Methods of redundancy of technical systems. Reservation types. Classification of system redundancy methods
A system without redundancy has a significant drawback - its reliability is always less than the reliability of the most unreliable element of the system.
Redundancy is a method of increasing reliability by introducing spare (redundant) elements that are redundant in relation to the minimum structure of the system.
Equipment with redundant elements is called redundant. The efficiency of redundancy is determined by the fact that, due to redundancy, it is possible to create reliable equipment even from relatively unreliable elements.
The redundancy ratio is the number of reserve elements per one redundant.
Redundancy efficiency is evaluated using reliability improvement factors, defined as the ratio of the reliability index before and after the system transformation:
Reliability ratio:
Durability factor:
The classic methods are permanent redundancy and replacement redundancy.
With constant redundancy, the backup elements are connected to the main ones through the communication elements (EC). Reserve elements operate in the same mode as the main ones during the entire period of system operation. This is a way to reserve elements and simple nodes (Fig. 2.10).
When redundant by substitution, the functions of the main element are transferred to the backup only if the main element fails (Fig. 2.11). This method is used when backing up large units or entire systems.
|
Rice. 2.10. Permanent redundancy Fig. 2.11. Reservation by replacement
Recently, functional redundancy has become widespread. Functional redundancy can operate in both always-on redundancy mode and replacement redundancy mode. Functional redundancy is based on:
- on the ability of individual elements of the system to perform, in addition to the main ones, also additional functions;
- the ability of various elements of the system to perform the same functions, but in different physical ways.
The first direction includes methods for synthesizing redundant circuits from modules of the same type or multifunctional elements that allow rebuilding their structure in the event of failure of individual elements, for example, structures of the same type.
The second direction can be attributed, for example, duplication of some blocks of equipment by others with different physical actions: the electric drive of the aircraft is duplicated by a hydraulic drive, the magnetic compass on the ship is duplicated by a radio compass.
Related information:
- Reservation types. To improve the reliability of systems and elements, redundancy is used, based on the use of one or another type of redundancy.
1. The main positive property of redundancy is that it allows designing reliable systems from unreliable elements. This property of redundancy favorably distinguishes it from other methods of increasing reliability.
2. The reliability gain in terms of the probability of failures G Q (t) always starts from zero and asymptotically tends to one, regardless of the reliability of the redundant system and its elements. In this case, the growth rate G Q (t) is the higher, the less reliable the main system and the lower the redundancy ratio. With non-slip redundancy with a fractional multiplicity, starting from a certain value of the time of continuous operation of the system, the values of G Q (t) can be greater than one. This means that such a reservation may not be practical at all.
3. The reliability gain of a redundant system compared to a non-redundant one is higher, the shorter the time of continuous operation of the redundant system and the more reliable the system is redundant. This is the basic contradiction of any reservation.
Reliability gain in terms of failure rate G λ t) does not qualitatively differ from G Q (t) . Therefore, the properties of redundant systems, if their reliability is assessed by the failure rate, will be the same as when reliability is assessed by the probability of failures. Dependences G λ (t) have the same form as G Q (t) .
4. The MTBF for fractional-fold redundancy and non-rolling redundancy can be less than the MTBF for a non-redundant system. This may be provided that the number of backup systems is less than the number of main ones. With an increase in the redundancy ratio, the reliability gain G Q (t) in terms of mean time between failures increases, and the growth rate decreases significantly with an increase in the redundancy ratio. This property is also inherent in the general and element-by-element redundancy with always-on redundancy.
With redundancy by substitution and ideal switches, the gain G T grows linearly with the increase in the multiplicity of redundancy with total redundancy, and with separate (element-by-element) or sliding - faster. The circuit implementation of replacement redundancy requires the use of switches. The presence of switches significantly reduces the growth rate of the payoff G T (m). The current switches have such low reliability that in many cases an increase in the replacement redundancy ratio in its circuit implementation leads to a decrease in the growth rate of gain G T .
It follows from the foregoing that a significant increase in the redundancy ratio, and hence the weight, dimensions and cost of the equipment, leads to a less significant increase in the mean time of failure-free operation.
This is the second main contradiction of any redundancy, which limits its application to improve the reliability of complex automatic systems designed for long continuous operation.
5. With an increase in the time of continuous operation of the redundant system, its availability factor K g and gain G K g fall.
Thus, redundancy increases the availability of the system for action only under certain limiting conditions.
4. A characteristic feature of complex automatic disposable systems is that most of the time they are in a state of storage. Obviously, at the moment such a system is put into operation, all its elements must be in good order. This means that the failure of at least one element of a redundant system during its storage should be considered a failure of the entire system. Since the number of elements of a redundant system is always higher than the number of elements of a non-redundant system, the former will always have a greater risk of failure.
If the multiplicity of redundancy (m) , then the probability of failure-free operation will be:
non-redundant system
reserved
Obviously, with a large redundancy ratio (m), the probability of failure-free operation P c (t) will be low.
An important conclusion follows from the above reasoning: the reliability of a redundant system during its storage is always lower than the reliability of a non-redundant system of the same purpose. An increase in the number of failures of a redundant system during its storage requires an increase in the frequency of checks by m times, an increase in the number of spare elements. All this leads to an increase in the cost of operation.
The noted redundancy properties allow us to draw the following important conclusions:
1) redundancy as a means of improving reliability is most appropriate to use to improve the reliability of complex systems designed for a short time of continuous operation, often requires a high redundancy ratio. This limits its use in systems where there are weight, size, or cost restrictions;
2) increasing the reliability of equipment through its redundancy is carried out due to the deterioration of its characteristics such as weight, dimensions, cost, operating conditions (increase in the frequency of checks, the number of spare elements, assemblies and individual devices, etc.).
Classification of redundancy methods
IMPROVE RELIABILITY
RESERVATION AS A WAY
Reservation- this is one of the main means of ensuring a given level of reliability (especially non-failure operation) of an object with insufficiently reliable elements.
Redundancy is the use of additional tools and (or) capabilities in order to maintain the operable state of an object in case of failure of one or more elements. That. is a method of increasing the reliability of an object by introducing redundancy. In turn, redundancy is additional means and (or) capabilities that are super-minimal necessary for the object to perform the specified functions. The task of introducing redundancy is to ensure the normal functioning of the object after the occurrence of a failure in its elements.
According to the type of redundancy, the following classification of redundancy methods is adopted(Fig. 10.1).
