The failure rate is the ratio of the number of failed hardware samples per unit time to the number of samples initially set for testing, provided that the failed samples are not restored and replaced with serviceable ones.

Since the number of failed samples in a time interval may depend on the location of this interval along the time axis, the purity of failures is a function of time. This characteristic is further denoted.

Time interval;

The number of equipment samples initially set for testing

Expression (10) is a statistical definition of the failure rate. This quantitative characteristic of reliability is easy to give, a probabilistic definition. Let us calculate in expression (10), i.e., the number of samples that failed in the interval.

Obviously:

where N () is the number of samples that are working properly at the time;

The number of samples that are working properly at the time;

With a sufficiently large number of samples, the following ratios are valid:

Substituting (11) into (10) and taking into account (12), (13), we obtain:

Tending to zero and passing to the limit, we get:

or taking into account (4):

It can be seen from this expression that the failure rate is the distribution density of the operating time of the equipment before its failure. Numerically, it is equal to the derivative of the probability of no-failure operation taken with the opposite sign. Expression (16) is a probabilistic definition of the failure rate.

Thus, there are unambiguous dependencies between the failure rate, the probability of failure-free operation and the probability of failures for any distribution law of the time of failure. Based on (16) and (4), these dependences have the form:

The average failure rate is the ratio of the number of failed samples per unit time to the number of tested samples, provided that all failed samples are replaced with serviceable ones (new or remanufactured).

Failure rate

The failure rate is the ratio of the number of failed hardware samples per unit time to the average number of samples that are working properly in a given period of time, provided that the failed samples are not restored and are not replaced with serviceable ones.

where is the number of failed samples in the time interval from to;

Time interval;

Average number of properly working samples in the interval;

The number of samples working properly at the beginning of the interval;

The number of samples working properly at the end of the interval.

Expression (19) is a statistical definition of the failure rate. For a probabilistic representation of this characteristic, let us establish the relationship between the failure rate, the probability of failure-free operation, and the failure rate.

Substitute in expression (19) instead of its value from (11) and (12). Then we get:

Given, we will find:

We tend to zero and passing to the limit, we get:

Integrating, we get:

MTBF

MTBF is called the mathematical expectation of the MTBF. MTBF is determined by the relationship:

To determine MTBF from static data, use the formula:

where is the uptime of the i-th sample;

N0 is the number of samples to be tested.

Let us substitute in expression (25) instead of the derivative of failure-free operation with the opposite sign and perform integration by parts. We get:

Since it cannot have a negative value, it will be replaced by 0, since and then:

Reliability criterion is called a feature by which you can quantify the reliability of various devices. Some of the most widely used reliability criteria are:

Probability of non-failure operation over a certain period of time P(t);

Tav;

MTBF tcr;

Failure rate f(t) or a(t);

Failure rate λ ( t);

Failure flow parameter ω (t);

Ready function K G( t);

Availability ratio K G.

Reliability characteristic the quantitative value of the reliability criterion of a particular device should be named. The choice of quantitative characteristics of reliability depends on the type of object.

2.1.2. Reliability criteria for non-recoverable objects

Consider the following model of device operation. Let it be at work (on trial) N 0 items and the work is considered complete if all of them fail. Moreover, repaired elements are not installed instead of failed elements. Then the criteria for the reliability of these products are:

Probability of uptime P(t);

Failure rate f(t) or a(t);

Failure rate λ ( t);

Mean time to first failure Tav.

Probability of uptime is the probability that under certain operating conditions in a given time interval or within a given operating time, not a single failure will occur.

By definition:

P(t) = P(T> t), (4.2.1)

where: T- the operating time of the element from its activation to the first failure;

t- the time during which the probability of failure-free operation is determined.

Probability of uptime according to statistics about failures is evaluated by the expression:

where: N 0 - the number of elements at the beginning of work (tests);

n(t) is the number of failed elements during t;

Statistical evaluation of the probability of failure-free operation. With a large number of elements (products) N 0 statistical score P(t) practically coincides with the probability of failure-free operation P(t). In practice, sometimes a more convenient characteristic is the probability of failure. Q(t).

