Failure rate is the average time between failures. Quantitative characteristics of reliability. Probability of failure-free operation

Failure rate is the ratio of the number of failed samples of equipment per unit of time to the average number of samples that work properly in a given period of time, provided that the failed samples are not restored or replaced with serviceable ones.

This characteristic is designated .According to definition

where n(t) is the number of failed samples in the time interval from to ; – time interval, - average number of properly working samples in the interval ; N i is the number of properly working samples at the beginning of the interval, N i +1 is the number of properly working samples at the end of the interval.

Expression (1.20) is a statistical determination of the failure rate. To provide a probabilistic representation of this characteristic, we will establish a relationship between the failure rate, the probability of failure-free operation and the failure rate.

Let us substitute into expression (1.20) the expression for n(t) from formulas (1.11) and (1.12). Then we get:

.

Taking into account expression (1.3) and the fact that N av = N 0 – n(t), we find:

.

Aiming towards zero and passing to the limit, we get:

. (1.21)

Integrating expression (1.21), we obtain:

Since , then based on expression (1.21) we obtain:

. (1.24)

Expressions (1.22) – (1.24) establish the relationship between the probability of failure-free operation, the frequency of failures and the failure rate.

Failure rate as a quantitative characteristic of reliability has a number of advantages. It is a function of time and allows one to clearly establish characteristic areas of equipment operation. This can significantly improve the reliability of the equipment. Indeed, if the running-in time (t 1) and the end of work time (t 2) are known, then it is possible to reasonably set the time for training the equipment before the start of its operation.

operation and its service life before repair. This allows you to reduce the number of failures during operation, i.e. ultimately leads to increased equipment reliability.

The failure rate as a quantitative characteristic of reliability has the same drawback as the failure rate: it allows one to fairly simply characterize the reliability of equipment only up to the first failure. Therefore, it is a convenient characteristic of the reliability of disposable systems and, in particular, the simplest elements.

Based on the known characteristic, the remaining quantitative characteristics of reliability are most easily determined.

The indicated properties of the failure rate allow it to be considered the main quantitative characteristic of the reliability of the simplest elements of radio electronics.

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What is a site's bounce rate?

Let's look at an example. During the month, only 140 visitors visited the site, 60 of them viewed only one page and closed your resource, the remaining 80 viewed two or more pages. Divide 60 by 140 and multiply by 100%. As a result, we get a failure rate on the site of 43%.

What is the normal bounce rate on a website?

Achieving a zero level is almost impossible. Even popular online stores have a failure rate of 30-40%. The average for different sites varies greatly, and we need to be sure to take this into account:

• for a portal site or service site this value is approximately 10% to 30%;
• For online stores, the normal percentage of failures on the site is already higher - 20-40%;
• even more for information sites - 40-60%.

You should not focus on any specific number. It is more important that the bounce rate is lower than that of competitors.

Reasons for refusal on the site: how to keep visitors on the site?

1. Download speed

The average user strives to obtain all the required information as soon as possible. Believe me, a few seconds of waiting can be a good reason why the site will be bypassed. Put yourself in the visitor's shoes. It is unlikely that you will wait more than 10 seconds. You should look for site errors that affect this parameter. Also, remove ads before content. Many ad servers are extremely slow, so the likelihood of immediately saying goodbye to the site is very high.

2. Too much advertising

Remember forever: the site is not a Christmas tree.

Flashing and sparkling elements really attract the eye, but at the same time they cause persistent disgust among visitors. Stupid tabloid-style headlines and pop-ups lead to this effect. Is your resource filled with really interesting content? Feel free to launch pop-up ads a minute after the visitor logs in - this will help reduce bounces on the site.

3. Clear navigation, competent search

Do you think intuitive algorithms are only important in computer games? Give the guest the opportunity to feel like a fool; you will never see him again. Of course, the desire for uniqueness and originality is commendable. However, such originality will have a negative impact on your bounce rate if you force visitors to search for information.

We should also mention an effective tool - search. Its absence on sites with a large number of pages and products causes a lot of inconvenience; an ordinary visitor will prefer to quickly leave the site and look for the necessary information on another resource.

4. Music and video are obvious enemies

Unlike supermarket customers, where there is no way to hide from the background music, your guests can always say goodbye to it instantly. People are tired of unnecessary pictures and sounds. Do you like a beautiful melody playing endlessly in a circle? The only desire is for her to stop. Desperate to turn off the music, the visitor will leave the site.