Structural(hardware, element, circuit) provides for the use of reserve elements of the object structure. The essence of structural redundancy is that additional elements are introduced into the minimum required version of the object.
The elements in the block diagram are divided into main(an element necessary for the object to perform the required functions in the absence of failures of its elements and reserve(an element designed to perform the functions of the main element in the event of a failure of the latter). The definition of the main element is not related to the concept of minimality of the main structure of the object, since the element, which is the main one in some operating modes, can serve as a backup in other conditions. Reserved element - the main element, in case of failure of which the object provides a reserve element.
Temporary reservation associated with the use of spare time. At the same time, it is assumed that the time required for the object to perform the necessary work is obviously greater than the minimum required. Time reserves can be created by increasing the productivity of the object, the inertia of its elements, etc. For chemical engineering facilities, this type of redundancy is implemented using the following methods and operations:
1) an increase in operating conditions of the estimated operating time necessary to fulfill the set goal or to produce a given amount of chemical products;
2) devices and machines are developed for a higher productivity value than is required by calculation, and, consequently, objects can complete the task in a shorter period of time than is established by the plan;
3) introduction into the structure of the technological scheme of intermediate tanks (reservoirs and bunkers for the accumulation of the product) between individual production devices. This technique creates conditions that allow the operation of the technological scheme to continue, even if part of the equipment up to the intermediate tank or bunker is stopped. A similar function is also performed by gas tanks, warehouses, etc.;
4) the functional inertia of objects, for example, the thermal inertia of furnaces, due to lining arrays, prevents a rapid decrease in the temperature of the furnace during a break in the supply of fuel. The inertia of objects makes it possible to eliminate an accident in a short possible period of time by switching the process to some backup object or by performing some other operations.
Information redundancy- this is a redundancy with the use of information redundancy. Examples of information redundancy are multiple transmission of the same message over a communication channel; the use of various codes in the transmission of information over communication channels that detect and correct errors that appear as a result of equipment failures and the influence of interference; the introduction of redundant information symbols in the processing, transmission and display of information. The excess of information makes it possible to compensate for the distortions of the transmitted information or to eliminate them.
Functional Redundancy– redundancy in which a given function can be performed different ways and technical means.
For example, to manufacture a part, a group of machines is used, each of which can perform one of the sequential machining operations. In this case, functional redundancy will be the introduction of a universal or multi-operational machine into the production line. As another example, we can cite the creation of structurally combined reaction-mass transfer processes occurring in one apparatus of chemical technology. Functional redundancy also includes production and excess redundancy (for example, the manufacture of products with an increased accuracy class), which is often used to ensure and improve the reliability of chemical engineering facilities. At the same time, conditions are created for increasing reliability and durability, since, first, in the process of operation, the object wears out to the traditional accuracy class, and then already underway normal wear process.
load(or regime) redundancy - redundancy with the use of load reserves - involves the use of the facility's ability to perceive additional, or redundant, loads. In chemical engineering, it is implemented by introducing safety factors, reducing the permissible operating parameters (pressure, speed).
Redundancy in the chemical industry is widely used to improve the reliability of power supply systems (electricity, heat, water supply), devices that ensure the safety of the process are redundant (several safety valves are installed on one high-pressure tank).
Rice. 10.1 Classification of redundancy methods
Redundancy allows you to create objects whose reliability is higher than the reliability of their constituent elements, however, the possibilities of using redundancy are limited due to the increase in the mass and production area of the system and due to the increase in the cost of a unit of product compared to non-redundant. This leads to the problem of choosing the optimal redundancy method and the optimal number of redundant elements.
To analyze the structural reliability of technical systems, it is of interest structural redundancy– introduction of additional elements into the structure of the object that perform the functions of the main elements in case of their failure.
The classification of various methods of structural redundancy is carried out according to the following criteria:
1) according to the reserve switching scheme :
- general reservation, in which the object as a whole is reserved;
- separate redundancy, in which individual elements or their groups are reserved;
- mixed redundancy, in which different types of redundancy are combined in one object;
2) according to the method of inclusion of the reserve :
- permanent redundancy, without restructuring the object structure in the event of a failure of its element;
- dynamic redundancy, in which, in the event of an element failure, the circuit structure is rebuilt. In turn, dynamic subdivided into a:
a) redundancy by replacement, in which the functions of the main element are transferred to the backup only after the failure of the main one;
b) sliding redundancy, in which several main elements are backed up by one or more reserve ones, each of which can replace any main one (that is, the groups of main and reserve elements are identical).
3) according to the mode of operation of the reserve :
Loaded redundancy, in which the backup elements (or one of them) are in the mode of the main element;
Lightweight redundancy, in which the backup elements (at least one of them) are in a less loaded mode compared to the main ones;
Unloaded redundancy, in which the backup elements are in an unloaded mode before they begin to perform their functions.
4) according to the conditions of restoration of working capacity during operation:
Redundancy with recovery;
Backup without recovery.
The main characteristic of structural redundancy is the multiplicity of redundancy - the ratio of the number of reserve elements to the number of reserved (main) elements. Reservation can be with integer and fractional multiplicity (such as 2:3; 4:2, etc.).
The reservation of one main element with one reserve (i.e. with a multiplicity of 1: 1) is called duplication.
When redundant with a fractional multiplicity, the normal operation of a redundant connection is possible provided that the number of serviceable elements is not less than necessary for normal operation. When redundant with a fractional multiplicity, one backup element of the system falls on two or more main elements. A reservation with a fractional multiplicity also includes a reservation with a sliding (floating) reserve.
In chemical engineering, the reliability of non-recoverable redundant devices and production lines, as a rule, is increased by:
- general and separate redundancy with permanently switched on reserve;
- general and separate reservation by the method of substitution;
– redundancy of the system with a sliding (floating) reserve.
The use of this type of structural redundancy as a sliding one is possible only if there is a special diagnostic device that allows you to find a faulty element and connect a backup one instead. In this case, the reserve elements must be of the same type. However, this type of redundancy gives the greatest gain in reliability.
The increase in system reliability as a result of redundancy or the use of highly reliable elements can be quantified by the reliability gain factor, defined as the ratio of the reliability indicator before and after the system conversion
Structural diagram of the reserve group, consisting of one main and m reserve elements, is shown in fig. 20.3.3.1.