Probability of failure is called the probability that under certain operating conditions in a given time interval, at least one failure occurs. Failure and uptime are inconsistent and opposite events, therefore:

Failure rate on statistics is the ratio of the number of failed elements per unit time to the initial number of working (tested), provided that all failed products are not restored. By definition:

where: n(Δ t) is the number of failed elements in the time interval from ( t– Δ t) / 2 to ( t+ Δ t) / 2.

Failure rate there is a probability density (or distribution law) of the product's operation time until the first failure. That's why:

Failure rate on statistics called the ratio of the number of failed products per unit of time to the average number of products that are working properly in a given period of time. According to the definition

where: is the average number of properly working elements in the interval Δ t;

Ni- the number of products that work properly at the beginning of the interval Δ t;

Ni+1 is the number of elements working properly at the end of the interval Δ t.

The probabilistic estimate of the characteristic λ ( t) is found from the expression:

λ( t) = f(t) / P(t). (4.2.7)

Failure rates and uptime are related to

addiction:

Mean time to first failure is the mathematical expectation of the operating time of an element to failure. Like expectation Tav calculated through the failure rate (density of the uptime distribution):

Because t positively and P(0) = 1 and P(∞) = 0, then:

By statistics about failures, the mean time to first failure is calculated by the formula

where: t i - uptime i th element;

N 0 - the number of investigated elements.

As can be seen from formula (4.2.11), in order to determine the mean operating time to the first failure, it is necessary to know the moments of failure of all tested elements. Therefore, it is inconvenient to use this formula to calculate the mean time between failures. Having data on the number of failed elements ni in every i-th time interval, it is better to determine the mean operating time to the first failure from the equation:

In expression (4.2.12) tсрi and m are found by the following formulas:

t cpi = (t i –1 + t i) / 2, m= t k / Δ t,

where: t i–1 - start time i th interval;

t i - end time i th interval;

t k - the time during which all elements were out of order;

Δ t= (t i –1 – t 1) - time interval.

From the expressions for assessing the quantitative characteristics of reliability, it can be seen that all characteristics, except for the mean operating time to the first failure, are functions of time. Specific expressions for a practical assessment of the quantitative characteristics of the reliability of devices are discussed in the section "Laws of Failure Distribution".

The considered reliability criteria make it possible to fairly fully assess the reliability of non-refurbished products. They also allow you to evaluate reliability of remanufactured products to the first failure ... The presence of several criteria does not mean at all that it is always necessary to assess the reliability of elements according to all criteria.

The most complete reliability of products is characterized by failure rate f(t) or a(t). This is because the failure rate is the density of the distribution, and therefore carries all the information about a random phenomenon - uptime.

Mean time to first failure is a fairly obvious characteristic of reliability. However, the application of this criterion for assessing the reliability of a complex system is limited in cases where:

System uptime is much less than MTBF;

The law of distribution of the time of failure-free operation is not one-parameter and for a sufficiently complete assessment, moments of higher orders are required;

The system is redundant;

The failure rate is not constant;

The operating time of individual parts of a complex system is different.

Failure rate- the most convenient characteristic of the reliability of the simplest elements, since it makes it easier to calculate the quantitative characteristics of the reliability of a complex system.

The most appropriate criterion for the reliability of a complex system is an uptime probability... This is due to the following features of the probability of failure-free operation:

It enters as a factor in other, more general characteristics of the system, such as efficiency and cost;

Characterizes the change in reliability over time;

It can be obtained relatively simply by calculation in the process of designing a system and evaluated during its testing.

2.1.3. Reliability Criteria for Recoverable Objects

Consider the following model of work. Let it be at work N elements and failed elements are immediately replaced with serviceable ones (new or repaired). If we do not take into account the time required to restore the system, then the quantitative characteristics of reliability can be the parameter of the flow of failures ω (t) and MTBF tcr.

Failure flow parameter is the ratio of the number of failed products per unit of time to the number of tested ones, provided that all failed products are replaced with serviceable ones (new or repaired). Statistical definition is the expression:

where: n(Δ t) is the number of failed samples in the time interval from t– Δ t/2

before t+Δ t/2;

N- the number of tested elements;

Δ t- time interval.

