point tw in time. Otherwise the sudden failure may be costly in 27 28 consequences. 7
  8. 8. 01-02-2012 Constant Hazard Rate The useful life period of bath tub curve during which catastrophic failures are dominant is often characterized by constant hazard rate thus is best modeled by the exponential failure distribution. The failure process is memoryless and does not recognize the time already elapsed already during the life span. 29 30Decreasing /Increasing Hazard rates Decreasing /Increasing Hazard rates • Decreasing or increasing hazard rates can be modeled Decreasing/increasing hazard rates can also be modeled using by Weibull distribution by suitably choosing the value Gamma distribution. When  < 1, decreasing hazard rates of parameter . When  < 1, we get reliability function would be described whereas for  > 1, we can obtain corresponding to decreasing Hazard rate. However, if  increasing hazard rates with the help of Gamma distribution. > 1, we get reliability function corresponding to , g y p g increasing Hazard rate. 31 32 8
  9. 9. 01-02-2012Thanks 33 9