Rice. 10.2. Structural diagram of the system from one main and m reserve elements
If a system with a permanently switched on reserve is given, consisting of two elements operating in parallel (Fig. 10.2, m= 1) with the probability of failure-free operation of the main element R 1 , reserve - R 2 , then the probability of failure-free operation of such a system is equal to
P(t) = 1 – (1 – p 1 (t)) (1 – p 2 (t))
In the case of equally reliable elements:
P(t) = 1 – (1 – p 1 (t)) 2 = 2R 1 – R 1 2 = R 1 (2 – R 1). (10.1)
For the exponential distribution of failures of each of the two parallel operating elements p 1 (t) = p 2 (t) = exp(–l t) taking into account (10.1), the probability of failure-free operation of the system is defined as P(t) = 2e-l t – e-2l t.
Since the mean time between failures of a single non-redundant element is:
The mean time to failure of the system will be:
Then the reliability gain for the system. consisting of two elements operating in parallel, compared with one non-reserved element is equal to:
The probability of failure-free operation of a system consisting of one main and m reserve non-equivalent elements (Fig. 10.2), is determined by the formula:
In the case of equally reliable elements, this formula will take the form:
For an exponential distribution of the probability of failure-free operation of elements, i.e. p(t) = e-l t, we get:
At small t a simple estimate from below is valid:
where i– failure rate i-th element.
With identical elements, the previous formula becomes:
Reliability gain W T(2) by the mean time of failure-free operation of a system consisting of ( m+ 1) equally reliable elements operating in parallel, compared with the average uptime of one non-reserved element, provided that the law of distribution of the probability of failure-free operation of each element is exponential, is equal to:
Mean time to failure of the system T(Fig. 10.2) in the general case can only be found by numerical integration using the formula
For identical elements, the mean time to failure with the probability of no-failure operation of the elements R(t) = exp(–l t) given is defined as:
For large values m :
10.2.1 According to the reserve switching scheme at general reservation the object as a whole is reserved. At separate reservation separate elements (subsystems) of the object or their groups are reserved.
Examples of general redundancy (Fig. 10.3, a) are standby production lines or units of large unit capacity. With separate reservation (Fig. 10.3, b), individual elements of the object are reserved.
Rice. 10.3. Redundancy schemes of a system consisting of n main elements: a) general redundancy with a permanently switched on reserve (number of reserve circuits m= 1); b) separate (element-by-element) duplication with a permanently switched on reserve
For serial connection system n elements with a common redundancy (duplication) (Fig. 10.3, a) the probability of failure-free operation is equal to:
With separate redundancy (duplication) (Fig. 10.3, b):
The system reliability gain coefficients in terms of the probability of failure-free operation for these two cases are respectively equal to:
It follows that separate redundancy is more efficient than the general one: for example, for a system of three identical elements ( n= 3) at R = 0,9 R = 0,9 3 = 0,729; R (1) = 0,729(2 – 0,729) = 0,9266, R(2) \u003d 0.729 (2 - 0.9) \u003d 0.9703. Then: Gp (1) = 1,27; Gp (2) =1,33.
Separate redundancy, ceteris paribus, gives a greater gain in reliability than the general one. Separate redundancy is especially beneficial with a large number of elements in the system and with an increase in the redundancy ratio.
The probability of failure of a system consisting of n elements (one main and ( n- 1) reserve), without taking into account the reliability of the switches is calculated by the formula
Serial connection system n elements with common redundancy ( m reserve circuits) will function normally while maintaining the operability of at least one of them.
For a general redundancy scheme with a permanently switched on reserve, the probability of failure-free operation of the system (Fig. 10.4), PBR is equal to (the elements are equally reliable, the probability of failure-free operation of each element is equal to p(t)):
In turn, trouble-free work i-th chain ( i = 1, ..., m) will take place during the failure-free operation of each of the n elements. Then:
Here: p ij– probability of failure-free operation j-th element i-th chain ( j = 1, ..., n); n- the number of series-connected circuit elements.
Rice. 10.4. Block Diagram of a General Redundant Always On Redundant System with a Serial Connection n elements
For the case when all elements are equally reliable (with a probability of failure-free operation equal to R), the probability of failure-free operation of the main system from n elements (failures are random and independent) is equal to: .
Therefore, the probability of failure of the entire system, consisting of one main and m redundant systems will be equal to:
Then the probability of failure-free operation of a system with general redundancy is:
If the failure rate is constant, i.e. R(t) = exp(–l t), then, using (10.5), we can find the reliability gain W T(3) by the mean time of failure-free operation during the operation of a system consisting of ( m+ 1) redundant systems operating in parallel (Fig. 10.4) compared to the mean time of failure of a non-redundant system:
Reliability gain W T(4) in terms of the mean time of failure-free operation during the operation of a system consisting of ( m+ 1) redundant systems operating in parallel (Fig. 10.4), compared with the mean time of failure of one element:
Consider the case of a system with separate reservation with a permanently switched on reserve, assuming that all elements are equally reliable with the probabilities of failure-free operation p(t) (Fig. 10.5).
Rice. 10.5. Block diagram of a split redundant always-on redundant system with a serial connection n elements
For a system with separate redundancy, the probabilities of failure-free operation of individual elements with redundancy can be determined using formula (10.14). Then the overall probability of failure-free operation of a system with separate redundancy is determined by the formula:
For the case when all elements are equally reliable, the probability of failure-free operation of a system with separate redundancy is:
Reliability gain in terms of mean uptime during operation of the redundant system in comparison with the mean time of non-failure operation of the main system with an exponential distribution law:
10.2.2 By the method of switching on the reserve. Redundant elements can be constantly switched on for the entire period of operation - use permanent redundancy (redundancy with a permanently switched on reserve without switching) or only in case of failure of the main ones - redundancy by replacement.
With permanent redundancy, the backup elements are connected to the main ones during the entire operation time and are in the same operating mode as them. Permanent inclusion of the reserve is the only possible one in systems where even a short interruption in operation is unacceptable (for example, in control systems technological processes). Although it is simple (the absence of switches and short-term shutdowns in the operation of devices), the main disadvantage of permanent redundancy is the increased resource consumption of redundant elements. Pumps, filters, etc. are usually redundant in this way.
If it is not possible to apply the constant parallel operation of apparatuses in chemical engineering, then it is necessary to use replacement redundancy ("substitution with an unloaded reserve"). Substitution is done automatically or manually.