The failure flow parameter and the failure rate for ordinary flows with limited aftereffect are related by the Voltaire integral equation of the second kind:

According to the famous f(t) you can find all the quantitative characteristics of the reliability of non-recoverable products. Therefore, (4.2.14) is the basic equation that relates the quantitative characteristics of the reliability of non-recoverable and recoverable elements with instant recovery.

Equation (4.2.14) can be written in operator form:

Relations (4.2.15) allow one to find one characteristic in terms of another if there exist Laplace transforms of functions f(s) and ω (s) and inverse transformations of expressions (4.2.15).

The failure flow parameter has the following important properties:

1) for any moment in time, regardless of the distribution law of the time of failure-free operation, the parameter of the flow of failures is greater than the frequency of failures, i.e., ω ( t) > f(t);

2) regardless of the type of functions f(t) parameter of the flow of failures ω ( t) at t→ ∞ tends to 1 / Tav... This important property of the parameter of the flow of failures means that during long-term operation of the repaired product, the flow of its failures, regardless of the distribution law of the time of failure-free operation, becomes stationary. However, this does not mean at all that the failure rate is constant;

3) if λ ( t) is an increasing function of time, then λ ( t) > ω( t) > f(t) if λ ( t) is a decreasing function, then ω ( t) > λ( t) > f(t);

4) for λ ( t) ≠ const parameter of the system failure flow is not equal to the sum of the failure flow parameters of the elements, i.e.:

This property of the parameter of the flow of failures makes it possible to assert that when calculating the quantitative characteristics of the reliability of a complex system, it is impossible to summarize the currently available values ​​of the failure rate of elements obtained from statistical data on product failures under operating conditions, since these values ​​are actually parameters of the flow of failures;

5) for λ ( t) = λ = const the parameter of the failure flow is equal to the failure rate

ω( t) = λ( t) = λ.

From consideration of the properties of the intensity and the parameter of the flow of failures, it is clear that these characteristics are different.

Currently, statistics on failures obtained in the operating conditions of equipment are widely used. Moreover, they are often processed in such a way that the given reliability characteristics are not the failure rate, but the parameter of the failure flow ω ( t). This introduces errors in reliability calculations. In some cases, they can be significant.

To obtain the failure rate of elements from statistical data on failures of repaired systems, it is necessary to use formula (4.2.6), for which it is necessary to know the history of each element of the technological scheme. This can significantly complicate the collection of failure statistics. Therefore, it is advisable to determine λ ( t) with respect to the parameter of the flow of failures ω ( t). The calculation method is reduced

to the following computational operations:

According to statistical data on failures of elements of repaired products and formula (4.2.13), the parameter of the failure flow is calculated and a histogram ω i (t);

The histogram is replaced by a curve that is approximated by an equation;

Find the Laplace transform ω i (s) functions ω i (t);

According to the well-known ω i (s) based on (4.2.15), we write the Laplace transform f i (s) failure rates;

According to the famous f i (s) is the reverse conversion of the failure rate f i (t);

An analytical expression for the failure rate is found by the formula:

The graph λ i ( t).

If there is a section where λ i (t) = λ i = const, then a constant value of the failure rate is taken to assess the probability of failure-free operation. In this case, the exponential law of reliability is considered to be valid.

The above technique cannot be applied if it is not possible to find by f(s) reverse conversion of the failure rate f(t). In this case, one has to apply approximate methods for solving the integral equation (4.2.14).

MTBF the average value of the time between adjacent failures is called. This characteristic is determined by statistics about refusals according to the formula:

where: t i - the time of the correct operation of the element between ( i- 1) -m and i-m refusals;

n- the number of refusals for some time t.

From the formula (4.2.18) it can be seen that in this case the MTBF is determined according to the test data of one sample of the product. If the test is N samples over time t, then the MTBF is calculated by the formula:

where: t ij - time of good work j th sample of the product between ( i- 1) -m and i th refusal;

n j - the number of refusals over time tj-th sample.

MTBF is a fairly obvious characteristic of reliability, therefore it has become widespread in practice. The failure flow parameter and MTBF characterize the reliability of the product being repaired and do not take into account the time required for its restoration. Therefore, they do not characterize the readiness of the device to perform its functions at the right time. For this purpose, criteria such as availability and downtime are introduced.

