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DOI: 10.24412/2413-2527-2022-432-71-77
Improving the Reliability of Electronic Elements Based on Volumetric Redundancy
Grand PhD V. A. Smagin International Informatization Academy Saint Petersburg, Russia va_smagin@mail. ru
Abstract. One of the possible methods of increasing the reliability of electronic elements based on volumetric redundancy is considered. A further study of the Shannon bridge from the standpoint of probability theory has been carried out. The construction of examples of some more complex bridge structures of elements is proposed. They can be the basis for building more complex systems with a large number of states. In the context of the development of nanotechnology, the principles of building systems down and up are applicable.
Keywords: redundancy, element self-restoration, Shannon bridge, Shannon elements, probability of working capacity, reliability structure of Sedyakin.
Introduction
The purpose of the article is an in-depth study of the properties of known hardware redundancy in the context of the introduction of the concept of nanotechnology at the present time and obtaining new additional information in this regard [1, 2].
Primarily we will be interested in examples of redundancy that can recover itself when its individual elements fail. In our opinion, one of the first examples of such redundancy is the bridge of C. E. Shannon [3].
In the development of information theory. This element is characterized by the fact that it contains the fifth unloaded element. If one of the main four values of the elements fails, this diagonally positioned element automatically turns on, taking over part of the energy load, helping other elements to reduce its value. It is known that elements can fail for reasons of their breakage or the electrical short, leading to various functional or parametric properties of the entire bridge [4].
Fig. 1. Schematic diagram of the Shannon bridge
Assume that all elements of the bridge have only one type of failure—breakage. We consider the bridge to be operational only in two cases: when it does not have a single failure or one failure of a functional element out of four occurred in it, the diagonal element started working, taking over part of the load. In this case, the parametric side is not taken into account.
Grand PhD V. P. Bubnov, D. V. Barausov Emperor Alexander I St. Petersburg State Transport University Saint Petersburg, Russia bubnov1950@yandex.ru, daniil_bara95@mail.ru
Simply put, the bridge is operable with the assumption of no more than one failure. The problem of calculating reliability is solved when these two hypotheses are fulfilled. The question is what effect in the reliability of the bridge is obtained when the diagonal element is included in the work. We will accept the following initial data with a normal distribution of elements for calculation: m = 50 h, c = 10 h, f(t) = dnorm(t, m, c). Then the probability of failure-free element:
P(t) = I f(z)dz, Jt
and the resource of its reliability in the sense of N. M. Sedya-kin [3-5].
Will be equal to r(t) = — ln(P(t)). Graphs in Figures 2-4 illustrate expressions for the density, probability, and reliability resource of an element.
0.03 f(t) 0.02 0.01
° 20 40 60 so :oo [
Fig. 2. Probability density of failure-free operation of the Shannon bridge
Fig. 3. Probability of failure-free operation of the Shannon bridge
1
Fig. 4. Element reliability resource
Failure is a breakage of an element in the bridge Figure 5 shows the breakage scheme of the element in the bridge. In the absence of a breakage, the serviceability of the bridge (hypothesis 1) is equal to
Pi(t) = (P(t))4 .
Breakage (hypothesis 2), diagonal element is not included,
4x (P(t))3 x (1 -P(t)).
The reliability resources are equal at the same time:
r±(t) = - ln(P!(t)), r2(t) = - ln(P2(t)).
Fig. 5. The failure is a breakage scheme of the element in the bridge
The resulting probability of bridge serviceability when the sum of the two hypotheses is fulfilled, taking into account the inclusion of a diagonal element under the load, will be determined by the sum:
,, f' (P(t + t))3 X P(t- t) P3(t)= P1(t)+ I ai(T)X^--i dT, (1)
Jo p2(T)
where a1 (t) = P1 (t) is the probability density of failure
of one working element.
The reliability resource for (1) is defined as
?3 (t) = - ln( P3 (t)), and the reliability resource of a workable bridge will be equal to
ri3(t) = ri (t)+ ?3 (t).
Figure 6 shows the graphical dependence of the values of the reliability resource of N. M. Sedyakin for the bridge circuit in the absence of an element failure in it by breakage, and Figure 7 shows the comparative dependences of the values of the bridge operability resources in the absence and presence of a failure by breakage.
Fig. 6. Graphical dependence of the values of the reliability resource of N. M. Sedyakin for a bridge circuit in the absence of the element failure in it by breakage
Fig. 7. Comparative dependences of values of the bridge operability resources in the absence and presence of a breakage failure
The dotted curve characterizes the contribution to the reliability of the bridge of the inclusion of a diagonal element in the form of an additional reliability resource in the form of an additional resource of N. M. Sedyakin, distributed over the remaining operable elements of the Shannon Bridge.