When redundant by replacement (or “replacement with an unloaded reserve”), the system is designed in such a way that when an element fails, it is rebuilt and restores its operability by replacing the failed element with a redundant one. This does not require adjustment at the time of switching on the backup element; the backup device can be in a “warm” or “cold” state before it is put into operation - this saves the reliability resource of each of the devices and increases the overall reliability of the entire system. In the case of elements of the same type, several reserve (or one) can be used to replace the main elements in case of failure.
Rolling redundancy is replacement redundancy, in which a group of primary elements is backed up by one or more reserve elements, each of which can replace any of the failed elements of this group.
Sliding redundancy is used to back up several identical or interchangeable system elements with one or more redundant ones, and the redundancy can be either loaded or unloaded. The system will fail if the number of failed main elements exceeds the number of standby ones. With a sliding (floating) reserve, any of the reserve elements can replace any main element of the system (for example, refrigerators, pumps). The sliding reserve gives the greatest gain in increasing reliability, but its significant drawback is that it is possible only for the same type of elements (subsystems).
The sliding redundancy scheme in the monoethanolamine purification unit is shown in Fig. 10.6.
Rice. 10.6. Scheme of sliding redundancy in the monoethanolamine purification unit:
1 - absorber; 2, 3 - pumps; 4 - regeneration unit; 5 - standby pump
With a loaded sliding redundancy with ideal switches, the calculation of system reliability is similar to the calculation of a system of the type " m from n". If the failure rates of the main and reserve elements are constant and the same, then the probability of failure-free operation of the system consisting of n basic and m reserve elements, in the loaded reserve mode can be determined by the formula:
If the probability of failure-free operation of elements obeys an exponential law, then it is possible to calculate the mean time between failures of the system:
With unloaded sliding redundancy, in the general case, the reliability characteristics of the system are expressed by complex formulas. However, if the failure rates of the main and reserve elements are constant and the same, i.e., the probability of failure-free operation of the elements obeys an exponential law, then the probability of failure-free operation of a system consisting of n basic and m reserve elements, in the unloaded reserve mode can be determined by the Poisson formula:
Since, with an unloaded sliding redundancy, the total failure rate is equal to n and the failure of the system will occur at the moment of failure ( m+ 1)-th element, mean time between failures of the system:
10.2.3 According to the mode of operation of the reserve. Calculation of systems with loaded redundancy is carried out according to the formulas for series and parallel connection of elements. At the same time, it is considered that the failure of the reserve group, consisting of the main and reserve elements, will occur when its last element fails, and the reserve elements operate in the main mode both before and after their failure, therefore, the reliability of the reserve elements does not depend on the moment of their failure. transition from the reserve to the main state.
When redundant by substitution, there are three types of operating conditions for redundant elements until they are put into operation:
a) loaded (hot) standby . The external conditions of the reserve completely coincide with the conditions in which the working apparatus is located. Backup elements operate in the same mode as the main element, their reliability (probability of failure-free operation) does not depend on the moment at which they switched on to replace the main one. In this case, the resource of the reserve elements of the object begins to be consumed from the moment the entire system is put into operation;
b) idle (cold) standby . The reserve elements are turned off and, by condition (until they are switched on in place of the main one), they cannot fail. The external conditions in which the reserve is located are so much lighter than the working ones that, in practice, the reserve elements begin to consume their resource only from the moment they are put into operation instead of the failed element.
in) light (warm) reserve . The external conditions affecting the device until it is put into operation are light. Reserve elements are in light mode until they are switched on in place of the main one. While waiting in reserve, they may fail, but with a lower probability than the probability of failure of the main element (the reserve, which is in easier conditions than the main element).
System MTBF with m-fold of the total loaded reserve can be found from the expression:
In the case of the exponential law of reliability of elements, we obtain:
where L = 1/ T O– circuit failure rate.
After integration, we represent (10.27) as a finite difference: .
Substituting into this equation successively m= 1, 2, 3, ..., we get:
With unloaded redundancy by replacement, the backup elements are put into operation when the primary fails, then the first backup, etc., so the reliability of the elements at any given time depends on the moment of their transition from the backup state to the primary one. In this case, it is considered that the replacement of the failed element by the reserve element occurs instantly, the system will fail when the last element fails. In the non-working state, the element cannot fail and its reliability does not change.
Unloaded redundancy is common, since it is similar to replacing failed elements (parts, assemblies, assemblies) with spare ones.
Classification of system redundancy methods
The level of reliability of the element base of electronics, radio engineering, mechanical elements, electrical engineering achieved at present is characterized by the values of the failure rate λ=10 -6 ...10 -7 1/h. In the near future, this level should be expected to rise to λ= 10 -8 1/h. This will make it possible to raise the time between failures of a system consisting of N = 10 6 elements up to 100 hours, which is clearly not enough. The necessary reliability of complex systems can only be achieved by using various kinds reservations .
Redundancy is one of the main means of ensuring a given level of reliability (especially reliability) of an object with insufficiently reliable elements.
In accordance with GOST 27.002-89 reservation called the use of additional tools and (or) capabilities in order to maintain the operable state of the object in case of failure of one or more of its elements. Thus, redundancy is a method of increasing the reliability of an object by introducing redundancy. In its turn, redundancy - these are additional means and (or) capabilities that are superminimal necessary for the object to perform the specified functions. The task of introducing redundancy is to ensure the normal functioning of the object after the occurrence of a failure in its elements.
There are various backup methods. It is advisable to divide them according to the following criteria (Figure 4.7): type of redundancy, method of connecting elements, multiplicity of redundancy, method of switching on the reserve, mode of operation of the reserve, recoverability of the reserve.
Figure 4.7 - Classification of redundancy methods
Structural redundancy, sometimes called hardware (element, circuit), provides for the use of reserve elements of the structure of the object. The essence of structural redundancy is that additional elements are introduced into the minimum required version of the object. The elements of a redundant system have the following names. main element- element of the structure of the object, necessary for the object to perform the required functions in the absence of failures of its elements. Reserve element - element of the object, intended to perform the functions of the main element in case of failure of the latter.
The definition of the main element is not related to the concept of minimality of the main structure of the object, since the element, which is the main one in some operating modes, can serve as a backup in other conditions.
Reserved element- the main element, in case of failure, which is provided in the facility as a backup element.
Figures 4.8 - 4.10 show the connection diagrams of the main and reserve elements, the so-called parallel connection of elements. A system with parallel connection of elements is a system that fails only if all its elements fail.
