failure rate calculator

A comparison between the approximation and the actual probability of failure is shown in Table 1, where the value of the failure rate is 0.001 failing/hour (which equates to a mean time to failure of 1000 hours). For example, a reliability of 97.5% at 50 hours means that if 1000 new components are put into the field, then 975 of those components are expected to last at least 50 hours of operation. Please let us know by emailing [emailprotected]. In special processes called renewal processes, where the time to recover from failure can be neglected and the likelihood of failure remains constant with respect to time, the failure rate is simply the multiplicative inverse of the MTBF (1/). . Geoff Macangus-Gerrard, in Offshore Electrical Engineering Manual (Second Edition), 2018. endobj Ten identical components are each tested until they either fail or reach 1000 hours, at which time the test is terminated for that component. (5.2). There is, however, one more item to take care of before the confidence limits can be established. Decisions may require strategic trade-offs with cost, performance and, security, and decision makers will need to ask questions beyond the system dependability metrics and specifications followed by IT departments. These figures are often provided in the instruction manuals for equipment, to give owners, operators and technicians a rough measure of the reliability of the machine. By detecting changes in system performance or operation early, you can schedule maintenance at a convenient time and repair problems before they turn into unplanned downtime or cause collateral damage to the whole system. Using MTBF to make predictions for a specific device has limited accuracy, and so is better used to estimate how many spares are needed to support a given number of assets, rather than to predict when a specific asset will fail. HWr6^1@h The failure rate of an electronic assembly is roughly equal to the sum of the failure rates of all its components. x=rIr?#>6IZJm2B ,)2(:v^~uUvo/{zwz}z;17eE^/F*yny_}/.4:@9 iIvRrKFBpBk|byr~YEBOe.KBQKi`-iy"C>)y./M~/v.gM|J/*v!XU.5 LyYBx/ESq2*!JhVB?-B7+wK;AvgVI` The most common means are: Given a component database calibrated with field failure data that is reasonably accurate[1] As these defects are eliminated, the curve levels off into the second zone. In this article, we will provide a brief overview of each of these four functions, followed by a discussion of how to obtain the pdf, CDF and reliability functions from the failure rate function usingReliaSoft Weibull++. Similarly, a manufacturer can also provide a specified failure rate for an assembly. endobj This is not necessarily a valid assumption, but to assume otherwise you would have to work out the assembly's failure modes for each component failure and for combinations of failures, which is not practical unless your customer is prepared to pay for a great deal of development work. Failure mechanisms that are more random in naturethird-party damages or most land movements for exampletend to drive the failure rate in this part of the curve. (The average time solely spent on the repair process is called mean time to repair.). Firstly, it can be used retrospectively as a measure of reliability and availability, as discussed previously. 1 The average failure rate of 11% also ticked down slightly from last year. Copyright 2023 Elsevier B.V. or its licensors or contributors. (E.g. For example, a 99.999% (Five-9s) availability refers to 5 minutes and 15 seconds of downtime per year. A decreasing failure rate (DFR) describes a phenomenon where the probability of an event in a fixed time interval in the future decreases over time. A primary goal for all businesses is to maximize output and minimize downtime and mean time between failures is a useful metric to assess the reliability of the systems that support your operations. Failure rate is defined as how often a system or piece of equipment fails unexpectedly during normal operation. Total uptime The total amount of time that the system or components were operating correctly under normal conditions. {\displaystyle t} It is a continuous representation of a histogram that shows how the number of component failures is distributed in time. IT systems contain multiple components connected as a complex architectural. t 3 0 obj ( The {\displaystyle F(t)} Failure rates vs. failure mechanisms. From this, we understand that our conveyor belts have typically run for around 2012 hours on average before failing, or around 12 weeks. However, the failure rate versus time plot is an important tool to aid in understanding how a product fails. , which is a cumulative distribution function that describes the probability of failure (at least) up to and including time t. where Yi Xiao-Jian, Mu Hui-Na, in Goal Oriented Methodology and Applications in Nuclear Power Plants, 2020. It can only provide an estimate of the likelihood of future failures, and only when used with appropriate statistical models. Improve ROI with enhanced asset management. The failure rate can be used interchangeably with MTTF and MTBF as per calculations described earlier. It does in this case only relate to the flat region of the bathtub curve, which is also called the "useful life period". Figure 8.1.11. F The MTBF of a system or piece of equipment can also be predicted by analysing known factors. If any one of the four functions presented above is known, the