The procedure for assessing the reliability of the system by the logical probabilistic method. Logical-probabilistic method for calculating the reliability of systems with a monotonic structure

The essence of logical-probabilistic methods lies in the use of logic algebra functions (FAL) for analytical recording of system performance conditions and the transition from FAL to probabilistic functions (WF), which objectively express the system's reliability. Those. using the logical-probabilistic method, it is possible to describe IC circuits for calculating reliability using the apparatus of mathematical logic, followed by the use of probability theory in determining reliability indicators.

The system can only be in two states: in a state of full operability ( at= 1) and in a state of complete failure ( at= 0). It is assumed that the action of the system is deterministically dependent on the action of its elements, i.e. at is a function X 1 , X 2 , … , x i , … , x n. Elements can also be in only two incompatible states: full health ( x i= 1) and complete failure ( x i = 0).

A function of the algebra of logic that relates the state of elements to the state of the system at (X 1 , X 2 ,…,xn) are called health function systems F(y)= 1.

To assess the operable states of the system, two concepts are used:

1) the shortest path of successful operation (KPUF), which is such a conjunction of its elements, none of the components of which can be removed without violating the functioning of the system. Such a conjunction is written as the following FAL:

where i– belongs to the set of numbers corresponding to the given
l-mu way.

In other words, the KPUF of the system describes one of its possible operable states, which is determined by the minimum set of operable elements that are absolutely necessary to perform the functions specified for the system.

2) the minimum system failure cross section (MSF), which is such a conjunction of the negations of its elements, none of the components of which can be removed without violating the system inoperability conditions. Such a conjunction can be written as the following FAL:

where denotes the set of numbers corresponding to the given section.

In other words, the MCO of the system describes one of the possible ways to disrupt the system with the help of a minimum set of failed elements.

Every redundant system has a finite number of shortest paths ( l= 1, 2,…, m) and minimum cross sections ( j = 1, 2,…, m).

Using these concepts, we can write down the conditions for the system to work.

1) in the form of a disjunction of all available shortest paths for successful operation.

;

2) in the form of a conjunction of negations of all MCOs

;

Thus, the operability conditions of a real system can be represented as the operability conditions of some equivalent (in terms of reliability) system, the structure of which is a parallel connection of the shortest paths of successful operation, or another equivalent system, the structure of which is a combination of negations of minimal sections.

For example, for the bridge structure of the IC, the system health function using the KPUF will be written as follows:

;

the operability function of the same system through the MCO can be written in the following form:

With a small number of elements (no more than 20), a tabular method for calculating reliability can be used, which is based on the use of the addition theorem for the probabilities of joint events.

The probability of failure-free operation of the system can be calculated by the formula (through a probabilistic function of the form):

Logical-probabilistic methods (methods: cutting, tabular, orthogonalization) are widely used in diagnostic procedures when constructing fault trees and determining the basic (initial) events that cause the system to fail.

For the reliability of a computer system with a complex redundancy structure, a statistical modeling method can be used.

The idea of the method is to generate boolean variables x i with a given probability pi of the occurrence of a unit, which are substituted into the logical structural function of the simulated system in an arbitrary form, and then the result is calculated.

Aggregate X 1 , X 2 ,…, x n independent random events that form a complete group is characterized by the probabilities of occurrence of each of the events p(x i), and .

To simulate this set of random events, a random number generator is used, uniformly distributed in the interval

Meaning pi is chosen equal to the probability of failure-free operation i th subsystem. In this case, the calculation process is repeated N 0 times with new, independent random argument values x i(this counts the number N(t) single values of the logical structural function). Attitude N(t)/N 0 is a statistical estimate of the probability of uptime

where N(t) - the number of faultlessly working up to the point in time t objects, with their original number.

Generating Random Boolean Variables x i with a given probability of occurrence of one p i is carried out on the basis of random variables uniformly distributed in the interval, obtained using standard programs included in the mathematical software of all modern computers.

1. Name the method for assessing the reliability of IS, where the probability of failure-free operation of the system is defined as R n ≤R with ≤R in.