Let's discuss the video, here the situation is even worse than with music. Many users refuse to pay for the traffic of the imposed video. This behavior of a webmaster is directly associated with a thief picking his pocket. Do you like this kind of role? Then give up unnecessary attributes.

How to keep a visitor on the site? Don't force him to listen and watch something he doesn't want.

5. Unregister

You know about the high competition on the Internet. Have you ever seen the free use of numerous sites without the slightest hint of registration? Many sites offer registration through social network accounts. But mentality and natural laziness force us to look for warmer places where “registration” is completely absent. If you remove a feature that irritates guests today, you will stop being surprised by the number of refusals tomorrow.

6. Update information

Prices from two years ago, a catalog of clothes that lost relevance 10 years ago are good reasons for refusal on the site. If phone numbers or terms of delivery of goods have changed, immediately update the site data. Is your creation well-designed and up-to-date? Then feel free to add interesting articles. New visitors often study the dates of the latest publications, try to please the audience.

7. Use your 404 page correctly

It is impossible to insure against software errors, so the appearance of a 404 page should be provided for. Thanks to Google's suggestions, improving this page is easy by using Google Webmaster Tools. Simply adding a link to the main page and a search box will help smooth out the awkward situation with the 404 page. It remains to be generous with humor, design and the problem can be considered solved.

8. Add contrasts, sort out fonts

Minimum steps are needed to make it easier for visitors to read the information offered. It is the contrasting background and bright pictures that will help highlight areas of the site that need special attention.

Choosing the perfect font is quite easy. You should layout the article and read it carefully. If your eyes feel comfortable while reading, then everything was done correctly. It is also necessary to take into account the impact of content color, font type, line spacing, background color, and the presence of paragraphs on readability.

9. Improve your design

Only a beginner can afford cheap, unprofessional design. Such savings will make visitors doubt the seriousness of the resource owner and the veracity of the information posted on the site.

Imagine walking into an untidy office or store that hasn't been wallpapered for decades. Nice? Also, visitors rush to neat, beautifully designed websites.

10. Get rid of gray sheets, improve text quality

No matter how interesting and unique the text posted on the page is, its design should be given at least minimal attention. Bright headings, sensible lists, and correctly highlighted paragraphs will help convey the necessary information to the reader.

Use the advice above. Format your articles correctly and your visitors will read them to the end!

In addition, you should get rid of clumsily entered key phrases, spelling and punctuation errors. If you are working with a highly specialized topic, then try to use terms carefully. Be generous by compiling a mini-dictionary or simply providing clear definitions in articles.

11. Offer additional content

If you are familiar with the term "related products", half the battle is done. Imagine the process of purchasing beer in a store. Fish, crackers, and chips are perfect as complementary products. This principle also applies when working on website content. For example, a woman chooses a stylish dress in a store, invite her to look at the section of modern jewelry and luxury underwear. The simplest technique will help increase the number of pages viewed and make the entire resource more attractive.

12. Extremely useful information

Competent, unique, but absolutely useless texts are also among the reasons for refusal on the site. A visitor who comes to look at the cost of orthopedic mattresses will be disappointed to see lengthy discussions about their relevance, high quality and health benefits. Give specific answers to a specific request, stop pouring water.

Of course, the provided list of factors that irritate visitors is not complete. But you have plenty of work ahead of you. Using the suggested tips, you can significantly reduce the site’s bounce rate.

Failure Rate- the ratio of the probability distribution density of failures to the probability of failure-free operation of an object:

where is the probability density of failures and is the probability of failure-free operation.

In simple words, the failure rate expresses the chance of an object (for example, a device) that has already worked without failure for a certain time to fail at the next moment in time.

Statistically, the failure rate is the ratio of the number of failed equipment samples per unit time to the average number of samples operating properly over the interval:

Where is the average number of properly working samples

on the interval.

Relation (1) for small ones follows directly from the formula for the probability of failure-free operation (3)

and formulas for the distribution density of failure-free operation (failure rates) (4)

Based on the definition of failure rate (1), the following equality holds:

Integrating (5), we get:

Failure rate is the main indicator of the reliability of elements of complex systems. This is explained by the following circumstances:

• the reliability of many elements can be assessed with one number, because the failure rate of elements is a constant value;

• the failure rate is not difficult to obtain experimentally.

Experience in operating complex systems shows that changes in the failure rate of most objects are described by a shaped curve.

Time can be divided into three characteristic sections: 1. Run-in period. 2. Period of normal operation. 3. The aging period of the object.