remaining three can be obtained. It only looks at correct operation time under typical conditions. For component or system manufacturers, testing of samples can be done to create an estimate of MTBF for the given asset. approaches to zero: A continuous failure rate depends on the existence of a failure distribution, ScienceDirect is a registered trademark of Elsevier B.V. ScienceDirect is a registered trademark of Elsevier B.V. In Lees' Loss Prevention in the Process Industries (Fourth Edition), 2012. Under certain engineering assumptions (e.g. Failure rate is the frequency with which an engineered system or component fails, expressed in failures per unit of time. This theory is the basis of the ubiqui-tously discussed bathtub curve. Values of the Percentage Point of Distribution tc for 95% Confidence Interval. New, digital terminals will have very low failure rates, whereas first-generation addressable products often failed at several percent per month. The failure rate of a system usually depends on time, with the rate varying over the life cycle of the system. Combining MTBF-based maintenance approaches, with other strategies, such as condition-based monitoring and programmed maintenance, will help avoid costly break downs. Time is taken to repair it, then the system is switched back on, runs for a while, and then fails unexpectedly again. f~3e[i4_U4u[|q/5eYq8*&rizkM ZH_>kw;U)rQfURfVzzomcy:nX;1]X opYyZ8`Eiphn_(x)Js.{F=9vq>udVpx$_ZlGr8=7h 2cZGX@! Quality-One uses this calculator to t 2 WebThis calculator estimates the FIT rate and MTBF for SiTime products under specific operating conditions. This first portion of the curve is called the burn-in phase or infant mortality phase. These postings are my own and do not necessarily represent BMC's position, strategies, or opinion. The failure rate of 1.0 per year means that if 100 instruments are checked over a period of a year, 100 failures will be found, i.e. H.W. For a repairable machine, it may fail multiple times within its useful life, so the two metrics are not generally closely related. What are Smart Contracts and How are Enterprises Using Them? This general shape represents the failure rate for many manufactured components and systems over their lifetimes. h Basic Excel percentage formula Enter the formula =C2/B2 in cell D2, and copy it down to as many rows as you need. over a time interval First, the sample variance is given by. WebFailure Rate (Weibull Distribution) Probability Calculator Weibull distribution calculator, formulas & example work with steps to estimate the reliability or failure rate or life-time Because MTBF does not include the time taken to repair a piece of equipment after a failure, it is a measure of how long a machine typically runs correctly before failing but doesnt reflect how long it is out of operation. 1 0 obj The failure rate can be defined as the following: Although the failure rate, Based on field failure data from several systems over a number of years, we suggest that a default value of 7% be used, lacking more device-specific data. R Failure modes, effects, and diagnostic analysis, List of important probability distributions, Reliability theory of aging and longevity, "Bounds, Inequalities, and Monotonicity Properties for Some Specialized Renewal Processes", "DFR Property of First-Passage Times and its Preservation Under Geometric Compounding", "Further Monotonicity Properties for Specialized Renewal Processes", "A note on comparing response times in the M/GI/1/FB and M/GI/1/PS queues", "A Realistic Evaluation of Memory Hardware Errors and Software System Susceptibility", "Calculating Failure Rates of Series/Parallel Networks", "Mission Reliability and Logistics Reliability: A Design Paradox", Reliability Engineering and Risk Analysis: A Practical Guide, "Approximation of Mean Time Between Failure When a System has Periodic Maintenance", Multivariate adaptive regression splines (MARS), Autoregressive conditional heteroskedasticity (ARCH), https://en.wikipedia.org/w/index.php?title=Failure_rate&oldid=1142602546, Wikipedia articles needing clarification from April 2015, Creative Commons Attribution-ShareAlike License 3.0. By tracking how often software fails to perform as expected under normal use, we can calculate an estimate for MTBF, and use this to improve performance. Because of this, it is incorrect to extrapolate MTBF to give an estimate of the service lifetime of a component, which will typically be much less than suggested by the MTBF due to the much higher failure rates in the "end-of-life wearout" part of the "bathtub curve". Alternatively, analytical methods can also be used to perform these calculations for large scale and complex networks. The configuration can be series, parallel, or a hybrid of series and parallel connections between system components. t Far into the life of the component, the failure rate may begin to increase. It can be observed that the reliability and availability of a series-connected network of components is lower than the specifications of individual components. If the observed value is larger than the true value, the percentage error will be positive. <> For hybrid systems, the connections may be reduced to series or parallel configurations first. For a life test unit whose basic failure rates are evaluated by reliability data analysis of Bayes method, the evaluating steps are as follows: first, determine the total time test according to selected data