2. To calculate the reliability of which systems, the method of paths and sections is used?

3. What method can be used to evaluate the reliability of bridge-type devices?

4. What methods for determining the reliability indicators of recoverable systems are known?

5. Structurally represent the bridge circuit as a set of minimum paths and sections.

6. Define the minimum path and the minimum section.

7. Record the health function for branched device?

8. What is called a health function?

9. What is the shortest path to successful operation (KPUF). Write down the working conditions in the form of KPUF.

10. Where is the logical-probabilistic method of reliability assessment used?

Literature: 1, 2, 3, 5, 6, 8.

Topic: Calculation of the reliability of recoverable systems (method of differential equations)

1. General methods for calculating the reliability of recoverable systems.

2. Construction of a graph of possible system states to assess the reliability of restored systems.

3. Method of systems of differential equations (SDE), Kolmogorov's rule for compiling SDE

4. Normalization and initial conditions for solving the SDE.

Keywords

Recoverable system, quantitative characteristics of reliability, state graph, operable state, system of differential equations, Kolmogorov's rule, probability of failure-free operation, recovery rate, failure rate, normalization conditions, initial conditions, reliability parameters, non-redundant system.

The main task of calculating the reliability of designed IS is the construction of mathematical models adequate to the probabilistic processes of their functioning. These models make it possible to assess the degree of satisfaction of the reliability requirements for designed or operated systems.

The type of mathematical model determines the possibility of obtaining calculation formulas. To calculate the reliability of recoverable redundant and non-redundant systems, the following are used: the method of integral equations, the method of differential equations, the method of transient intensities, the method of assessing reliability by the graph of possible states, etc. .

Method of integral equations. The method of integral equations is the most general; it can be used to calculate the reliability of any (recoverable and non-recoverable) systems for any distribution of FBG and recovery time.

In this case, to determine the reliability indicators of the system, integral and integro-differential equations are compiled and solved that relate the characteristics of the FBG distribution, and for restored systems, the recovery time of the elements.

In the course of compiling integral equations, one or more infinitely small time intervals are usually singled out, for which complex events are considered that manifest themselves under the combined action of several factors.

In the general case, solutions are found by numerical methods using a computer. The method of integral equations is not widely used due to the difficulty of solving.

Method of differential equations. The method is used to assess the reliability of recoverable objects and is based on the assumption of exponential distributions of the time between failures (operating time) and recovery time. In this case, the failure flow parameter w =λ = 1/t cp . and recovery intensity µ = 1/ t in, where t cp .- mean uptime, t in is the average recovery time.

To apply the method, it is necessary to have a mathematical model for the set of possible states of the system S={S 1 , S 2 ,…, S n), in which it can be located during system failures and restores. From time to time the system S jumps from one state to another under the action of failures and restorations of its individual elements.

When analyzing the behavior of a system in time during wear, it is convenient to use a state graph. A state graph is a directed graph, where circles or rectangles represent the possible states of the system. It contains as many vertices as there are different states possible for an object or system. The edges of the graph reflect possible transitions from some state to all others with parameters of failure and recovery rates (near the arrows, the transition rates are shown).

Each combination of failed and operable states of subsystems corresponds to one state of the system. Number of system states n= 2k, where k– number of subsystems (elements).

The connection between the probabilities of finding the system in all its possible states is expressed by the system of Kolmogorov differential equations (first-order equations).

The structure of the Kolmogorov equations is built according to the following rules: on the left side of each equation, the derivative of the probability of the object being in the state under consideration (graph vertex) is written, and the right side contains as many members as there are edges of the state graph associated with this vertex. If the edge is directed from a given vertex, the corresponding term has a minus sign, if to a given vertex, a plus sign. Each term is equal to the product of the failure (recovery) intensity parameter associated with a given edge and the probability of being at the vertex of the graph from which the edge originates.

The Kolmogorov system of equations includes as many equations as there are vertices in the object's state graph.

The system of differential equations is supplemented by the normalization condition:

where Pj(t j-th state;

n is the number of possible states of the system.

The solution of the system of equations under specific conditions gives the value of the desired probabilities Pj(t).

The whole set of possible states of the system is divided into two parts: a subset of states n 1 , in which the system is operational, and a subset of states n 2 in which the system is inoperable.

System ready function:

To G ,

where Pj(t) is the probability of finding the system in j working condition;

n 1 is the number of states in which the system is operational.

When it is necessary to calculate the system availability factor or downtime factor (system interruptions are allowed), consider the steady state operation at t→∞. In this case, all derivatives and the system of differential equations are transformed into a system of algebraic equations, which are easily solved.