Availability factor is called the ratio of the time of good work to the sum of the time of good work and the forced downtime of the device, taken for the same calendar period. This characteristic is statistics is determined by:

where: t R - the total time of the correct operation of the product;

t NS - total forced downtime.

Time tp and tp calculated by the formulas:

where: t pi - the operating time of the product between ( i- 1) -m and i th refusal;

t pi - time of forced downtime after i th refusal;

n- the number of product failures (repairs).

To pass to the probabilistic interpretation of the quantity tp and tp are replaced by the mathematical expectations of the time between adjacent failures and the recovery time, respectively. Then:

K r = t cp / (t cp + t v ), (4.2.22)

where: t Wed - MTBF;

t v is the average recovery time.

Forced downtime ratio called the ratio of the time of forced downtime to the sum of times of serviceable work and forced downtime of the product, taken for the same calendar period.

By definition:

K NS = t p / (t p + t NS ), (4.2.23)

or, passing to average values:

K NS = t v / (t cp + t v ). (4.2.24)

The availability factor and the forced downtime factor are interconnected by the dependence:

K NS = 1– K G . (4.2.25)

When analyzing the reliability of restored systems, the availability factor is usually calculated by the formula:

K G =T cp / (T cp + t v ). (4.2.26)

Formula (4.2.26) is true only if the flow of failures is the simplest, and then t Wed = T Wed .

Often the availability factor, calculated by the formula (4.2.26), is identified with the probability that at any moment of time the restored system is operational. In fact, these characteristics are unequal and can be identified under certain assumptions.

Indeed, the probability of a failure of the repaired system at the beginning of operation is small. With the rise of time t this probability increases. This means that the likelihood of finding the system in good working order at the beginning of operation will be higher than after some time has elapsed. Meanwhile, on the basis of formula (4.2.26), the availability factor does not depend on the operating time.

To clarify the physical meaning of the availability factor Kg we write down the formula for the probability of finding the system in good condition. In this case, we will consider the simplest case when the failure rate λ and the recovery rate μ are constant values.

Assuming that for t= 0 the system is in good condition ( P(0) = 1), the probability of finding the system in good condition is determined from the expressions:

where λ = 1 / T cp ; μ = 1 / t v ; K G =T cp / (T cp + t v ).

This expression establishes the relationship between the availability of the system and the probability of finding it in good condition at any time t.

It is seen from (4.2.27) that at t→ ∞, that is, in practice, the availability factor makes sense of the probability of finding a product in good working order during a steady process of operation.

In some cases reliability criteria for recoverable systems can be criteria for non-recoverable systems, for example: probability of operation, failure rate, mean time to first failure, failure rate ... Such the need arises:

When it makes sense to assess the reliability of the restored system before the first failure;

In the case when redundancy is used with the restoration of failed backup devices during the operation of the system, and the failure of the entire redundant system is not allowed.

Failure rate is the ratio of the number of failed samples of equipment per unit time to the number of samples initially set for testing, provided that the failed samples are not restored and not replaced with serviceable ones.

Since the number of failed samples in a time interval may depend on the location of this interval along the time axis, the failure rate is a function of time. This characteristic is further denoted by α (t).

According to the definition

where n (t) is the number of failed samples in the time interval from to; N 0 - the number of equipment samples initially set for testing; - time interval.

Expression (1.10) is a statistical definition of the failure rate. This quantitative characteristic of reliability is easy to give a probabilistic definition. Let us calculate n (t) in expression (1.10), i.e. the number of samples that failed in the interval. Obviously,

n (t) = -, (1.11)

where N (t) is the number of samples that are working properly by the time t; N (t +) is the number of samples that are working properly by the time t +.

With a sufficiently large number of samples (N 0), the following relations are valid:

N (t) = N 0 P (t);

N (t +) = N 0 P (t +). (1.12)

Substituting expression (1.11) into expression (1.10) and taking into account expression (1.12), we obtain:

,

and taking into account expression (1.4) we get:

α (t) = Q / (t) (1.13)

From expression (1.13) it is seen that the failure rate characterizes the density of the distribution of the operating time of the equipment before its failure . Numerically, it is equal to the derivative of the probability of no-failure operation taken with the opposite sign. Expression (1.13) is a probabilistic definition of the failure rate.