Figure 4.8 - Example of parallel connection of elements
a - circuit diagram, b – calculation scheme
Figure 4.9 - An example of a parallel-serial connection of the elements of the SUHTP
a - functional diagram, b – calculation scheme
Figure 4.10 - An example of a bridge connection of elements
Temporary reservation associated with the use of time reserves. At the same time, it is assumed that the time allotted for the object to perform the necessary work is obviously greater than the minimum required. Time reserves can be created by increasing the productivity of the object, the inertia of its elements, etc.
Information redundancy- this is a redundancy with the use of information redundancy. Examples of information redundancy are multiple transmission of the same message over a communication channel; the use of various codes in the transmission of information over communication channels that detect and correct errors that appear as a result of equipment failures and the influence of interference; the introduction of redundant information symbols in the processing, transmission and display of information. The excess of information makes it possible, to some extent, to compensate for the distortions of the transmitted information or to eliminate them.
Functional Redundancy redundancy, in which a given function can be performed in various ways and technical means. For example, the function of transmitting information to the automated control system can be performed using radio channels, telegraph, telephone and other means of communication. Therefore, the usual average reliability indicators (mean time between failures, the probability of failure-free operation, etc.) become uninformative and insufficiently suitable for use in this case. The most appropriate indicators for assessing functional reliability are: the probability of performing a given function, the average time to complete a function, the availability factor for performing a given function.
Load redundancy- this is a redundancy with the use of load reserves. Load redundancy, first of all, consists in ensuring optimal reserves of the ability of elements to withstand the loads acting on them. With other methods of load redundancy, it is possible to introduce additional protective or unloading elements.
The listed types of redundancy can be applied either to the system as a whole, or to individual elements of the system or to their groups. In the first case, the reservation is called general, in the second - separate. The combination of different types of reservation in the same object is called mixed.
According to the method of including reserve elements, there are permanent, dynamic, replacement reservation, sliding and majority reservation. Permanent reservation- this is redundancy without restructuring the structure of the object in the event of a failure of its element. For permanent redundancy, it is essential that in the event of a failure of the main element, no special devices are required to put the reserve element into operation, and there is also no interruption in operation (Figures 4.11 - 4.13). Permanent redundancy in the simplest case is a parallel connection of elements without switching devices.
Figure 4.13 - Mixed redundancy with permanently switched on reserve
Dynamic Redundancy- this is a redundancy with the restructuring of the object structure in the event of a failure of its element. Dynamic redundancy has a number of varieties.
Reservation by replacement- This is a dynamic redundancy, in which the functions of the main element are transferred to the backup only after the failure of the main element. The inclusion of a reserve by replacement (Figures 4.14, 4.15) has the following advantages:
- does not violate the mode of operation of the reserve;
- retains the reliability of the backup elements to a greater extent, since during the operation of the main elements they are in a non-operating state;
- allows you to use a reserve element for several main elements.
A significant disadvantage of replacement redundancy is the need for switching devices. With separate redundancy, the number of switching devices is equal to the number of main elements, which can greatly reduce the reliability of the entire system. Therefore, it is beneficial to reserve large nodes or the entire system by replacement, and in all other cases - with high reliability of switching devices.
rolling reservation- this is redundancy by replacement, in which a group of the main elements of an object is backed up by one or more backup elements, each of which can replace any failed main element in this group (Figure 4.16).
Figure 4.16 - Sliding reservation of the same type (a) and heterogeneous (b) elements
Widely used in control systems majority reservation(using "vote"). This method is based on the use of an additional element called the majority or logical element. The logical element allows you to compare the signals coming from the elements that perform the same function. If the results match, then they are transferred to the output of the device.
Figure 4.17 shows a 2 out of 3 redundancy, i.e. any two out of three matching results are considered true and passed to the output of the device. According to this principle, many schemes of subsystems of control and protection systems (CPS) are built. It is possible to apply the ratios "3 out of 5", etc. The main advantage of this method is to ensure an increase in reliability for any types of element failures and an increase in the reliability of information-logical objects.
Figure 4.17 - Majority reservation
The degree of redundancy is characterized by the multiplicity of redundancy. Reserve ratio- this is the ratio of the number of reserve elements of the object to the number of the main elements reserved by them, expressed as a non-reduced fraction. Integer redundancy occurs when one primary element is backed up by one or more reserve elements.
Fractional Redundancy – this is such a reservation, when two or more elements of the same type are reserved by one or more reserve elements. The most common redundancy with fractional multiplicity is when the number of main elements exceeds the number of reserve ones. Reservation, the multiplicity of which is equal to one, is called duplication.
Depending on the mode of operation of the reserve, loaded, light and unloaded reserves are distinguished. loaded reserve - it is a reserve that contains one or more standby elements that are in the mode of the main element. At the same time, it is assumed that the elements of the loaded reserve have the same level of reliability, durability and persistence as the main elements of the object reserved by them. Lightweight Reserve - this is a reserve that contains one or more reserve elements that are in a less loaded mode than the main one. Lightweight reserve elements usually have more high level reliability, durability and persistence than the main elements. Unloaded reserve- this is a reserve that contains one or more backup elements that are in an unloaded mode before they begin to perform the functions of the main element. For the elements of an unloaded reserve, it is conditionally assumed that they never fail and do not reach the limit state.
Redundancy, in which the operability of any one or more redundant elements in the event of failures is subject to restoration during operation, is called redundancy with recovery, otherwise there is redundancy without recovery. The recoverability of the reserve is ensured in the presence of monitoring the health of the elements. In the presence of redundancy, this is especially important, since in this case the number of hidden failures may be greater than in the absence of redundancy. AT ideal the failure of any element of the object is detected without delay, and the failed element is immediately replaced or repaired.
CHAPTER V. SYSTEM RESERVATION
One of the fundamental tasks of the theory of reliability is the task of developing methods for improving the reliability of systems. System redundancy is such a method.
Reservation - a method of increasing the reliability of the object by introducing redundancy.
Redundancy - additional means or capabilities in excess of the minimum required for the object to perform the specified functions.
There are the following types of redundancy:
1.Temporal Redundancy . Provides for the use of excess time by the object to perform the specified functions. That is, with this kind of redundancy predefined functions can be performed by the object, generally speaking, in a shorter period of time. Example: The computer can continuously perform a number of tasks, but in order to improve reliability, failure diagnostics can be performed.
2.Information redundancy . Provides for the use of redundant information. For example:
a) repetition of sending messages in a noisy channel in order to increase the reliability of information transmission,
b) retention of an excess number of significant figures in calculations,
c) error-correcting redundant coding,
3.Load redundancy occurs when the object is operating in a mode lighter than normal. for example: element load factor Kn< I.