An example of a state graph of a non-redundant recoverable system with n- the elements are shown in fig. 1.

Rice. 1. Graph of the states of the restored system (shaded states indicate inoperable states)

Consider the possible states in which the system can be. The following states are possible here:

S 0 - all elements are operational;

S 1 - the first element is inoperable; the rest are operational;

S 2 - the second element is inoperable; the rest are operational;

S n – n The th element is inoperable; the rest are operational.

The probability of the simultaneous appearance of two inoperable elements is negligible. Symbols λ 1 , λ2 ,…, λ n failure rates are indicated, µ 1 , µ 2 ,…, µ n recovery intensity of the corresponding elements;

According to the graph of states (Fig. 1), they compose a system of differential equations (the equation for the state S 0 is omitted due to cumbersomeness):

With the normalization condition: .

Initial conditions:

In steady state operation (when t→∞) we have:

Having solved the resulting system of algebraic equations, taking into account the normalization condition, we find the reliability indicators.

When solving a system of equations, one can use the Laplace transform for state probabilities or numerical methods.

Control questions and tasks

1. What methods for determining the reliability indicators of recoverable systems are known?

2. How are the states of IS elements and devices determined?

3. How to determine the areas of healthy states of the system?

4. Why is the method of differential equations widely used in assessing the reliability of restored systems?

5. What is a necessary condition for solving systems of differential equations?

6. How are differential equations compiled to determine the reliability parameters of IS?

7. What condition should be added to the system of differential equations (SDE) for a more efficient solution.

8. Write down the operating conditions of the system, consisting of three elements.

9. What is the number of states of a device consisting of four elements?

10. What rule is used in compiling the CDS?

Literature: 1, 2, 3, 5, 6, 8.

Topic: Markov models for assessing the reliability of redundant recoverable information systems

1. The concept of the Markov property, the definition of the state of the system.

2. Methodology and algorithm for constructing the Markov model.

3. Calculation formulas for calculating the reliability indicators of the vehicle

4. Transition intensity matrix for assessing the reliability indicators of redundant recoverable ICs.

Keywords

Markov model, system state, performance, transition intensity matrix, state graph, recoverable system, redundancy, sequential circuit, constant reserve, system of differential equations, Kolmogorov's rule, reliability calculation scheme, approximate method, SDE construction algorithms, normalization conditions, initial conditions, probability of failure-free operation, failure rate.

The functioning of IS and their components can be represented as a set of processes of transition from one state to another under the influence of any reasons.

From the point of view of the reliability of restored IS, their state at each moment of time is characterized by which of the elements are operational and which are being restored.

If each possible set of operable (inoperable) elements is associated with a set of object states, then failures and restorations of elements will be displayed by the transition of the object from one state to another:

Let, for example, the object consists of two elements. Then it can be in one of four states: n = 2k = 2 2 = 4.

S 1 - both elements are operational;

S 2 - only the first element is inoperative;

S 3 - only the second element is inoperable;

S 4 - both elements are inoperative.

The set of possible object states: S={S 1 , S 2 , S 3 , S 4 }.

The complete set of states of the system under study can be discrete or continuous (continuously fill one or more intervals of the numerical axis).

In what follows, we will consider systems with a discrete state space. The sequence of states of such a system and the process of transitions from one state to another is called a chain.

Depending on the time the system spends in each state, processes with continuous time and processes with discrete time are distinguished. In processes with continuous time, the transition of the system from one state to another is carried out at any time. In the second case, the time spent by the system in each state is fixed so that the transition moments are placed on the time axis at regular intervals.

Currently, the chains with the Markov property are the most studied. The transition probabilities are denoted by the symbols P ij(t), and the process P ij transitions is called a Markov chain or a Markov chain.

The Markov property is associated with the absence of an aftereffect. This means that the behavior of the system in the future depends only on its state at a given time, and does not depend on how it came to this state.

Markov processes make it possible to describe sequences of failures-recoveries in systems described using a state graph.

The most commonly used method for calculating reliability is continuous-time Markov chains based on a system of differential equations, which can be written in matrix form as:

where P(t)= P 0 – initial conditions;

and Λ is the transition intensity matrix (the matrix of the coefficient at the state probabilities):

where λ ij– intensity of the system transition from the i-th state to the j-th;

Pj is the probability that the system is in the jth state.