Thus, there are unambiguous dependencies between the failure rate, the probability of failure-free operation and the probability of failures for any distribution law of the time of failure. Based on (1.13) and (1.4), these dependences have the form:

. (1.15)

The failure rate, being the distribution density, most fully characterizes such a random phenomenon as the time of failure. Probability of no-failure operation, mathematical expectation, variance, etc. are only convenient characteristics of the distribution and can always be obtained if the failure rate α (t) is known. This is its main advantage as a characteristic of reliability.

The characteristic α (t) also has significant disadvantages. These shortcomings become clear upon detailed consideration of expression (1.10). When determining a (t) from the experimental data, the number of failed samples n (t) is recorded over a period of time, provided that all previously failed samples are not replenished with serviceable ones. This means that the failure rate can be used to assess the reliability of only such equipment that, after a failure occurs, is not repaired and subsequently not used (for example, single-use equipment, simple elements that cannot be repaired, etc.). Otherwise, the failure rate characterizes the reliability of the equipment only up to its first failure.

It is difficult to estimate the reliability of repairable durable equipment using the failure rate. For this purpose, it is necessary to have a family of curves α (t) obtained: before the first failure, between the first and second, second and third failures, etc. However, it should be noted that in the absence of aging of the equipment, the indicated failure rates will coincide. Therefore, α (t) well characterizes the reliability of the hardware also in the case when the failures obey an exponential distribution.

The reliability of equipment for long-term use can be characterized by the frequency of failures obtained under the condition of replacing the failed equipment with a serviceable one. In this case, outwardly the formula (1.10) does not change, but its internal content changes.

The failure rate obtained under the condition of replacing the failed equipment with serviceable (new or refurbished) is sometimes called the average failure rate and is denoted.

Average failure rate is called the ratio of the number of failed samples per unit time to the number of tested samples, provided that all samples that have failed are replaced with serviceable ones (new or remanufactured).

Thus,

where n (t) is the number of failed samples in the time interval from to, N 0 is the number of tested samples (N 0 remains constant during testing, since all failed samples are replaced with serviceable ones), is the time interval.

The average failure rate has the following important properties:

1) . This property becomes obvious when we consider that;

2) regardless of the type of function α (t) at, the average failure rate tends to some constant value;

3) the main advantage of the average failure rate as a quantitative characteristic of reliability is that it allows a fairly complete assessment of the properties of equipment operating in the mode of changing elements. Such equipment includes complex automatic systems designed for long-term use. Such systems are repaired after failures and then re-operated;

4) the average failure rate can also be used to assess the reliability of complex systems of one-time use during their storage;

5) it also quite simply allows you to determine the number of elements of a given type that have failed in the equipment. This property can be used to calculate the required number of elements for the normal operation of the equipment during the time t. Therefore, it is the most convenient characteristic for repair enterprises;

1) knowledge also allows you to correctly plan the frequency of preventive measures, the structure of repair organs, the required number and range of spare elements.

The disadvantages of the average failure rate include the difficulty of determining other reliability characteristics, and in particular the main of them is the probability of failure-free operation, if known.

A complex system consists of a large number of elements. Therefore, it is of interest to find the dependence of the average failure rate. Let us introduce the concept of the total failure rate of a complex system.

Total failure rate is the number of hardware failures per unit of time per one of its copies.

Lecture number 3

Topic number 1. EMC reliability indicators

Reliability indicators characterize such important properties of systems as reliability, vitality, fault tolerance, maintainability, preservation, durability and are a quantitative assessment of their technical condition and the environment in which they operate and operate. The assessment of reliability indicators of complex technical systems at various stages of the life cycle is used to select the structure of the system from a variety of alternative options, assign warranty periods of operation, select a strategy and tactics for maintenance, and analyze the consequences of failures of system elements.

Analytical methods for assessing the reliability indicators of complex technical control systems and decision-making are based on the provisions of the theory of probability. Due to the probabilistic nature of failures, the assessment of indicators is based on the use of methods of mathematical statistics. In this case, statistical analysis is carried out, as a rule, in conditions of a priori uncertainty regarding the distribution laws of random values ​​of the system operating time, as well as on samples of a limited volume containing data on the moments of failure of system elements during testing or under operating conditions.

