Questions tagged [reliability]

is for Reliability Engineering and Reliability Theory that describes the probability of a system to complete its expected function during an interval of time. The engineering is an area focused on optimising the reliability or the probability of successful functioning, of systems such as graphs, percolations and mechanical systems such as airplanes. Originally a tool for insurance companies but nowadays more general interdisciplinary research area.

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Failure time distributions from an uncertain starting point

I have a data analysis problem in the laboratory which I think can be reframed as a kind of reliability question. Question: Suppose a warehouse is filled with light bulbs. The probability distribution ...
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What is the reliability of k-out-of-n system with fixed repair time?

In k-out-of-n system, a system is functional if $k$ out of $n$ components are functional. Given each component have exponential distribution of $\lambda e^{-\lambda t}$ until it is broken. Let us say ...
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Probability of not happening an event in some years, if probability of it happening in a minute is given

I came across a problem for which I found two answers and I don't know which is the correct one. At a particular site, the probability of occurrence of moderate earthquake in a single minute is ...
84 views

Weighted average of MTBF vs failure rates

Background Let's say I have $m$ machines. Machine $i$ was seen running for a total of $u_i$ time in some observation window and the number of failures observed for it was $n_i$. Now, I want an ...
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Series system: why does the failure rate not depend on component repair rates?

If we have a series system with $n$ components (so if any of the components fails, the system fails), the failure rate of the system becomes simply the sum of the failure rates of the components. It ...
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How to calculate the reliability (probability of success) time-dependent equation given the probability of success for a one-hour time interval?

I've encountered such a probability problem in a reliability engineering textbook: A reliability function can be derived directly from probability theory. Assume that the probability of successful ...
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MTTR and MTBF formulas for series system must be associative

On page $614$ of "Introduction to probability models" by Sheldon Ross ($9$th ed.), he describes the MTTR of a series system as a function of the failure rates ($\lambda_i$) and repair rates (...
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Parallel system when the time to repair of the components becomes very large.

A component has a mean to to failure $U$ and a mean time to repair $D$. Two of these components are combined into a parallel system. In other words, we need just one of the components to be working ...
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1 vote
87 views

Reliability of a system with non-independent parallel components

Consider a system $Z$ that consists of two components $X$ and $Y$, whose failure rates per time unit are $\lambda_x$ and $\lambda_y$ respectively. Assume that if $X$ fails, then the failure rate of $Y$...
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How to find the partial derivatives of the following nested expression?

I want to find the partial derivatives of the expression for $v_3(\boldsymbol{u})$ with respect to $u_1$, $u_2$ and $u_3$ from the expressions below. Here $\Phi$ denotes the cumulative distribution ...
1 vote
84 views

Approximate way to find function of multiple random variables

I want to know if there are approximate solutions to the joint distribution of functions of random variables, where the variables can be only normal or lognormal. The functions would be using only ...
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How compute overall reliability based on two related (dependent) inter-rater reliability coefficients

The Situation In many disability programs, a physician, psychologist, or other health professional conducts an exam with the claimant. Although it's not easy, a research study could determine an inter-...
84 views

Probability that the connection broke down in this grid

See below chart. The yellow blocks are two islands. It's connected by a grid of cables. There are 16 vertical cables and 9 horizontal cables and 12 nodes (blue highlighted) in between. Hurricane ...
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Striking symmetry in series and parallel system mean time to failure and mean time to recovery

I'm looking at the chapter on reliability theory (chapter 9) in the book on Introduction to probability models by Sheldon Ross: https://fac.ksu.edu.sa/sites/default/files/introduction-to-probability-...
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Reliability of an $l$ pod $k$ of $n$ system.

We have $n$ machines that form a system. The system works if $k$ or more of the $n$ machines are running. The machines are hosted in pods. They are evenly distributed across $l$ of the pods (best ...
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probability of 2 elements failure

I am a little bit confused by an exercise. Given a system consisting of 6 elements with reliabilities: $p_2=p_3=p_4=p_5=1/2, p_1 = 2/3, p_6 = 3/4.$ Find a probability of exactly two elements failure ...
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How can we use Birnbaum importance measure for complicated systems in reliability?