of the life test unit; second, develop the likelihood function according to the test data of test sample, as shown in Eq. Safety is one benefit of MTBF that is best understood in the context of critical systems, for example an aircraft. If the failure rate is constant with time, then the product exhibits a random or memoryless failure rate behavior. Although it was a useful approximation when it was first presented, it applies only for a constant failure rate Histograms of the data were created with various bin sizes, as shown in Figure 1. The relationship of FIT to MTBF may be expressed as: MTBF = 1,000,000,000 x 1/FIT. <>>> Availability is related to reliability and is a measure of how much of the time a system is performing correctly, when it needs to be. The calculation of MTBF can also be skewed by biased selection of time periods or assets. Refer to the equations below for clarification. FIT stands for Failure in Time and indicates the frequency at which a system component fails every 1000000000 hours. The average time elapsed between the occurrence of a component failure and its detection. By measuring MTBF for components, we can reduce the chances of an unexpected failure of a critical system that could endanger the lives of everyone. (248) 280-4800 | information@quality-one.com, F M E A Training Virtual Workshop March 2023, RCA 8D Training Virtual Workshop March 2023, Core Tools Training Virtual Workshop March 2023. They can also use MTBF to look ahead and have the necessary parts and skills available for when unexpected failures occur. t Learn more about BMC . This information can be used to measure the decrease in reliability that can occurs as an asset ages and determine when a decision is made to replace a piece of equipment. endobj This distribution is related to the normal distribution and depends on a parameter known as the number of degrees of freedom (DF). Johnson, Barry. Hazard rate refers to the rate of death for an item of a given age (x), and is also known as the failure rate. Instead, what we need to focus on is calculating MTBF for our specific equipment or systems, to begin to develop an estimate of reliability. MTTR is used to measure the average time it takes to repair the system after it has failed, which measures how long the equipment is offline due to unplanned maintenance. Ip`cluv^"rBnBqDhd5f The failure rates of a loop with a pneumatic flow indicator controller, as calculated from the data in Table 13.7 (UKAEA), as calculated from the data in Table 13.8 (Anyakora, Engel, and Lees), and as given by Skala, are shown in Tables 13.18 and 13.19. For more complex arrangements a truth table may be used. WebHow do you calculate failure rate in Excel? The two end points of the confidence interval are called the confidence limits. t Improve ROI with better inventory management. (5.1); finally, obtain the point estimation of the basic failure rate for the life test unit by solving the likelihood equation, which is obtained by using a logarithm derivation for the likelihood function, as shown in Eq. HWKsF}TvI#Fcf0xrpV9@P When the failure rate tends to increase with time and is logically linked with an aging effect, the underlying mechanism is time dependent. {\displaystyle t} Note that the pdf is always normalized so that its area is equal to 1. If, for example, the measured value varies from the expected value by 90%, there is likely an error, or the method of measurement may not be accurate. The failure rate for a pressure measurement system used in factory A is found to be 1.0 per year while the system used in factory B is 3.0 per year. While most of these defects will be eliminated in the final sorting process, a 1 0 obj All Rights Reserved. [clarification needed] To clarify; the more promptly items are repaired, the sooner they will break again, so the higher the ROCOF. {\displaystyle t_{2}} Those of particular interest here are as follows. These The best way to calculate an accurate MTBF for a specific piece of equipment or software is to measure its performance in day-to-day use in an organisation. Concepts & Best Practices, PMP & Other Project Management Certifications, The Chief Data Officer (CDO) Role & Responsibilities. The shortcomings of the part count method are many: It assumes a constant failure rate, memory-less failure rate A new part fails n = sample size The calculations below are computed for reliability and availability attributes of an individual component. Therefore, it is recommended that the CDF should be used for calculations of unreliability at a given time and the time at which a given unreliability occurs, and the failure rate function should be used only as an aid to understand if the model used to fit the data is consistent with the types of failure modes observed or expected for the component. It can also be used in calculations of operational efficiency and performance and used to identify ways to decrease costs and increase output and profits. WebThis approximation still exists in some reliability textbooks and standards. Note that this is a conditional probability, where the condition is that no failure has occurred before time Failure rate data can be obtained in several ways. Once an MTBF has been determined for a system, it is generally used in one of two ways. Availability refers to the probability that a system performs correctly at a specific time instance (not duration). t For the life test unit, according to the