Probability of no-failure operation (FBR) Is the probability that under certain operating conditions in a given time interval not a single failure will occur. Probability P(t) - function decreasing see Fig. 1 and,

FBG according to the statistics on refusals is estimated by the expression

(1)

where is the statistical estimate of the WBG; - the number of products at the beginning of testing, with a large number of products, the statistical assessment practically coincides with the probability P(t) ; –Number of failed products over time t.

Figure 1. Probability of uptime and probability of failure curves

Probability of failure Q ( t ) Is the probability that under certain operating conditions in a given time interval, at least one failure will occur. Failure and uptime are opposite and incompatible events

(2)

Failure rate a ( t ) - there is a ratio of failed products per unit of time to the initial number of tested products

(3)

where is the number of failed products in the time interval D t.

The failure rate or probability density of failures can be defined as the time derivative of the failure probability

The sign (-) characterizes the rate of decrease in reliability over time.

Mean time to failure - the average value of the duration of the operation of a non-repairable device until the first failure:

where is the duration of work (operating time) to failure i-go device; - the number of monitored devices.

Example. Observations of the operation of 10 electric motors revealed that the first worked to failure for 800 hours, the second - 1200 and further, respectively; 900, 1400, 700, 950, 750, 1300, 850 and 1500 hours. Determine the operating time of the engines to a sudden failure,

Solution... By (5) we have

Failure rate l ( t ) - conditional density of the probability of failure, which is defined as the ratio of the number of failed products per unit of time to the average number of products that are working properly in a given period of time

, (6)

where is the number of devices that have failed over a period of time; - the average number of devices operating properly during the observation period; - observation period.

Probability of uptime P (t) through express

. (8)

Example 1. During the operation of 100 transformers for 10 years, two failures occurred, and each time a new transformer failed. Determine the failure rate of the transformer over the observation period.

Solution. By (6) we have open / year.

Example2... The change in the number of BJI failures due to the production activities of third-party organizations by month of the year is presented as follows:

Determine the average monthly failure rate.

Solution. ; open / month

The expected calculated intensity is l = 7.0.

Mean time between failures - the mean value of the operating time of the device being repaired between failures, defined as the arithmetic mean:

, (9)

where is the operating time before the first, second, n th refusal; n- the number of failures from the start of operation to the end of observation. MTBF, or MTBF, is the mathematical expectation:

. (10)

Example. The transformer failed after about a year. After eliminating the cause of the failure, he worked for another three years and again failed. Determine the mean MTBF of the transformer.

Solution... By (1.7), we compute of the year.

Failure flow parameter - the average number of failures of the repaired device per unit of time, taken for the considered moment in time:

(11)

where is the number of failures i-th device as of the considered moments of time - and t respectively; N- number of devices; - the considered period of work, moreover.

The ratio of the average number of failures of the restored object for its arbitrarily small operating time to the value of this operating time

Example... The electrical device consists of three elements. During the first year of operation, two failures occurred in the first element, one in the second, and no failures in the third. Determine the parameter of the flow of failures.

Solution

From where according to (1.8)

Average resource value calculated from operating or test data using the already known expression for operating time:

.

Average recovery time - the average time of forced or regulated downtime caused by the detection and elimination of one failure:

where is the serial number of the refusal; Is the average time to detect and eliminate a failure.

Availability ratio - the likelihood that the equipment will be operational at a randomly selected time in the intervals between scheduled maintenance. With an exponential distribution law of uptime and recovery time, the availability factor is

.

Forced downtime ratio Is the ratio of forced downtime to the sum of uptime and forced downtime.

Technical utilization rate Is the ratio of the operating time of equipment in units of time for a certain period of operation to the sum of this operating time and the time of all downtime caused by maintenance and repairs during the same period of operation:

.

In addition, [GOST 27.002-83] defines durability indicators, in terms of which the type of actions should be indicated after the onset of the limiting state of the object (for example, the average resource before overhaul; gamma-percentage resource before the average repair, etc.). If the limiting state determines the final decommissioning of the object, then the indicators of durability are called: full average resource (service life), full gamma-percentage resource (service life), full assigned resource (service life).

Average resource Is the mathematical expectation of the resource.

