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     Reliability  Definition The reliability of a component or system is the probability that it will be able to perform its function when required, for a specified time and in a particular environment.  Function? A component performs its function if it does not fail. However, there may be many levels of functioning (or failing). A motor that will not start clearly has failed by most people’s understanding? But what if a indicator light / gauge does not work? Thus the required level of functioning must be clarified.  Environment? The conditions wherein the component are to be used must be considered. i.e.    the physical environment (temperature, humidity, dust)    the user (their skills\knowledge\experience)    maintenance (servicing schedule etc.) Time? The longer a component is in use, the more likely it is to fail. However, time need not be measured in conventional time units (hours / months / years), it may be distance travelled (cars), machine cycles etc.  Probability? Only in the most trivial of situations will we able to assign a deterministic reliability measure. More typically, there is a degree of uncertainty and this is quantified using probability. Further concepts and terminology  Design life or mission time The design life is the period of time (however measured) that a component is required to last, or is stipulated to last, e.g. the time period for which it is guaranteed. Guarantee is something of a misnomer though as the  probability of a component surviving for its design life or longer, will usually be less than 100%. Components V System (and sub-systems) In simple terms, we will take a component to one of a collection of atomic unit that collectively make up our system (or sometimes, device). For example, our system might be a desk top computer base station and the components might be the motherboard, the hard disk, the CR-ROM drive, the cooling fan and the power supply. Of course from the perspective of the manufacturer of these parts, they would regard their particular part as the system, made up from a different set of components. And similarly in turn with the makers of these components. Conversely, the manager of a production line might see a complete PC as just a single component component amongst many that make up his system of interest.  Mean time to failure (MTTF) and mean time between failures (MTBF) A frequently used metric for a component’s reliability is the MTTF (average life for non -reparable device) or MTBF (average up-time for repairable device). There may be a subtle difference between the two as some devices upon repair, may not be restored to their full “health” and thus the MTBF may be getting shorter with time.   2    Overview of topics considered in this module  Example To give an overview of the material covered on this module we will consider a hypothetical example of a manufacturer who produces cooling fans for computers. Producer Consumer 1.   Design It should be intuitively obvious that “building in” reliability at the design stage should be an objective in the development of any new product. FMECA (Failure modes, effects and criticality analysis) is a technique used to identify, prioritise and eliminate potential failures from a component or system. 2.   Prototype testing By testing a product (until it fails) the reason for failure can be ascertained and used to inform an improvement in the design, leading to a longer MTTF. The longer the testing, the more the reliability “grows”. How long should the testing programme be if a specified MTTF is to be achieved? (Duane models) 3.   Guarantee The MTTF is not sufficient information to quantify the reliability of a component. A light bulb and a cell  battery may both have MTTF of 1,000 hours but their lifespan would be described using very different mathematical models or probability density functions (PDF). In addition to the PDF, the related function of reliability and hazard are useful in understanding the reliability of a component. Failure test data (possible censored) might be obtained to inform the selection of a suitable lifespan model. From this it might be determined that a component has say a 99% probability of surviving for 2 years or longer and so the producer may be happy to offer a two-year guarantee. Of course the consumer may wish to conduct their own investigations to determine the reliability of a component. 4.   Sale Frequently it is not possible to individually test all items output from a producer. The consumer may devise a sampling plan. i.e. Select a sample of output from a batch and purchase the entire batch if the number of defective items in the sample is sufficiently small. 5.   System Reliability The consumer may wish to assemble a system consisting of many components, each with their own reliability measure. It is of interest to know the overall reliability of the system. If this is unacceptably low, the system may incorporate a degree of redundancy. For example, a dual powered calculator (solar + battery) utilizes redundancy as the device will continue to function if either (but not both!) the solar panel or battery fails. 6.   Machine breakdown Machines frequently deteriorate from their optimal operating state to ones where the contribute less value. If we know the likelihood of machines changing state (from one shift/day to the next) we could calculate the average time a machine spends in each of its possible states, and hence its average worth.   3    Density, distribution, reliability and hazard function  Example Consider an electronic device and a study on its life span in years. We might track the life of 50 such devices and compile the following table and histogram. Years 1 2 3 4 5 6 7 8 Failed 4 8 14 11 7 3 2 1 We could use this histogram to estimate survival probabilities of similar devices. Eg. Pr(failure in 2 years) = 12/50. Pr(survive for 5 of more years) = 13/50. If we convert our frequencies to probabilities Years 1 2 3 4 5 6 7 8 Failed 0.08 0.16 0.28 0.22 0.14 0.06 0.04 0.02 Our probability estimates amount to taking an area under the histogram. We could imagine repeating this procedure for a large number of devices with their survival time being recorded as a continuous rather than a discrete quantity. This would smooth our histogram giving something like that shown opposite. This is a probability density function (PFF), sometimes (incorrectly) called a distribution function f(t)  Note that the only requirement for a function to be a valid density function is that  Notes o   Taking the lower limit to be 0 rather than - ∞ assumes t ≥ 0, which is usually the case in reliability.   o   The upper limit of ∞ really means the maximum age of the device, which sometimes is not bounded.  Rather than the empirical (based on data) development of a PDF outlined above, sometimes a parametric PDF can be arrived at and this is the approach that we will usually take. For each of the following, verify that these are valid density function and sketch the function. Evaluate the  probability of a component with life T given by this density surviving for the time specified. Problem 1   f(t) = 0.25 0 ≤ t ≤ 4 and evaluate Pr(1.5 ≤ T ≤ 2.3) (This is the uniform PDF.) Problem 2   f(t) = ½ t 0 ≤ t ≤ 1  and evaluate Pr(T > 1) (This is the triangular PDF) 1 / 6  (4-t) 1 < t ≤ 4     4 Problem 3   f(t) = t 2   0 ≤ t ≤ 2  This is NOT a valid density Problem 4   f(t) = 2e -2t   0 ≤ t <   and evaluate Pr(T < 3) This last PDF is the (negative) exponential and is used a lot in reliability    The distribution function or cumulative distribution function . Thus F  (t) = f(t) The cumulative distribution function F(t) is Pt(T < t) = Problem 5   Find F(t) for the valid density functions above. For a component, F(t) is Pr(survival time < t).    The reliability function In reliability, it is usually more useful to work with R(t) = Pr(survival time > t). Clearly R(t) = 1- F(t). Also, R   (t) = -f(t)  Note that F(0) = 0 and F(  ) = 1 and conversely for R(t). I.e. R(0) = 1 and R(  ) = 0 Sketches of the reliability functions A sketch of a reliability function can be very helpful when selecting from two competing components. Opposite are two sketches, each showing two reliability functions. In the left sketch, A is clearly is better than B for all t. In the right sketch C is superior to D early in its life but the situation is reversed as the components age. Which one might be preferred overall would depend on the context. MTTF or MTBF MTTF Problem 6   Find the MTTF for f(t) = 0.25 0 ≤ t ≤ 4   Solution  Exercise Find the MTTF for the valid density functions above
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