test type of the life test unit to evaluate its failure rate, the corresponding steps are as follows: Step 1: To select the reliability data analysis method of the unit to evaluate the basic failure rate of the life test unit. 1 on average each instrument is failing once. Figure 8.1.9. Sample sizes of 1 are typically used due to the high cost of prototypes and long lead times for testing. {\displaystyle t} for In this article, we discussed the probability density function, unreliability function, reliability function, failure rate function and the relationships between them. These measures of a Failure rates are statistical values based on samples of known population. For parallel connected components, use the formula: For hybrid connected components, reduce the calculations to series or parallel configurations first. Theorized failure rate curve for pipelines. Because this is a forward-looking approach, it can only ever be approximate, and needs to take into account all factors affecting the situation and use appropriate predictive modelling methods. endstream = endobj The following formulae are used to calculate MTBF: The average time duration to fix a failed component and return to operational state. Failure rates are further discussed in Chapter 14. The sample standard deviation s is obtained by taking the square root of the variance. 4 0 obj A number of the items are put into normal operating conditions and run until they fail, giving values for total operating time and total number of failures that can be used to calculate an MTBF. For example, if a component has an MTBF value of 500,000 h, and the failure rate is desired in failures per million hours, the failure rate would be: For an existing product MTBF can be found by studying field failure data, but for a new product or if significant changes are made to the design, it may be required to estimate MTBF before any field data is available. These types of failures are typically caused by mechanisms like design errors, poor quality control or material defects. David Large, James Farmer, in Broadband Cable Access Networks, 2009. An examination of the failure data of a particular system may suggest such a curve and theoretically tell the evaluator what stage the system is in and what can be expected. *8k>Qji#)FPHpkBj?/]c?k"GvS6`[fQ.vZO Je=8KaONZ >5V.6nknp}4P+&j7zCCiI)C)e6?A_..-j/ Uptime for the purposes of MTBF is calculated as the duration from the start of uptime to the start of the next unplanned downtime. Inventory levels can be more effectively managed when tracking MTBF, which can help predict which components and systems will fail and when, increasing the chances that technicians will have the right parts on hand when required. Sure, it might have just been a worn out part or a random occurrence, but take the time to look for systemic issues that might have contributed to the failure, that you can address. (Learn more about availability metrics and the 9s of availability.). Thus, in the context of an experiment, a negative percentage error just means that the measured value is smaller than expected. t Failures in similar components will tend to reach a peak after a certain time and then tail off, again producing a bell-shaped characteristic called a normal distribution (see Fig. Figure3.4 shows the bathtub curve of a nonrepairable product, in which the first part shows a decreasing failure rate, known as early failure; the second part is a constant failure rate, known as random failure; and the third part is an increasing failure rate, known as wear-out failure. In these cases, it might be more meaningful to express the failure rates in days or even weeks. Knowing how reliable these systems and their components are helps businesses run more efficiently and profitably, with minimal downtime and damage. It represents the probability that a brand-new component will survive longer than a specified time. In other words, the histogram shows the number of failures per bin, while the pdf is scaled to show the probability of failure per unit time. Calculating the failure rate for ever smaller intervals of time results in the .mw-parser-output .vanchor>:target~.vanchor-text{background-color:#b1d2ff}hazard function (also called hazard rate), Percentage error is a measurement of the discrepancy between an observed (measured) and a true (expected, accepted, known etc.) From: Practical E-Manufacturing and Supply Chain Management, 2004, In Practical E-Manufacturing and Supply Chain Management, 2004. MTBF is generally calculated over a period of time that includes multiple failures either multiple failures of a single asset or single failures of multiple assets of the same type so that an arithmetic mean or average of the time between disruptions can be determined. The hazard rate function for this is: Thus, for an exponential failure distribution, the hazard rate is a constant with respect to time (that is, the distribution is "memory-less"). As a statistic, its also important to collect enough data to ensure the accuracy of the calculation, as short time periods or few failures may lead to distorted and inaccurate MTBF figures. Here, defects that developed during initial manufacture of a component cause failures. = ) We have a total time of 4 weeks x 7 days x 24 hours x 150 belts = 100,800 hours minus the 200 hours of repair time = 100,600 hours of uptime, with 50 failures in total. <>/ExtGState<>/ProcSet[/PDF/Text]>> However, neither the total population, the mean value of failure rate for all components of a particular type, nor the way the values vary over the range from the worst to the least is known. A mistake that is often made when calculating reliability metrics is trying to use the failure rate function instead of the probability of failure function (CDF). 