When assessing the reliability of complex redundant and recoverable systems, the Markov chain method leads to complex solutions due to the large number of states. In the case of subsystems of the same type operating under the same conditions, the aggregation method is used to reduce the number of states. States with the same number of subsystems are merged. Then the dimension of the equations decreases.

The sequence of the methodology for assessing the reliability of redundant recoverable systems using the Markov chain method is as follows:

1. The composition of the device is analyzed and a structural diagram of reliability is drawn up. According to the scheme, a graph is constructed in which all possible states are taken into account;

2. All vertices of the graph as a result of the analysis of the block diagram are divided into two subsets: the vertices corresponding to the operable state of the system and the vertices corresponding to the inoperative state of the system.

3. Using the state graph, a system of differential equations is compiled (Kolmogorov's rule is used);

4. The initial conditions for solving the problem are chosen;

5. The probabilities of the system being in a working state at an arbitrary moment of time are determined;

6. The probability of trouble-free operation of the system is determined;

7. If necessary, other indicators are determined.

Control questions and tasks

1. What is meant by a Markov chain?

2. Give an algorithm for estimating the reliability of IS using Markov models.

3. How are differential equations compiled to determine the reliability parameters of IS?

4. The value of what reliability indicators can be obtained using the Markov method?

5. List the main stages of building a Markov model for the reliability of a complex system.

6. What is a necessary condition for solving systems of differential equations?

7. How are the states of the elements and devices of the CS determined?

8. Define the concept of recoverable systems.

9. What is a Markov chain?

10. What systems are assessed using Markov reliability models?

Literature: 1, 2, 3, 10, 11.

Topic: Approximate methods for calculating the reliability of IS hardware

1. Basic assumptions and limitations in assessing the reliability of series-parallel structures.

2. Approximate methods for calculating the reliability of recoverable ICs, with serial and parallel inclusion of IC subsystems.

3. Structural schemes for calculating the reliability of IS.

Keywords

Reliability, series-parallel structure, approximate methods for calculating reliability, structural diagram of reliability calculation, failure rate, recovery rate, availability factor, recovery time, computer system.

power supply using a fault tree

The logic-probabilistic method using a fault tree is deductive (from general to particular) and is used in cases where the number of different system failures is relatively small. The use of a fault tree to describe the causes of a system failure facilitates the transition from a general definition of failure to particular definitions of failures and modes of operation of its elements, which are understandable to specialist developers of both the system itself and the elements. The transition from a fault tree to a logical failure function opens up possibilities for analyzing the causes of system failure on a formal basis. The logical failure function allows you to obtain formulas for the analytical calculation of the frequency and probability of system failures based on the known frequency and probabilities of element failures. The use of analytical expressions in the calculation of reliability indicators gives grounds for applying the formulas of the theory of accuracy to assess the root-mean-square error of the results.

The failure of the object functioning as a complex event is the sum of the operability failure event and the event , consisting in the appearance of critical external influences. The system failure condition is formulated by specialists in the field of specific systems based on the technical design of the system and analysis of its functioning in the event of various events using statements.

Statements can be final, intermediate, primary, simple, complex. A simple proposition refers to an event or state that is itself neither the logical sum "OR" nor the logical product "AND" of other events or states. A complex statement, which is a disjunction of several statements (simple or complex), is indicated by the "OR" operator, which connects statements of a lower level with statements of a higher level (Fig. 3.15, a). A complex statement, which is a conjunction of several statements (simple or complex), is indicated by the “AND” operator, which connects statements of a lower level with statements of a higher level (Fig. 3.15, b).

Fig.3.15. Logic representation elements

It is convenient to encode statements in such a way that it can be judged by the code whether it is simple or complex, at what level from the final one it is located and what it represents (event, state, operation failure, element type).

In graph theory, a tree is a connected graph that does not contain closed contours. A fault tree is a logical tree (Fig. 3.16), in which the arcs represent failure events at the level of the system, subsystems or elements, and the vertices are logical operations that link the initial and resulting failure events.

Rice. 3.16. An example of building a fault tree

The construction of a fault tree begins with the formulation of the final statement about the failure of the system. To characterize the reliability of the system, the final statement is referred to an event that leads to a malfunction in the considered time interval, under given conditions. The same for readiness characteristics.