Gamma Percentage Resource- the operating time during which the object does not reach the limit state with a given probability g, expressed as a percentage.

Assigned resource- the total operating time of the object, upon reaching which the intended use should be terminated.

Average service life- mathematical expectation of service life.

Gamma Percentage Life- calendar duration from the start of operation of the object, during which it will not reach the limit state with a given probability g, expressed as a percentage.

Assigned service life- the calendar duration of the object's operation, upon reaching which the intended use should be terminated.

The indicators of maintainability and preservation are determined as follows.

The likelihood of restoration of an operational state- this is the probability that the recovery time of the operational state of the object will not exceed the specified one.

Average recovery time Yaniya is the mathematical expectation of the recovery time.

Average shelf life Is the mathematical expectation of the shelf life.

Gamma Percentage Shelf Life Is the shelf life achieved by an object with a given probability expressed as a percentage.

Distinguish between probabilistic (mathematical) and statistical indicators of reliability. The mathematical indicators of reliability are derived from the theoretical distribution functions of the probability of failures. Statistical indicators of reliability are determined empirically when testing objects on the basis of statistical data on equipment operation.

Reliability is a function of many factors, most of which are random. Hence, it is clear that a large number of criteria are needed to assess the reliability of an object.

Reliability criterion is a feature by which the reliability of an object is assessed.

The criteria and characteristics of reliability are probabilistic in nature, since the factors affecting the object are random in nature and require a statistical assessment.

The quantitative characteristics of reliability can be:
the likelihood of failure-free operation;
average uptime;
failure rate;
failure rate;
various safety factors.

1. Probability of uptime

Serves as one of the main indicators in calculating the reliability.
The probability of failure-free operation of an object is called the probability that it will maintain its parameters within specified limits for a certain period of time under certain operating conditions.

In the future, we assume that the operation of the object occurs continuously, the duration of the object's operation is expressed in units of time t, and the operation started at the moment of time t = 0.
We denote by P (t) the probability of an object's no-failure operation over a period of time. Probability, considered as a function of the upper bound of the time interval, is also called the reliability function.
Probabilistic estimate: P (t) = 1 - Q (t), where Q (t) is the probability of failure.

It is obvious from the graph that:
1. P (t) is a non-increasing function of time;
2. 0 ≤ P (t) ≤ 1;
3. P (0) = 1; P (∞) = 0.

In practice, sometimes a more convenient characteristic is the probability of an object malfunctioning or the probability of failure:
Q (t) = 1 - P (t).
Statistical characteristic of the probability of failure: Q * (t) = n (t) / N

2. Failure rate

The failure rate is the ratio of the number of failed objects to their total number before the start of the test, provided that the failed objects are not repaired or replaced with new ones, i.e.

a * (t) = n (t) / (NΔt)
where a * (t) is the failure rate;
n (t) is the number of failed objects in the time interval from t - t / 2 to t + t / 2;
Δt is the time interval;
N is the number of objects participating in the test.

The failure rate is the density of the distribution of the operating time of the product before its failure. Probabilistic determination of the failure rate a (t) = -P (t) or a (t) = Q (t).

Thus, there is an unambiguous relationship between the failure rate, the probability of failure-free operation and the probability of failures for any law of failure time distribution: Q (t) = ∫ a (t) dt.

Failure is interpreted in the theory of reliability as a random event. The theory is based on the statistical interpretation of probability. Elements and systems formed from them are considered as mass objects belonging to one general population and operating in statistically homogeneous conditions. When we talk about an object, in essence they mean an object taken at random from the general population, a representative sample from this population, and often the entire general population.

For mass objects, a statistical estimate of the probability of no-failure operation P (t) can be obtained by processing the results of reliability tests of sufficiently large samples. The way in which the score is calculated depends on the test plan.

Let the tests of a sample of N objects be carried out without replacements and restorations before the failure of the last object. Let's designate the duration of time until the failure of each of the objects t 1, ..., t N. Then the statistical estimate is:

P * (t) = 1 - 1 / N ∑η (t-t k)

where η is the Heaviside unit function.

For the probability of no-failure operation on a certain segment, it is convenient to estimate P * (t) = / N,
where n (t) is the number of objects that have failed by time t.