Birnbaum importance measure is used when we are provided the survival probabilities of all the components in a coherent system. It is the rate of increase in the system reliability with respect to the ...
44 views

Log-Likelihood function of log-Normal distribution with right censored observations and regression

EXERCISE Find the log-likelihood function for the regression model of log-Normal Distribution considering the right-censored observations ATTEMPT: So, we know that the probability density ...
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Reliability and Provability

I have a system of which 700 are out on the field and one of the components is causing an issue. A new component has been sourced and I want to test it to ensure reliability. So the test I've come up ...
1 vote
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Non parametric plot of $R(t)$

Here is the attached question from the book, introduction to reliability engineering. Here is the attached solution related to it, but i don't know whether it is correct or not. I just want ...
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Proof of likelihood ratio ordering implying hazard rate ordering

This question was originally asked in stats.stackexchange (the link is given below). I'm reposting it here as I did not get any answer to the said post. https://stats.stackexchange.com/questions/...
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Proof of expected time to failure for a "k out of n" system, each component IID exponentially distributed

A "k out of n" system functions if, as the name suggests, k of the n components of the system remain operational, i.e. the system fails upon failure of any n-k+1 components. My A/S/M MAS-I study ...
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How reliable is the linear problems like $\min \|Ax - b\|^2$?

The following linear optimization is common used $$\min_x \|Ax - b\|^2$$ Here, $A$ is the matrix; $x,b$ are the vectors. I am curious about how reliable is the solution $x$ by solving above ...
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Probability of system failure, as it requires two adjacent component to fail

I am stuck with a problem in probability. It says that the system consists of 16 components distributed in corners and intersections as the following: ...
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Combined probability distribution for two hazard functions

I have two items with two different failure rates. I need to find the combined hazard function corresponding to both items failing, from which I can find the combined probability distribution. I ...
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What density distribution could be used for time delays?

I am analyzing a process related with time delays produced in scheduled departures or arrivals of a distribution trucks. The process consists on: The definition of the problem is: Each truck has ...
534 views

What is the expected number of infected people?

I was recently in an interview and got asked an interesting problem which I want to know the answer to. Suppose we're in a party with $n$ people. At minute $t=0$, there is $1$ person who has a ...
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Failure Rate calculation based on manufacturing defects (electronics-capacitor)

Electronics failure rates typically follow the exponential distribution and models like Prism, Mil-Hdbk-217 or 217Plus (newer) are used to predict failure rates. I'm using 217Plus model for ...
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Analytic model for Preventative Maintenance to increase availability?

I would like to create a model that can increase Availability via preventative maintenance. My current model decreases the availability when I include preventative maintenance. Can you suggest a way ...
1 vote
108 views

system reliability, probabilities of rocket launch suspended

A rocket has 3 computers (a,b,c). If a fails than b will take over and so on. Each computer has 0.01 failure rate. These 3 computers are linked to an expensive sensor. If this sensor fails than all 3 ...
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k out of n probability

An engineering system consisting of n components is said to be a k-out-of-n system (k≤n) when the system functions if and only if at least k out of the n components function. Suppose that all ...
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Wilson Confidence Interval for binomial proportion, 99.9% level yields strange results for p=0.5

Fellow Stackers, I have a formal customer requirement for a mechanism to have a failure rate of 0.1% or less. After some study, I found various methods for estimating the confidence interval of a ...
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Historically key works on structural reliability theory?

I want to understand better the structural reliability theory. Is it related to the reliability of structures? Where does it originate and what are the most important work for it?
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1 vote
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Finnish-English Dictionary on Graph Theory and Reliability Engineering?

I need to translate an abstract to Finnish and I need to find a dictionary on Graph theory. Does there exist any canonical reference work in Finnish that could be used as a reference for finding the ...
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Is Site Percolation with Bernoulli variables i.i.d. independent and identically distributed?

I cannot understand the identically distributed part in the i.i.d assumption. Consider a site percolation where each event is a Bernoulli variable. Does this mean ...
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