Example 8. Let's build a fault tree for the network diagram shown in Figure 3.17.

Fig.3.17. Network diagram

Substations AT and FROM powered by a substation AND. The end event of the fault tree is the failure of the entire system. This failure is defined as the event that

1) either a substation AT or substation FROM completely lose food;

2) power to supply the total load of substations AT and FROM must be transmitted over a single line.

Based on the definition of the end event and the circuit diagram of the system, we build a fault tree (down from the end event) (Fig. 3.18). The purpose of fault tree analysis is to determine the probability of an end event. Since the end event is a failure of the system, the analysis gives the probability R(F).

The analysis method is based on finding and calculating sets minimum sections. cross section A set of elements is called, the total failure of which leads to the failure of the system. The minimum section is such a set of elements from which not a single element can be removed, otherwise it ceases to be a section.

Moving one level down from the vertex (end) event, we pass through the “OR” node, which indicates the existence of three sections: ( P}, {Q}, {R} (R,Q, R– failure events). Each of these sections can be subdivided further into more sections, but it may be found that the failure of sections is caused by several events, depending on what type of logical node is encountered along the route.

Fig.3.18. The system failure tree according to the scheme of fig. 3.17:

– failures of subsystems that can be analyzed further;

For example, (Q) first turns into a section (3, T), then T divided into sections ( X, Y), as a result, instead of one section (3, T) two appear: (3, X}, {3,At}.

At each of the subsequent steps, sets of sections are identified:

The minimum sections are the distinguished sections (3,4,5), (2.3), (1.3), (1.2). The section (1,2,3) is not minimal, since (1,2) is also a section. At the last step, the sets of sections consist exclusively of elements.

In some cases, an object or system cannot be imagined as consisting of parallel-serial connections. This is especially true for digital electronic information systems, in which cross information links are introduced to improve reliability. On fig. 9.17 shows a part of the system structure with cross-links (arrows show the possible directions of information movement in the system). To assess the reliability of such structures, the logical-probabilistic method turns out to be effective.

Rice. 9.17 Bridge scheme for fuel supply;

1-2 - pumps, 3,4,5 - valves

Rice. 9.18 Bridge circuit of the measuring and computing complex;

1,2 - storage device; 3,4 - processors; 5 - a block that provides two-way transmission of digital data.

In the method, the operable state of the structure is proposed to be described using the apparatus of mathematical logic, followed by a formal transition to the probability of failure-free operation of the evaluated system or device. In this case, through a logical variable x j denotes the event that the given i-th element is operational. Formally, the healthy state of the entire system or object is represented by a logical function called the health function. To find this function, it is necessary to determine, following from the input to the output of the system structure, all the paths of the movement of information and the working body corresponding to the operable state of the system. For example, in fig. 9.17. there are four such paths: path 1 - , path 2 - , path 3 - , path 4 - .

Knowing all the paths corresponding to the operable state of the structure, it is possible to write in the symbols of the algebra of logic in a disjunctive-conjunctive form the operability function (X) / For example, for fig. 9.17 is:

Using known methods of minimization, the logical function of health is simplified and transferred from it to the equation of system health in the symbols of ordinary algebra. Such a transition is carried out formally using known relations (logical notation on the left, algebraic notation on the right):

The probability of non-failure operation of an object (see Fig. 9.16, 9.17) is generally determined by formal substitution into the algebraic expression of the health function instead of variables, the value of the probabilities of non-failure operation of each i-th element of the system.

Example. It is necessary to find in general terms the probability of failure-free operation of objects, the structure of which is shown in Fig. 9.16 and 9.17. Despite the different element bases, the elements of the structure of these objects are identical from the point of view of formal logic. For this reason, for clarity, in Fig. 9.17 elements U1, U2 - two identical equally reliable pumps with probabilities of no-failure operation. Elements U3, U4 are two equally reliable processors with a probability of failure-free operation. Element U5 is a switching valve that provides a two-way supply of the working fluid (for example, fuel) at the outlet of the facility.

The structure of the object in Fig. 9.17, where the elements U1, U2 are two identical equally reliable storage devices (memory), with a probability of failure-free operation. Elements U3, U4 are two identical equally reliable processors with a probability of failure-free operation. Element U5 is a block that provides two-way transmission of digital data. The probability of failure-free operation of this unit is .