4.Structural redundancy is that the object includes redundant elements. For example, a digital computer usually includes several input and output devices.
§ 5.1 Classification of redundancy methods
Let us agree for convenience in what follows to speak of the reservation of an element, meaning by the word both the element itself and any part of the system, including the entire system.
We give the following definitions.
main element - the element is the minimum necessary to ensure the operability of the system.
Reserve element - an element designed to ensure the operability of the system in the event of a failure of the main element. The set of the main and its reserve elements will be called the reserve group.
Example: A digital computer with multiple input and output devices. One input device and one output device are the main elements, other input and output devices are redundant. All input devices and output devices are two redundant groups.
Reserve group - this is a combination of the main element and all its reserve ones.
Classification sign | Reservation type |
||
Use of a failed element (primary or backup) | Backup with recovery |
||
Backup without recovery |
|||
Method of switching on the reserve element | General reservation |
||
Separate reservation |
|||
Scheme of switching on the reserve element | Permanent redundancy (passive) |
||
Replacement redundancy (active) |
|||
Redundancy status (for active redundancy methods | Unloaded (cold) standby |
||
Loaded (hot) standby |
|||
Lightweight (warm) standby |
|||
Load sharing between non-failed elements (for passive redundancy methods) | With constant load |
||
With load sharing |
|||
Reserve fixation (for active reserve methods) | Fixed Reservation |
||
rolling reservation |
|||
Reservation uniformity | Homogeneous Redundancy |
||
Mixed redundancy |
If the main or backup element is subject to restoration after a failure, then the redundancy will be with restoration. Otherwise, no recovery.
General reservation - when a reserve is provided in case of failure of the entire system as a whole (Fig. 40).
Separate reservation - when a reserve is provided in case of failure of individual elements of the object or their groups (see Fig. 41).
Example: ECVM + ECVM - common redundancy.
input device + input device, AU + AU, UU + UU, ZU + ZU,
output device + output device - separate redundancy.
Permanent reservation - redundancy, in which the reserve elements participate in the operation of the object on an equal basis with the main ones. The block diagram of permanent redundancy is shown in fig. 40
Reservation by replacement redundancy, in which the functions of the main element are transferred to the backup only after the failure of the main element. The block diagram is shown in Fig. 42 (option a) - separate redundancy, option b) - general redundancy).
Example: A digital computer has several output devices (ATsPU). If the information is displayed immediately on everything (ATsPU), then we have a permanent reservation. If the backup ATsPU is connected only after the failure of the main one, then we have redundancy by replacement.
In replacement redundancy, the occurrence of an element failure causes a system rebuild. This restructuring is carried out using switches that turn off failed elements and connect healthy ones.
There are two types of permanent reservation:
1. With constant load when the failure of one or more elements of the redundant group does not change the load on the remaining serviceable elements.
Example: When the main and standby ADCs are connected all the time and the same material is output to each of them, display devices.
2. With load sharing when the failure of at least one element of the redundant group changes the load on the elements that remain serviceable.
Example: In the absence of failures, punched cards are entered evenly from several input devices. If at least one input device fails, the load on the remaining ones increases.
Depending on the state of the redundant elements before they are put into operation, active redundancy is divided into several types:
1. loaded reserve- when the backup elements are in the same mode as the main element.
2. Unloaded reserve- when the redundant elements are in the off state. Until the moment of switching on, the reserve cannot fail.
3. Light Reserve- when the reserve elements are less loaded than the main one. While waiting, the backup elements may fail, but with a lower probability than the probability of the main element.
Obviously, the lightweight reserve is the most common type of active reserves, since the 1st and 2nd are obtained as private ones from the lightweight ones.
Fixed Reservation - redundancy by substitution, in which the connection point of each backup element is strictly defined in advance (Fig. 42a).
https://pandia.ru/text/78/494/images/image005_73.gif" width="77" height="25 src=">
the system is unrecoverable
elements (main and reserve) are equally reliable and the reliability function =
We will compare the reliability of redundant and non-redundant systems according to the indicator
https://pandia.ru/text/78/494/images/image008_44.gif" width="114" height="28 src="> - redundant and non-redundant system reliability functions.
§ 5.2 System reliability with loaded active redundancy and passive redundancy without load sharing
Let the system contain N serially connected basic elements.
1. Shared Reservation Case
https://pandia.ru/text/78/494/images/image010_42.gif" width="344" height="386 src="> Consider the timing diagram of the operation of a redundant system in the special case N=2, M=1. It is shown in Fig. 45. On it - the time to failure of the n -th element in the m -th reserve group, in the general case
a) Consider the case active redundancy.
Let us find the system reliability function. It can be seen that its structural reliability scheme is series-parallel and has M + 1 parallel connected groups, each of which contains N elements. Then from (4.25) the reliability of the redundant system
where https://pandia.ru/text/78/494/images/image015_29.gif" width="49" height="28 src="> will be determined from (5.1)
From (5.1) it follows:
1. The reliability of the system does not depend on the order in which the redundant elements are switched on.
2. The reliability of the system at the time t is determined by the values of the reliability of the elements at the same time t and does not depend at all on how the reliability changed before the time.
3. The reliability of a redundant system is higher than that of a non-redundant one. Indeed, it is easy to check
Where is the operating time to failure, m is the number of the redundant group, n is the number of the element in the redundant group
Task 1. Let the reliability of the element be given and it is required to determine such M number of groups of redundant elements, in which the reliability of the redundant system will be at least https://pandia.ru/text/78/494/images/image019_21.gif 28">
https://pandia.ru/text/78/494/images/image021_22.gif" width="212" height="31 src=">
https://pandia.ru/text/78/494/images/image023_20.gif" width="193" height="52 src=">
https://pandia.ru/text/78/494/images/image006_62.gif" width="52 height=29" height="29">.gif" width="87" height="28">
https://pandia.ru/text/78/494/images/image022_17.gif" width="303" height="31 src=">
https://pandia.ru/text/78/494/images/image026_18.gif" width="199" height="32 src=">
The system has N redundant groups, each of which contains 1 main and N reserve elements. The main element will further be conditionally considered as a zero reserve element (in the reserve group). Consider the timing diagram of the operation of a redundant system in the particular case N=2, M=1 (see Fig. 42-a). It is shown in fig. 46.
a) Consider the case active redundancy .