2 0 obj One does not expect to replace an exhaust pipe, overhaul the brakes, or have major transmission problems in a new vehicle. where. Reliability is also an important consideration during the product design process, where MTBF estimates can help improve reliability before a product is even made. Lets say you have a very expensive piece of medical equipment such as an EKG machine in a large hospital thats in use 16-hours a day, 7 days a week, measuring patients heart signals. A component manufacturer may sometimes provide a specified failure rate usually based on field or laboratory test data. Normal operation an electronic assembly is roughly equal to the sum of the or. Larger than the specifications of individual components systems and their components are businesses. X 1/FIT low failure rates, whereas first-generation addressable products often failed at percent! Mtbf as per calculations described earlier laboratory test Data of prototypes and long lead times for testing:... Depends on time, with other strategies, such as condition-based monitoring programmed! Is equal to the high cost of prototypes and long lead times for testing is constant with time, the. Has been determined for a system or piece of equipment can also provide a specified failure rate is as... Burn-In phase or infant mortality phase determined for a repairable machine, it be... For large scale and complex networks how often a system component fails every hours! It down to as many rows as you need process, a negative percentage error means! Terminals will have very low failure rates in days or even weeks system depends. Under typical conditions 5 minutes and 15 seconds of downtime per year smaller... On the repair process is called mean time to repair. ) specific time (. Skills available for when unexpected failures occur process is failure rate calculator mean time to repair. ) many. Testing of samples can be done to create an estimate of the failure rate constant... Or component fails, expressed in failures per unit of time periods or assets the variance they also... Is lower than the specifications of individual components 's position, strategies, or opinion s is by... A truth table may be expressed as: MTBF = 1,000,000,000 x 1/FIT future,. Particular interest here are as follows specified failure rate of 11 % also ticked down slightly last. Fails, expressed in failures per unit of time that the system components. Here, defects that developed during initial manufacture of a failure rates vs. failure mechanisms analytical can... The formula =C2/B2 in cell D2, and copy it down to as many rows as you need down from! Than the true value, the Chief failure rate calculator Officer ( CDO ) &..., expressed in failures per unit of time periods or assets failure and its detection how a product.! Access networks, 2009 life, so the two metrics are not closely! So the two end points of the ubiqui-tously discussed bathtub curve, with minimal and... Is larger than the specifications of individual components that its area is equal the... Rates in days or even weeks some reliability textbooks and standards useful life, so the two points. Important tool to aid in understanding how a product fails once an has... Means that the pdf is always normalized so that its area is equal 1... Is an important tool to aid in understanding how a product fails curve is called mean time to repair )! Occurrence of a system or components were operating correctly under normal conditions in Broadband Access... Points of the curve is called mean time to repair. ) similarly, a 1 0 (... ( CDO ) Role & Responsibilities and their components are helps businesses run more efficiently profitably... Complex architectural minimal downtime and damage for a system component fails, expressed in failures per of! Of availability. ) to series or parallel configurations first under specific operating conditions can be that. Only when used with appropriate statistical models used retrospectively as a measure of reliability availability... Amount of time periods or assets these postings are my own and do not necessarily represent BMC 's position strategies... Its components a complex architectural eliminated in the final sorting process, a manufacturer also! Components connected as a measure of reliability and availability, as discussed previously observed the. That a system component fails every 1000000000 hours its licensors or contributors, defects that developed during manufacture. ), 2012 ) availability refers to the probability that a system component fails expressed!, defects that developed during initial manufacture of a system performs correctly at a time! Larger than the true value, the remaining three can be obtained interval are called the burn-in phase or mortality... Note that the measured value is larger than the specifications of individual components a time interval first, Chief. Necessarily represent BMC 's position, strategies, such as condition-based monitoring and programmed maintenance, help. Are Enterprises Using Them larger than the true value, the failure rates whereas! Particular interest here are as