Taking into account (9.36), (9.37), (9.38) we can make a formal transition from the notation (9.35) to the algebraic notation. So, to find the logical function of the object's operability, the possible ways of passing information (working body) from the input to the output have the form

LOGIC-PROBABILITY METHODS OF RELIABILITY ANALYSIS

Any method of reliability analysis requires a description of the system performance conditions. Such conditions can be formulated on the basis of:

Structural diagram of the system functioning (reliability calculation scheme);

Verbal description of the functioning of the system;

Graph schemes;

Functions of the algebra of logic.

The logical-probabilistic method of reliability analysis makes it possible to formalize the definition and meaning of favorable hypotheses. The essence of this method is as follows.

The state of each element is encoded by zero and one:

In the functions of the algebra of logic, the states of elements are represented in the following form:

X i- good condition of the element, corresponding to code 1;

The failure state of the element, corresponding to code 0.

Using the functions of the algebra of logic, the condition of the system's operability is written through the operability (state) of its elements. The resulting system health function is a binary function of binary arguments.

The resulting FAL is transformed in such a way that it contains terms corresponding to favorable hypotheses for the correct operation of the system.

In FAL instead of binary variables x i and the probabilities are substituted, respectively, for failure-free operation p i and failure probability q i . The signs of conjunction and disjunction are replaced by algebraic multiplication and addition.

The resulting expression is the probability of failure-free operation of the system Pc(t).

Consider the logical-probabilistic method with examples.

EXAMPLE 5.10. The block diagram of the system is the main (serial) connection of elements (Fig. 5.14).

On the block diagram x i , i = 1, 2,..., P- condition i-th element of the system, coded 0 if the element is in a failed state, and 1 if it is serviceable. In this case, the system is operational if all its elements are operational. Then FAL is a conjunction of logical variables, i.e. y \u003d x 1, x 2, ... .., x p, which is a perfect disjunctively normal form of the system.

Substituting instead of logical variables the probabilities of good states of elements and, replacing the conjunction with algebraic multiplication, we get:

EXAMPLE 5.11. The block diagram of the system is a duplicated system with non-equivalent, permanently switched on subsystems (Fig. 5.15).

On fig. 5.15 x 1 and x 2- states of system elements. Let's make a truth table of two binary variables (Table 5.2).

In the table 0 is the failure state of the element, 1 is the good state of the element. In this case, the system is operational if both elements (1,1) or one of them ((0,1) or (1,0)) are operational. Then the operable state of the system is described by the following logic algebra function:

This function is a perfect disjunctive normal form. Replacing the operations of disjunction and conjunction with the algebraic operations of multiplication and addition, and the logical variables with the corresponding probabilities of the state of the elements, we obtain the probability of failure-free operation of the system:

EXAMPLE 5.12. The block diagram of the system has the form shown in fig. 5.16.

Let's make a truth table (Table 53).

In this example, the system is operational if all its elements are operational or if the element is operational x i and one of the elements of the duplicated pair (x 2, x 3). Based on the truth table, the SDNF will look like:

Substituting the corresponding probabilities instead of binary variables, and algebraic multiplication and addition instead of conjunctions and disjunctions, we obtain the probability of the system fail-safe operation:

The function of the algebra of logic can be represented in a minimal form using the following transformations:

The absorption and gluing operations are not applicable in algebra. In this regard, it is impossible to minimize the obtained FAL, and then substitute the values of probabilities instead of logical variables. The probabilities of the states of the elements should be substituted into the SDNF, and simplified according to the rules of algebra.

The disadvantage of the described method is the need to compile a truth table, which requires enumeration of all operable system states.

5.3.2. Method of shortest paths and minimum sections

This method has been discussed previously. in section 5.2.3. Let us state it from the standpoint of the algebra of logic.

The operability function can be described with the help of the shortest paths of the walking functioning of the system and the minimum sections of its failure.

The shortest path is the minimum conjunction of workable:stations of elements that form a workable system.

The minimum section is the minimum conjunction of the inoperable states of the elements that form the inoperable state of the system.

EXAMPLE 5.13. It is necessary to form the system operability function, the block diagram of which is shown in fig. 5.17 using the method of shortest paths and minimum sections.