At the same time, the longer the bridge operation time, the greater the contribution value.
Failure is a short circuit of an element in the bridge
In the absence of the short circuit, the serviceability of the bridge (hypothesis 1) is equal to
Pi(t) = (P(t))4 .
Hypothesis 2 (the short circuit), its serviceability will be equal to
4 x (P(t))3 x (1 -P(t)). The reliability resources under these hypotheses are equal to
ri(t) = - ln(Pi(t)), r2(t) = - ln(P2(t)). The short circuit scheme is shown in Figure 8.
Fig. 8. The failure scheme is a short circuit of the element in the bridge
We make the assumption that the failures of breakage and short circuit in the cases considered here are equivalent.
The resulting probability of bridge serviceability when the sum of the two hypotheses is fulfilled, and in this case, taking into account the inclusion of a diagonal element under the load, will be determined by the sum (1), but the semantic content of this formula will be different. However, the parametric parameters of these two schemes in case of failures and short circuits will be different, which should be borne in mind. In this case, the result from the inclusion of a diagonal element in the work in both cases will be the same and it is determined by the formula (1) and Figures 6 and 7. Only the index elements will be different.
Shannon Bridge made of Shannon bridge elements
Remark. As for the bridge studies, when elements have two types of failures, breakage or closure of elements, and the output parametric properties of the bridge are determined by breakage or closure and the properties of its operability as a whole, such results have been already obtained and are well known from textbooks or lectures given in universities.
A cube of Shannon bridges
In the same case, when bridges are considered as elements of complex redundant structures, such studies of structures from elementary bridges have been carried out quite a few, in our opinion. For example, the author of this work published an article quite a long time ago [1].
It considered a cube consisting of elements of edges carrying an electrical or other load. It contains six flat faces; diagonal elements are included in these faces of the cube, which do not carry any load with a fully functional cube. Therefore, each such facet represents a Shannon bridge. The diagonal element of any bridge can carry the load only if any element of the bridge carrying the load, fails. Thus, the cube consists of 12 main elements and 6 diagonal elements.
Analytically, in a closed form, the reliability and throughput of the cube have not been investigated even in the case when the edges of the cube are resistors, and the only one type of their failure is breakage. Such cube was built by a young engineer, the resistance value of the cube along its main diagonal was studied.
However, generalizing conclusions were not obtained. After all, even with one type of failure of a cube element, 218 states should be considered, and if the cube elements have two types of failures, breakage and closure, then the number of different states of the cube becomes 318, so it becomes a very large number for research. Therefore, the method of statistical tests should be used to study the cube. However, judging by the publications in various sources of domestic and foreign publications, this was not observed. Only the author of the article has
qualitatively established the presence of harmonic oscillation of the main diagonal parameter, as well as its duration in time. This means that the long-term persistence in time of the value of the observed parameter of the main diagonal is inherent to the cube.
Based on this fact, it was concluded that for certain materials and achievements of modern industrial nanotechnology, for example, in the crystal structures of substance, as well as new scientific proposals, very highly reliable elements for long-term systems can be created.
The majority scheme of Shannon bridges
In a majority scheme, Shannon bridges can be independent or dependent on each other. In case of independence, calculations are performed in known ways. In the case of dependent elements, the calculation requires the construction of more complex models.
An example of a complex scheme with dependent Shannon bridges is a cube, which was described in the previous section, and more fully considered in the cited article [1].
Here, Figure 9 shows the simplest scheme and its probabilistic estimates for one and two types of element failures. With one type of failure, the expression for the probability of failure-free operation is represented as
Pout(t) = 3x p2 (t) — 2 x p3(t) ,
and for two types of failures of elements, (breakage and closure or failure of 0 and 1) in the form of
Pout (t) = 3 x p2(t) — 2 x p3(t) + 6x p(t) x qo(t) x qi(t).
Fig. 9. Failure scheme is a short circuit of the element in the bridge
Figure 10 shows the probabilities of proper operation of one element, schemes of proper operation with one and two types of failures of their elements. Initial data: m = 100 h, c = 10 h, a(t) = dnorm(t, m, c). The calculation is performed according to the formulas:
rrX) rt
P(t) = I a(z)dz, Q(t) = I a(z)dz, 't '0
Qo(t) = 0.5 x (1 — P(tj), Qi(t) = 0.5 x (1 — P(t)), PQi(t) = 3x(P(t))2— 2 x (P(t))3, PQ2(t) = 3x(P(t))2— 2 x (P(t))3 +
+ 6 x Qo(t)x Qi(t)x P(t).