Let us find the system reliability function. Its reliability structure will be series-parallel, containing N series-connected groups, each of which contains M + 1 parallel-connected elements. From (4.26)
https://pandia.ru/text/78/494/images/image006_62.gif" width="52" height="29 src="> element reliability function.
b) For the case passive redundancy without load sharing diagrams will be similar to Fig. 46 and will be determined from (5.2). From (5.2), conclusions similar to those given above for the case of general redundancy follow. Reservation gain
https://pandia.ru/text/78/494/images/image031_15.gif" width="236" height="35 src=">
5.3 System reliability with unloaded active redundancy
For an unloaded reserve, we will assume that the reliability of the reserve elements does not decrease in the idle state. We will also keep in mind the assumptions introduced earlier.
1. Shared Reservation Case
Consider the case of general redundancy of a system consisting of N series-connected main elements. The structure of the redundant system will be similar to Fig. 44. Consider the timing diagram of the operation of a redundant system in the particular case N=2, M=1 (see Fig. 42-b). It is shown in fig. 47.
System failure time:
https://pandia.ru/text/78/494/images/image034_18.gif" width="124" height="33 src=">, which will not depend on M since the elements (primary and backup) are equally reliable and the number elements in a group of series-connected main and reserve elements in the same way and = N.
https://pandia.ru/text/78/494/images/image036_16.gif" width="495" height="33 src="> (5.5)
1. gain in reliability
2. do not depend on the order of connected redundant groups
3. It follows from (5.5) that for the case of an unloaded reserve, as opposed to a loaded one, the reliability function of the redundant system at the time t is determined by the values of the reliability functions of the elements on the interval , i.e., the prehistory of operation.
Let's compare loaded and unloaded active reserves. It is difficult to make a quantitative comparison of (5.1) and (5.5), so we confine ourselves to qualitative conclusions.
Time to system failure:
-
https://pandia.ru/text/78/494/images/image011_38.gif" width="35" height="25 src="> time to failure of the n -th element of the m -th group of reserve elements.
-
https://pandia.ru/text/78/494/images/image039_13.gif" width="223 height=52" height="52"> i.e.
and therefore , an unloaded reserve is more reliable than a loaded one .
2. Case of split reservation
https://pandia.ru/text/78/494/images/image042_12.gif" width="104" height="35 src=">
Redundant System Reliability Function:
https://pandia.ru/text/78/494/images/image044_12.gif" width="119" height="52 src=">
That is, the flow of failures of elements in the nth reserve group is similar to the flow of failures for the MPE. Then from (3.7)
https://pandia.ru/text/78/494/images/image046_12.gif" width="52" height="29 src="> - function of distribution of time to failure of an element.
Substituting (5.7) into (5.6) we get
(5.8)
Let's compare loaded and unloaded reserves at a qualitative level.
Time to system failure:
-for loaded active reserve
https://pandia.ru/text/78/494/images/image011_38.gif" width="35" height="25 src="> - time to failure of the m -th element in the n -th reserve group.
- for unloaded active reserve
https://pandia.ru/text/78/494/images/image050_12.gif" width="215" height="52 src=">
i.e..gif" width="77" height="25"> if the same for loaded and unloaded reserves.
§ 5.4. Comparison of the reliability of systems with active loaded and unloaded redundancy
It is difficult to make a quantitative comparison of the reliability functions, so we confine ourselves to qualitative conclusions and make a comparison at the level of comparing the operating times to system failure.
1. General reservation
For loaded reserve
gif" width="251" height="61 src=">
It's obvious that. and, therefore, an unloaded reserve is more reliable than a loaded one.
2.Separate reservation
For loaded reserve
For unloaded reserve
Obviously, since always i.e., an unloaded reserve is more reliable than a loaded one.
Note that this conclusion holds for all active redundancy methods, including those with non-absolutely reliable switches, if DIV_ADBLOCK253">
Let us find the system reliability function for the case general redundancy of a system containing N series-connected elements (Fig. 44)
The system operation diagram for the case N=2 and M=1 will be the same as in Fig. 47, only until a working group of reserve elements is connected to the place of a failed group of main or reserve elements, it will be in a lightened state, in which the elements are less likely to fail than in working condition.
For the sake of simplicity of reasoning, but not at the expense of generality (due to the fact that the main and reserve elements are equally reliable), we assume that the numbers of groups of reserve elements correspond to the order in which they are connected.
Denote:
Failure time (M - 1)-th group of redundant elements
Failure time of the M -th group of redundant elements = system failure time.
Note that they are time-dependent, since it depends on the moment of transition of the m-th group m=1,M of reserve elements from a lightweight state to a working one, i.e. on
System reliability function:
https://pandia.ru/text/78/494/images/image070_8.gif" width="363" height="42 src="> (5.7)
https://pandia.ru/text/78/494/images/image072_8.gif" width="226" height="44 src=">
https://pandia.ru/text/78/494/images/image074_7.gif" width="314" height="38 src="> (5.8)
where https://pandia.ru/text/78/494/images/image076_6.gif" width="39" height="19">
are the probabilities that, respectively M-th group and an element of this group will not fail in the interval , provided that there was no failure before the moment of failure.
That is, (5.7), (5.8) determines in terms of . Similarly, it is determined through, etc. through - the distribution function of the group of basic elements.
§ 5.5. Effect of Redundancy Scale on System Reliability
The reserve may cover either individual main elements, or several main elements, or all the main elements of the system. The level at which a reservation is made is called the scale of the reservation. The greater part of the main elements of the system is covered by one reserve, the greater the scale of the reservation. The more redundant groups, the smaller the redundancy scale.
Let us consider the issues of the influence of the scale of redundancy on the reliability of the system with an absolutely reliable and absolutely unreliable switch.
1. Absolutely reliable switch.
Let us show that as the redundancy scale increases, the reliability of the system decreases. That is, the sequential combination of redundant elements belonging to different redundant groups (Fig. 49 a, b) leads to a decrease in reliability.