The period of running-in of an object has an increased failure rate caused by running-in failures caused by defects in production, installation and adjustment. Sometimes the end of this period is associated with warranty service of the object, when the elimination of failures is carried out by the manufacturer. During normal operation, the failure rate practically remains constant, while failures are random in nature and appear suddenly, primarily due to random load changes, non-compliance with operating conditions, unfavorable external factors, etc. It is this period that corresponds to the main operating time of the facility. An increase in failure rate refers to the aging period of an object and is caused by an increase in the number of failures due to wear, aging and other reasons associated with long-term operation. That is, the probability of failure of an element that survives for a moment in a certain subsequent period of time depends on the values ​​​​only in this period, and therefore the failure rate is a local indicator of the reliability of the element in a given period of time.

The average value of the operating time of products in a batch until the first failure is called the average time to the first failure. This term applies to both repairable and non-repairable products. For non-repairable products, instead of the above, the term mean time to failure can be used.

GOST 13377 - 67 for non-repairable products introduced another reliability indicator, called failure rate.

Failure rate is the probability that a non-repairable product, which worked without failure until moment t, will fail in the next unit of time, if this unit is small.

The failure rate of a product is a function of the time it takes to operate.

Assuming that the failure-free operation of a certain unit in the electronic control system of a vehicle is characterized by a failure rate numerically equal to the calculated one, and this intensity does not change throughout its entire service life, it is necessary to determine the time to failure TB of such a unit.

The control subsystem includes k series-connected electronic units (Fig. 2).

Fig.2 Control subsystem with sequentially connected blocks.

These blocks have the same failure rate, numerically equal to the calculated one. It is required to determine the failure rate of the subsystem λ P and its average time to failure, to plot the dependence of the probability of failure-free operation of one block RB (t) and the subsystem RP (t) on the operating time and to determine the probabilities of failure-free operation of the block RB (t) and the subsystem RP (t) to operating time t= T P.

The failure rate λ(t) is calculated using the formula:

, (5)

Where is the statistical probability of a device failure on an interval, or otherwise the statistical probability of a random variable T falling within a specified interval.

Р(t) – calculated in step 1 – probability of failure-free operation of the device.

Setpoint 10 3 h - 6.5

Interval =

λ(t) = 0.4 / 0.4*3*10 3 h = 0.00033

Let us assume that the failure rate does not change throughout the entire service life of the object, i.e. λ(t) = λ = const, then the time to failure is distributed according to an exponential (exponential) law.

In this case, the probability of failure-free operation of the unit is:

(6)

R B (t) = exp (-0.00033*6.5*10 3) = exp(-2.1666) = 0.1146

And the average operating time of a block to failure is found as:

1/0.00033 = 3030.30 hours.

When k blocks are connected in series, the failure rate of the subsystem they form is:

(8)

Since the failure rates of all blocks are the same, the failure rate of the subsystem is:

λ P = 4*0.00033 = 0.00132 hours,

and the probability of failure-free operation of the system:

(10)

R P (t) = exp (-0.00132*6.5*10 3) = exp (-8.58) = 0.000188

Taking into account (7) and (8), the average time to failure of the subsystem is found as:

(11)

1/0.00132 = 757.58 hours.

Conclusion: As we approach the limit state, the failure rate of objects increases.

Calculation of the probability of failure-free operation.

Exercise: For operating time t = it is necessary to calculate the probability of failure-free operation Рс() of the system (Fig. 3), consisting of two subsystems, one of which is a backup one.

Rice. 3 Scheme of a redundant system.

The calculation is carried out under the assumption that the failures of each of the two subsystems are independent.

The probabilities of failure-free operation of each system are the same and equal to R P (). Then the probability of failure of one subsystem is:

Q P () = 1 – 0.000188 = 0.99812

The probability of failure of the entire system is determined from the condition that both the first and second subsystems have failed, i.e.:

0,99812 2 = 0,99962

Hence the probability of failure-free operation of the system:

,

Р с () = 1 – 0.98 = 0.0037

Conclusion: In this task, the probability of failure-free operation of the system in the event of failure of the first and second subsystems was calculated. Compared to a sequential structure, the probability of failure-free operation of the system is less.

The most convenient for analytical description is the so-called exponential (or exponential) reliability law, which is expressed by the formula

where is a constant parameter.

The graph of the exponential reliability law is shown in Fig. 7.10. For this law, the distribution function of failure-free operation time has the form

and density

This is the exponential distribution law already known to us, according to which the distance between neighboring events in the simplest flow is distributed with intensity (see § 4 of Chapter 4).