PQ2(t)
100 110 120
Fig. 10. The probability of proper operation of single-channel version of the scheme, a majority scheme with one or two types of failures
Figure 10 shows the probabilities of proper operation of a single-channel version of the scheme, a majority scheme with one and two types of failures, and in Figure 11, the unconditional and conditional probabilities of failures of one channel.
The majority scheme with two types of equally probable channel failures has the greatest probability. With one type of failure, it has the lowest probability.
Q(t)
Q0(t) • • •
Q1(t)
70 80 90 100 110 120 t
Fig. 11. Unconditional and conditional probabilities of failures of one channel
Figure 12 shows the values of the reliability resources of Professor N. M. Sedyakin. The single-channel and three-channel circuits have the greatest resource for two types of channel failures. The three-channel circuit with one type of channel failures has the smallest resource.
200i—r
- ln(P(t)) 150
- ln(PQ1(t))100
- ln(PQ2(t))
50-
0
100 200 300 400 t
Fig. 12. Values of reliability resource Professor N. M. Sedyakin
An example of a majority scheme with dependent channels is a more complex structure in which the extreme channels of individual majority schemes are connected (dependent) with the channels of neighboring majority schemes.
A model of the Shannon bridge chain. Several bridges make up a sequentially triggered circuit in time. The first active bridge is in operation. It consists of four elements - the environment of existence, the fifth, diagonal element is not active as long as its external environment of four elements is in a balanced state, balanced.
At the moment when the balance of the environment is disturbed, the diagonal bridge is activated. The structure of this bridge is the same bridge as the first bridge, until the equilibrium balance is disturbed in it. Now it performs the role of a destroyed bridge until its balance is disturbed. If the balance of this bridge is disturbed, its diagonal bridge is activated. It performs the work until the moment of violation of its balance.
After a violation of the balance, its diagonal is included in the work — the next similarly balanced bridge. The work of the chain ends when the last bridge of the chain finishes its work. It is required to make a probabilistic model of the network functioning. Based on the section in which the model of a bridge with a breakage of an element of the medium is considered with its balanced state and an unloaded diagonal element, the probability of the initial state was equal to Pi(t) = (P(t))4. And the probability of maintaining its operability, provided that this hypothesis is fulfilled and the hypothesis that the element of the environment has failed, and the diagonal element has assumed the function of maintaining the bridge's operability becomes equal to
PPi(t) = Pi(t) + J a(z) x P(t - z)dz.
0
On condition that the bridge operability property continues to be maintained in the future when a new medium of four elements is balanced or when an element of this medium fails and goes into a working state due to the inclusion of a new diagonal element of the new bridge, the probability of continuing to remain operational continuously becomes equal:
l
PP2(t) = P1(t) + I a(z) x P1 (t - z)dz +
+
I b(z)x PP1(z)dz.
The probability densities of failures a(t) and b(t) by the following expressions:
d
a(t) = -YtPi(t) ^
^ 4 x ^J dnorm(z, 50,10)dz^j x dnorm(t, 50,10);
b(t) = -^PPi (t) (2)
Formula (2) is not disclosed here because of the rather large complexity. If the number of sequentially connected bridges in the circuit is more than three, then it is possible to make a recurrent dependence for calculating the parameters of a variable random process and, based on this calculation, evaluate more
t
accurately the quality of this process and the possibility of practical implementation and effectiveness of the considered type of redundancy. Based on the performed research, the following graphic material is presented for illustration in Figures 13-15.
The graphs shown in Figures 13 and 15 indicate an increase in the efficiency effect of the circuit with an increase in the number of transitions performed in it. Figure 16 shows graphs of the probabilities of operability of the three stages of the chain using a majority redundancy scheme.
100
200
300
Fig. 13. The initial probability of bridge operability, the probability of operability after the first failure of the environment, and the probability of operability after the second failure of the bridge environment
Fig. 14. The probability densities of the time of the first and second failures of the medium
Fig. 15. Graphs of reliability resources by N. M. Sedyakin
Figure 13 shows the initial probability of bridge operability, the probability of operability after the first failure of the environment and the probability of operability after the second failure of the bridge medium. Figure 14 shows the probability densities of the time of the first and second failures of the medium. Figure 15 shows graphs of N. M. Sedyakin's reliability resources calculated using the formulas
rPl(t) = - ln(P1(t)), rp2(t) = - ln(PPiit), rp3(t) = -ln( PP2(t)).
M1(t)
MM1(t) • • • ^
MM2( t)
50
100 t
150 200
Fig. 16. Probabilities of operability of three stages of the chain using a majority redundancy scheme
A three-dimensional bridge of three dependent simple Shannon bridges. At the beginning of the article, we put a simple, primary structure of the Shannon bridge, which was associated with the book [6].
He also proposed a decomposition formula for the analytical calculation of the probability of executing the algorithm for the functioning of the bridge with known probabilities of all its elements.