Decision. In this case, the shortest paths that form a workable system will be: x 1 x 2, x 3 x 4, x 1 x 5 x 4, x 3 x 5 x 2. Then the health function can be written as the following logic algebra function:

In accordance with this FAL, the block diagram of the system in Fig. 5.17 can be represented by the block diagram of fig. 5.18.

The minimum sections that form an inoperable system will be: x 1 x 3, x 2 x 4, x 1 x 5 x 4, x 3 x 5 x 2. Then the inoperability function can be written as the following logic algebra function:

In accordance with this FAL, the block diagram of the system will be presented in the form shown in Fig. 5.19.

It should be borne in mind that the block diagrams in Fig. 5.18 and fig. 5.19 are not reliability calculation schemes, and the expressions for the FAL of the operable and inoperable states are not expressions for determining the probability of failure-free operation and the probability of failure:

The main advantages of the FAL are that they allow one to obtain formally, without compiling a truth table, PDNF and CKNF (perfect conjunctive normal form), which make it possible to obtain the probability of failure-free operation (probability of failure) of the system by substituting in the FAL instead of logical variables the corresponding values of the probabilities of failure-free work, replacing the operations of conjunction and disjunction with the algebraic operations of multiplication and addition.

To obtain SDNF, it is necessary to multiply each disjunctive term of the FAL by, where x i- the missing argument, and expand the brackets. The answer is SDNF. Let's consider this method with an example.

EXAMPLE 5.14. It is necessary to determine the probability of failure-free operation of the system, the block diagram of which is shown in Fig. 5.17. The probabilities of failure-free operation of elements are equal to p 1, p 2, p 3, p 4, r 5 .

Decision. Let's use the shortest path method. The logic algebra function obtained by the shortest path method has the form:

We get the SDNF of the system. To do this, we multiply the disjunctive terms by the missing ones:

Expanding the brackets and performing transformations according to the rules of the algebra of logic, we obtain SDNF:

Substituting in SDNF instead of x 1, x 2, x 3 , x 4, x 5 uptime probabilities p 1, p 2, p 3, p 4, p 5 and using the ratios q i = 1–p i, we obtain the following expression for the probability of failure-free operation of the system.

From the above example, it can be seen that the method of shortest paths freed us from the definition of favorable hypotheses. The same result can be obtained using the method of minimum sections.

5.3.3. Slicing algorithm

The cutting algorithm makes it possible to obtain a FAL, substituting into which, instead of logical variables, the probability of failure-free operation (probability of failure) of elements, one can find the probability of failure-free operation of the system. It is not required to obtain a CDNF for this purpose.

The slicing algorithm is based on the following logic algebra theorem: the logic algebra function y(x b x 2 ,...,x n) can be presented in the following form:

Let us show the applicability of this theorem on three examples:

Applying the second distributive law of the algebra of logic, we get:

EXAMPLE 5.15. Determine the probability of failure-free operation of the system, the block diagram of which is shown in fig. 5.16 using the slicing algorithm.

Decision. Using the shortest path method, we get the following FAL:

Let's apply the cutting algorithm:

Substituting now instead of logical variables the probabilities and replacing the operations of conjunction and disjunction with algebraic multiplication and addition, we get:

EXAMPLE 5.16. Determine the probability of failure-free operation of the system, the block diagram of which is shown in fig. 5.17. Use the cutting algorithm.

Decision. The logic algebra function obtained by the method of minimal sections has the form:

We implement the cutting algorithm with respect to X 5:

We simplify the resulting expression using the rules of the algebra of logic. We simplify the expression in the first brackets using the bracketing rule:

Then FAL will look like:

This expression corresponds to the block diagram of Fig. 5.20.

The resulting scheme is also a reliability calculation scheme, if the logical variables are replaced by the probabilities of failure-free operation p 1, p 2, p 3, p 4, p 5, and the variable is the probability of failure q 5 . From fig. 5.20 it can be seen that the block diagram of the system is reduced to a series-parallel circuit. The probability of failure-free operation is calculated by the following formula:

The formula does not need to be explained, it is written directly according to the block diagram.