When considering questions of reliability, it is often convenient to imagine the matter as if the element were subject to the simplest flow of failures with intensity I; the element fails at the moment when the first event of this thread arrives.

The image of a “failure flow” takes on real meaning if the failed element is immediately replaced with a new one (restored).

The sequence of random moments in time at which failures occur (Fig. 7.11) represents the simplest flow of events, and the intervals between events are independent random variables distributed according to the exponential law (3.3),

The concept of “failure rate” can be introduced not only for the exponential, but also for any other reliability law about density; the only difference will be that with a non-exponential law, the failure rate R will no longer be a constant value, but a variable.

The intensity (or otherwise “danger”) of failures is the ratio of the distribution density of the time of failure-free operation of an element to its reliability:

Let us explain the physical meaning of this characteristic. Let a large number N of homogeneous elements be tested simultaneously, each until it fails. Let us denote - the number of elements that turned out to be serviceable by the time , as before, - the number of elements that failed in a short period of time. Per unit of time there will be an average number of failures

Let us divide this value not by the total number of tested elements N, but by the number of elements that are operational at time t. It is easy to verify that for large N this ratio will be approximately equal to the failure rate

Indeed, for large N

But according to formula (2.6)

In works on reliability, the approximate expression (3.5) is often considered as a definition of the failure rate, i.e., it is defined as the average number of failures per unit of time per one operating element.

The characteristic can be given another interpretation: this is the conditional probability density of the failure of an element at a given time t, provided that before moment t it worked without failure. Indeed, let us consider the element of probability - the probability that over time an element will move from the “working” state to the “not working” state, provided that it was working before moment t. In fact, the unconditional probability of failure of an element in a section is equal to This is the probability of combining two events:

A - the element worked properly until the moment

B - the element failed during a period of time. According to the rule of multiplication of probabilities:

Considering that we get:

and the value is nothing more than the conditional probability density of the transition from the “working” state to the “failed” state for moment t.

If the failure rate is known, then reliability can be expressed through it. Considering that we write formula (3.4) in the form:

Integrating, we get:

Thus, reliability is expressed through the failure rate.

In the special case when , formula (3.6) gives:

i.e., the exponential reliability law already known to us.

Using the image of a “failure flow”, one can interpret not only formula (3.7), but also a more general formula (3.6). Let us imagine (quite conventionally!) that an element with an arbitrary reliability law is subject to a flow of failures with variable intensity. Then formula (3.6) for expresses the probability that no failure will appear in the time interval (0, t).

Thus, both with the exponential and with any other law of reliability, the operation of the element, starting from the moment of switching on, can be imagined in such a way that the element is subject to a Poisson flow of failures; for an exponential reliability law it will be a flow with constant intensity, and for a non-exponential one - with variable intensity

Note that this image is only suitable if the failed element is not replaced with a new one. If, as we did before, we immediately replace the failed element with a new one, the failure flow will no longer be Poisson. Indeed, its intensity will depend not simply on the time t that has elapsed since the beginning of the entire process, but also on the time t that has elapsed since the random moment of inclusion of this particular element; This means that the flow of events has an aftereffect and is not Poisson.

If, throughout the entire process under study, this element is not replaced and can fail no more than once, then when describing a process that depends on its functioning, one can use the scheme of a Markov random process, but with a variable rather than constant intensity of the failure flow.

If the non-exponential reliability law differs relatively little from the exponential one, then, for the sake of simplification, it can be approximately replaced by an exponential one (Fig. 7.12). The parameter of this law is chosen so as to keep unchanged the mathematical expectation of failure-free operation time, equal, as we know, to the area limited by the curve and coordinate axes. To do this, you need to set the parameter of the exponential law equal to

where is the area limited by the reliability curve

Thus, if we want to characterize the reliability of an element by a certain average failure rate, we need to take as this intensity the value inverse to the average failure-free operation time of the element.

Above, we defined the value t as the area limited by the curve. However, if you only need to know the average time of failure-free operation of an element, it is easier to find it directly from statistical material as the arithmetic mean of all observed values ​​of the random variable T - the operating time of the element before its failure. This method can also be used in cases where the number of experiments is small and does not allow one to construct a curve accurately enough

Example 1. The reliability of an element decreases over time according to a linear law (Fig. 7.13). Find the failure rate and mean time between failures of the element

Solution. According to formula (3.4) in section ) we have:

According to the given reliability law 4