The idea of a bridge was used in communication technology and game theory.
However, our goal is to study the structure of the bridge to assess the probability of its functioning under the condition that it is balanced, so to the failure of any one of the four elements with an unloaded diagonal element, and then take into account the influence of the latter on the final result.
An interesting fact is that when its medium of four elements has a goal to perform some task, then the fifth element is not needed. If the balance of the medium is disturbed, the fifth element begins to perform the task of supporting the environment in which it is located.
This property is inherent to the Shannon bridge. However, it is possible to build more complex structures that use the auxiliary role of the diagonal elements of the constituent bridges of these structures to solve the target tasks.
Conclusion
The idea of building a bridge by C. E. Shannon is widely applicable in the science and technology of measuring and controlling systems, game theory and decision-making. It can perform the function of monitoring the state in systems and serve as a system for maintaining a state of balance in them.
The article further studies the Shannon bridge from the standpoint of probability theory.
The construction of examples of some more complex bridge structures of elements is proposed. They can be the basis for building more complex systems with a large number of states. For now, the deep analysis of the properties of these elements has not been obtained.
References
1. Smagin V. A. Nanotechnology — The Basis for the Creation of New High-Reliability Elements, Automatic Control and Computer Sciences, 2008. Vol. 42, Is. 2. Pp. 109-111. DOI: 10.3103/S0146411608020090.
0
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2. Smagin V. A. Novye voprosy teorii ekspluatatsii [New questions of the exploitation theory]. Saint Petersburg Mozhaisky Military Space Academy, 2010, 127 p. (In Russian)
3. Sedyakin N. M. Ob odnom fizicheskom printsipe teorii nadezhnosti i nekotorykh ego prilozheniyakh [About one physical principle of reliability theory and some of its applications]. Leningrad [Saint Petersburg], Mozhaisky Air Force Engineering Academy, 1965, 41 p. (In Russian)
4. Sedyakin N. M. Ob odnom fizicheskom printsipe teorii nadezhnosti [About One Physical Principle of Reliability
Theory], Izvestiya Akademii nauk SSSR. Tekhnicheskaya kiber-netika [Proceedings of the Academy of Sciences of the USSR. Technical Cybernetics], 1966, No. 3, Pp. 80-87. (In Russian)
5. Polovko A. M. Osnovy teorii nadezhnosti [Fundamentals of reliability theory]. Moscow, Nauka Publishers, 1964, 446 p.
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Б01: 10.24412/2413-2527-2022-432-71-77
Повышение надежности электронных элементов на основе объемного резервирования
д.т.н. В. А. Смагин Международная академия информатизации Санкт-Петербург, Россия va_smagin@mail.ru
Аннотация. Рассмотрен один из возможных методов повышения надежности электронных элементов, основанный на объемном резервировании. Проведено дальнейшее исследование моста Шеннона с точки зрения теории вероятностей. Предлагается построение примеров некоторых более сложных мостовых конструкций из элементов. Они могут стать основой для построения более сложных систем с большим количеством состояний. В контексте развития нанотехнологий применимы принципы построения систем снизу вверх.
Ключевые слова: избыточность, самовосстановление элемента, мост Шеннона, элементы Шеннона, вероятность работоспособности, структура надежности Седякина.
Литература
1. Smagin, V. A. Nanotechnology — The Basis for the Creation of New High-Reliability Elements // Automatic Control and Computer Sciences. 2008. Vol. 42, Is. 2. Рр. 109-111. DOI: 10.3103/S0146411608020090.
2. Смагин, В. А. Новые вопросы теории эксплуатации. — Санкт-Петербург: ВКА им. А. Ф. Можайского, 2010. — 127 с.
д.т.н. В. П. Бубнов, Д. В. Бараусов Петербургский государственный университет путей сообщения Императора Александра I Санкт-Петербург, Россия bubnov1950@yandex.ru, daniil_bara95@mail.ru
3. Седякин, Н. М. Об одном физическом принципе теории надежности и некоторых его приложениях: Доклад в порядке дискуссии: Выступления по докладу. — Ленинград [Санкт-Петербург]: ЛВВИА им. А. Ф. Можайского, 1965. — 41 с.
4. Седякин, Н. М. Об одном физическом принципе теории надежности // Известия Академии наук СССР. Техническая кибернетика. 1966. № 3. С. 80-87.
5. Половко, А. М. Основы теории надежности. — Москва: Наука, 1964. — 446 с.
6. Шеннон, К. Э. Работы по теории информации и кибернетике: Сборник статей: Пер. с англ. / Под ред. Р. Л. Доб-рушина и О. Б. Лупанова. — Москва: Изд-во иностранной литературы, 1963. — 829 с.