The failure rate, determined under the condition of replacing the failed products with serviceable ones, is sometimes called the average failure rate and is denoted by ω (t).

3. Failure rate

The failure rate λ (t) is the ratio of the number of failed objects per unit time to the average number of objects operating in a given period of time, provided that the failed objects are not restored and are not replaced with serviceable ones: λ (t) = n (t) /
where N cf = / 2 is the average number of objects that worked properly in the time interval Δt;
N i - the number of products that worked at the beginning of the interval Δt;
N i + 1 - the number of objects that worked properly at the end of the time interval Δt.

Resource tests and observations on large samples of objects show that in most cases the failure rate changes non-monotonically over time.

From the curve of dependence of refusals on time, it can be seen that the entire period of operation of the facility can be conditionally divided into 3 periods.
I - period - running-in.

Break-in failures are, as a rule, the result of defects and defective elements in the object, the reliability of which is significantly lower than the required level. With an increase in the number of elements in a product, even with the most stringent control, it is not possible to completely exclude the possibility of elements that have certain hidden defects entering the assembly. In addition, errors during assembly and installation, as well as insufficient development of the facility by the service personnel, can lead to failures during this period.

The physical nature of such failures is of a random nature and differs from sudden failures of the normal period of operation in that failures can occur here not with increased, but also with insignificant loads ("burning out defective elements").
The decrease in the value of the failure rate of the object as a whole, with a constant value of this parameter for each of the elements separately, is precisely explained by the "burning out" of the weak links and their replacement with the most reliable ones. The steeper the curve in this area, the better: fewer defective elements will remain in the product in a short time.

To improve the reliability of the facility, taking into account the possibility of break-in failures, you need to:
conduct a more stringent rejection of elements;
to carry out tests of the object in modes close to operational ones and to use only the elements that have passed the tests during assembly;
improve the quality of assembly and installation.

The average running-in time is determined during tests. For especially important cases, it is necessary to increase the running-in period several times compared to the average.

II - th period - normal operation
This period is characterized by the fact that break-in failures have already ended, and failures related to wear have not yet occurred. This period is characterized by extremely sudden failures of normal elements, the MTBF of which is very high.

The retention of the failure rate at this stage is characterized by the fact that the failed element is replaced with the same one, with the same probability of failure, and not the best one, as it happened at the running-in stage.

The rejection and preliminary running-in of the elements going to replace the failed ones is even more important for this stage.
The designer has the greatest capabilities in solving this problem. Often, a design change or a lightening of the operating modes of only one or two elements provides a sharp increase in the reliability of the entire facility. The second way is to improve the quality of production and even the cleanliness of production and operation.

III - period - wear
The period of normal operation ends when wear failures begin to occur. The third period in the life of the product begins - the period of wear.

The likelihood of failures due to wear increases as the service life approaches.

From a probabilistic point of view, system failure in a given time interval Δt = t 2 - t 1 is defined as the probability of failure:

∫a (t) = Q 2 (t) - Q 1 (t)

The failure rate is the conditional probability that a failure will occur in the time interval Δt, provided that it has not occurred before λ (t) = / [ΔtP (t)]
λ (t) = lim / [ΔtP (t)] = / = Q "(t) / P (t) = -P" (t) / P (t)
since a (t) = -P "(t), then λ (t) = a (t) / P (t).

These expressions establish the relationship between the probability of failure-free operation, the frequency and the rate of failure. If a (t) is a non-increasing function, then the following relation is true:
ω (t) ≥ λ (t) ≥ a (t).

4. MTBF

MTBF is the mathematical expectation of uptime.

Probabilistic definition: MTBF is equal to the area under the MTBF curve.

Statistical definition: T * = ∑θ i / N 0
where θ I is the operating time of the i-th object to failure;
N 0 - the initial number of objects.

Obviously, the parameter T * cannot fully and satisfactorily characterize the reliability of durable systems, since it is a characteristic of reliability only until the first failure. Therefore, the reliability of long-term systems is characterized by the average time between two adjacent failures or MTBF t av:
t cf = ∑θ i / n = 1 / ω (t),
where n is the number of failures during time t;
θ i is the operating time of the object between the (i-1) th and the i-th failures.

MTBF is the average value of the time between adjacent failures, subject to the restoration of the failed element.