Before proceeding to the proof, we note that it is sufficient to prove the formulated statement for the case of reserving two main elements with two reserve ones with different scales (Fig. 48-b). Indeed, when m-th elements of the reserve groups are sequentially combined, the groups of main and reserve elements , obtained in the previous join step, can be considered as one element. That is, it is necessary and sufficient for us to show that element-by-element redundancy (Fig. 49-a) provides greater reliability than general redundancy (Fig. 49-b).
a) active loaded redundancy
For element-by-element redundancy (Fig. 49a) from (5.2)
https://pandia.ru/text/78/494/images/image089_7.gif" width="12" height="23 src=">.gif" width="384" height="37 src=">. gif" width="478" height="38 src=">
https://pandia.ru/text/78/494/images/image095_7.gif" width="212" height="38 src=">
That is, an increase in the scale of redundancy leads to a decrease in reliability.
b) active idle redundancy
For element-by-element redundancy (Fig. 49a)
https://pandia.ru/text/78/494/images/image097_5.gif" width="349" height="41 src=">
For comparative analysis and all possible relationships between the failure times of the main and reserve elements should be considered.
Let https://pandia.ru/text/78/494/images/image101_6.gif" width="239" height="25">
Let DIV_ADBLOCK255">
https://pandia.ru/text/78/494/images/image105_5.gif" width="115" height="25 src=">
etc. If we analyze all cases, we get
Whence it follows that separate redundancy is more reliable .
We note that the proved result is valid for any law of reliability. It can be physically explained by the fact that with separate redundancy, the failure of the main element is compensated by only one reserve element, and not by a group of reserve elements, as in the case of general redundancy, i.e., there is a more rational consumption of reserve elements.
2. Not absolutely reliable switch.
a) Consider case of common active loaded reserve (fig.50)
With respect to each element of the redundant groups, the switches will behave as a series-connected element. Assuming that all N switches in the redundant groups are equally reliable, we obtain
Comparing (5.2) and (5.12), we get a similar conclusion.
Above, we came to the conclusion that with an absolutely reliable switch, the greatest redundancy reliability is ensured with the smallest redundancy scale. of 5 series-connected elements.
As the reservation scale decreases, the unreliability of the system due to the non-absolute reliability of the switch will increase, and the unreliability of the system itself due to the reduction in the reservation scale will decrease. Therefore, there will be some optimal reservation scale at which left">
1. loaded reserve . Let's consider the time diagram of the operation of the redundant system in the particular case N=2, M=1. It is shown in fig. 53.
Reliability function
https://pandia.ru/text/78/494/images/image118_4.gif" width="47" height="28 src="> - number of failed elements per .
2. https://pandia.ru/text/78/494/images/image120_4.gif" width="136" height="29">. This follows from the fact that with a sliding reservation, all reserve elements are used completely, i.e. The failure of the system occurs after there is not a single backup element left and the main one fails.In the case of separate redundancy, there may be an underexpenditure of the backup elements, due to the fact that the failure of the backup group causes the system to fail.At the same time, some of the backup elements in other backup groups may be underused .
The use of sliding redundancy in practice is limited by the complexity of switching devices.
With an absolutely reliable switch and with the same number of redundant elements, sliding redundancy has greater reliability than separate and even more common, therefore, it is necessary to strive to use sliding redundancy.
Restrictions:
When implemented in software, there are no restrictions on switches;
With hardware implementation, there is, because in addition to the switching function, the switch is additionally assigned the function of identifying the failed element.
§5.8. Backup with recovery
In practice, in order to improve reliability, they often resort to restoring redundant systems. In this case, for the most general situation, the following scheme of the system can be given (in the usual sense):
https://pandia.ru/text/78/494/images/image122_4.gif" width="133" height="30">(Here we assume that it does not depend on t.
Then the graph of system transitions from state to state can be represented as Fig.55. It is a directed graph.
In the general case (for an arbitrary number of redundant elements), the process of death and reproduction (which is Markovian) can be used to describe the behavior of the system. Here the Markov constraint is not derived.
According to the transition graph, a system of differential equations is compiled using the following regulations:
The system contains so many differential equations, how many states the analyzed system has (vertices of the graph)
The left side of the i -th equation of the system contains https://pandia.ru/text/78/494/images/image126_5.gif" width="39" height="29 src="> the probability of the i -th state, and the right one - as many terms as there are graph arcs associated with the i -th state.
Each term is the product of the intensity of the transition to the i-th or from the i-th state by the probability of the state from which the arc originates. If the arc is directed to the i -th state, then the term is taken with the sign "+", if it comes from the i -th state, then with the sign "-".
https://pandia.ru/text/78/494/images/image128_4.gif" width="33" height="23"> can be produced using the Laplace transform, reducing the system of differential equations to a system of algebraic equations. The probability of a healthy state at the moment or factor of readiness:
https://pandia.ru/text/78/494/images/image130_3.gif" width="157 height=23" height="23"> .And the system of differential equations turns into a system of algebraic equations. For example, from (4.11 )
https://pandia.ru/text/78/494/images/image132_3.gif" width="180" height="34 src="> (5.17)
§ 5.9 Majority reservation
This method is also called voting reservation. It owes its name to the presence in the reserve groups of a special element called the majority element or voting element (quorum element).
Majority redundancy is widely used in discrete (digital) systems, including computing ones.
Let a system consisting of N elements connected in series in the sense of reliability be reserved (Fig. 56-a). Each element of the system is discrete, producing 0 or 1 depending on 0 or 1 at the output. For the definition, let's assume that in a healthy state, a 0 at the output corresponds to a 0 at the input and a 1 at the output corresponds to a 1 at the input.
An example of such a system can be a delay circuit for the front (back or front) of a pulse of unit amplitude for a time ³ t. For small t, such a circuit can be implemented on logical elements of the ''AND-NOT'' type, each of which provides a delay for the time t0. Then the number of ’’AND-NOT’’ elements must be even and selected from the condition
Each main element of the system is replaced by a reserve group consisting of an odd number M of input elements and one majority element (ME). As input elements, elements similar to the main ones are usually used.
The majority element generally implements the function
https://pandia.ru/text/78/494/images/image135_2.gif" width="91" height="24"> - signal at the output of the m-th input element.
Upor - the threshold of operation of the majority element.
Y - output signal of the reserve group.
b) Adaptive majority reservation
Allows you to take into account the failures of the input elements. This is achieved by the fact that in (5.13) am=var (0 or 1) and Upor=var. The reserve group in this case will look like Fig.60. Disabling input elements occurs in pairs. At the same time, Upor changes.
Class 2" href="/text/category/2_klass/" rel="bookmark">Class 2 (with multiple links) Fig.54 d.
This majority redundancy method makes it possible to reduce the reliability requirements for the majority element, which must be done with non-adaptive majority redundancy.
We present the calculation of the reliability of the 1st reserve group. It is operational then (provided that the input elements of the 2nd reserve group are operational) when at least
the outputs of the majority elements will be the correct signal