follows rate and MTBF as per calculations described.... Than a specified time values of the failure rate behavior by analysing known factors, for,! As many rows as you need, defects that developed during initial manufacture of a,. Always normalized so that its area is equal to 1 of components lower... May fail multiple times within its useful life, so the two metrics are not generally closely.! Mean time to repair. ) how a product fails known population FIT to MTBF may be to. ( Learn more about availability metrics and the 9s of availability. ) process is called the confidence limits be... In some reliability textbooks and standards while most of these defects will be eliminated in the context an! Is equal to the high cost of prototypes and long lead times for testing analysing known factors h the rate! Other strategies, such as condition-based monitoring and programmed maintenance, will help avoid costly break downs under. The calculation of MTBF for SiTime products under specific operating conditions series-connected network of components is than. These calculations for large scale and complex networks estimates the FIT rate and MTBF for given... Run more efficiently and profitably, with other strategies, such as condition-based monitoring and programmed,! Standard deviation s is obtained by taking the square root of the confidence limits can be done to create estimate..., or a hybrid of series and parallel connections between system failure rate calculator a brand-new will! Failed at several percent per month is one benefit of MTBF that is best understood in the process (. Equal to 1 down to as many rows as you need rates, whereas first-generation addressable products often at! Project Management Certifications, the Chief Data Officer ( CDO ) Role & Responsibilities measured... Terminals will have very low failure rates of all its components by analysing known factors copy it down as! Of equipment fails unexpectedly during normal operation used with appropriate statistical models sometimes a. Of 11 % also ticked down slightly from last year all its components the Chief Data Officer ( ). Obj ( the average time elapsed between the occurrence of a series-connected network of components is lower than the value! 2 WebThis calculator estimates the FIT rate and MTBF as per calculations described earlier interchangeably with MTTF and as... That shows how the number of component failures is distributed in time and indicates the frequency which! Complex architectural } Note that the system or component fails every 1000000000.... Meaningful to express the failure rate can be observed that the measured value is smaller than expected a 99.999 (! Complex arrangements a truth table may be expressed as: MTBF = 1,000,000,000 x 1/FIT of. Interest here are as follows product fails available for when unexpected failures occur brand-new component will longer! Operation time under typical conditions true value, the percentage Point of Distribution tc 95... How a product fails in the final sorting process, a 99.999 % ( Five-9s availability... Piece of equipment fails unexpectedly during normal operation, whereas first-generation addressable products often failed several... Are Smart Contracts and how are Enterprises Using Them and their components are helps businesses more... An important tool to aid in understanding how a product fails rate behavior 2 } } Those of particular here... Described earlier ) } failure rates vs. failure mechanisms the variance t Far into the life of the rate. To express the failure rate is the frequency with which an engineered system piece... Initial manufacture of a series-connected network of components is lower than the true value, the sample deviation! Smart Contracts and how are Enterprises Using Them value, the remaining three can be.. Knowing how reliable these systems and their components are helps businesses run more efficiently and profitably with! Deviation s is obtained by taking the square root of the system be as. Retrospectively as a measure of reliability and availability, as discussed previously component cause failures unit of that. Variance is given by, poor quality control or material defects described earlier the.... The remaining three can be established however, one more item to take care of the! Can be series, parallel, or opinion selection of time are follows... The ubiqui-tously discussed bathtub curve typically caused by mechanisms like design errors, poor control. And systems over their lifetimes 1,000,000,000 x 1/FIT unexpected failures occur not necessarily represent BMC 's position,,... Manufacture of a failure rates, whereas first-generation addressable products often failed at several percent per month has determined! Typically caused by mechanisms like design errors, poor quality control or material defects the necessary parts skills. Basic Excel percentage formula Enter the formula: for hybrid systems, for example an.. The percentage error just means that the reliability and availability, as discussed previously fails, expressed in failures unit. D2, and only when used with appropriate statistical models Far into the life of... Are my own and do not necessarily represent BMC 's position, strategies, a. Rate is constant with time, with the rate varying over the of. Continuous representation of a component cause failures scale and complex networks rate of an failure rate calculator a.

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