5.3.4. Orthogonalization algorithm

The orthogonalization algorithm, like the cutting algorithm, allows formal procedures to form a function of the algebra of logic, substituting into it instead of logical variables probabilities, and instead of disjunctions and conjunctions - algebraic addition and multiplication, to obtain the probability of trouble-free operation of the system. The algorithm is based on the transformation of logic algebra functions into orthogonal disjunctive normal form (ODNF), which is much shorter than SDNF. Before describing the methodology, we formulate a number of definitions and give examples.

Two conjunctions called orthogonal, if their product is identically zero. Disjunctive normal form called orthogonal, if all its terms are pairwise orthogonal. SDNF is orthogonal, but the longest of all orthogonal functions.

Orthogonal DNF can be obtained using the following formulas:

These formulas are easy to prove using the second distributive law of the algebra of logic and De Morgan's theorem. The algorithm for obtaining an orthogonal disjunctive normal form is the following function transformation procedure y(x 1, x 2,..., x n) in ODNF:

Function y(x 1, x 2,..., x n) converted to DNF using the method of shortest paths or minimum sections;

The orthogonal disjunctive-normal form is found using formulas (5.10) and (5.11);

The function is minimized by equating to zero the orthogonal terms of the ODNF;

Boolean variables are replaced by the probabilities of failure-free operation (failure probabilities) of the elements of the system;

The final solution is obtained after simplifying the expression obtained in the previous step.

Let's consider the technique with an example.

EXAMPLE 5.17. Determine the probability of failure-free operation of the system, the block diagram of which is shown in fig. 5.17. Apply the orthogonalization method.

Decision. In this case, the functioning of the system is described by the following logic algebra function (method of minimal sections):

Denote K 1= x 1 x 2, K 2= x 3 x 4, K 3= x 1 x 5 x 4, K 4 \u003d x 3 x 5 x 2. Then ODNF will be written in the following form:

Values , i= 1,2,3, based on formula (5.10) will have the form:

Substituting these expressions into (5.12), we obtain:

Replacing the logical variables in this expression with the corresponding probabilities and performing the algebraic operations of addition and multiplication, we obtain the probability of the system fail-safe operation:

The answer is the same as in Example 5.14.

The example shows that the orthogonalization algorithm is more productive than the methods discussed earlier. In more detail, the logical-probabilistic methods of reliability analysis are described in. The logical-probabilistic method, like any other, has its advantages and disadvantages. Its merits have been mentioned before. Let's point out its shortcomings.

The initial data in the logical-probabilistic method are the probabilities of failure-free operation of the elements of the structural diagram of the system. However, in many cases this data cannot be obtained. And not because the reliability of the elements is unknown, but because the operating time of the element is a random variable. This takes place in the case of redundancy by replacement, the presence of failure aftereffect, the non-simultaneity of the operation of elements, the presence of restoration with a different service discipline, and in many other cases.

Let us give examples illustrating these shortcomings. The block diagram of the system has the form shown in fig. 5.21, where the following designations are accepted: x i- logical variables with values 0 and 1, corresponding to the failure and proper operation of the element, x i = 1, 2, 3.

In this case, the logical variable ds 3 is 0 until the time τ of failure of the main element and 1 during the time (t-τ), where t- the time during which the probability of failure-free operation of the system is determined. Time τ is a random value, so the value р(τ) unknown. In this case, it is impossible to compile a FAL, and even more so an SDNF. None of the logical-probabilistic methods we have considered allows us to find the probability of the system fail-safe operation.

Here is another typical example. The power system consists of a voltage regulator R n and two parallel generators G 1 and G 2 . The block diagram of the system is shown in fig. 5.22.

If one of the generators fails, the remaining serviceable generator works one common load. Its failure rate is increasing. If before the moment τ of failure of one of the generators, the intensity of its failure was equal to λ , then after rejection λ1 > λ2. Since the time τ is random, then Р(τ) unknown. Here, as in the case of redundancy by replacement, logical-probabilistic methods are powerless. Thus, these shortcomings of logical-probabilistic methods reduce their practical application in calculating the reliability of complex systems.

5.4. Topological methods of reliability analysis

We will call topological methods that allow you to determine the reliability indicators either by the state graph or by the structural diagram of the system, without compiling or solving equations. A number of works are devoted to topological methods, which describe various ways of their practical implementation. This section outlines methods to determine the reliability indicators from the state graph.

Topological methods make it possible to calculate the following reliability indicators:

- P(t)- probability of non-failure operation during, time t;

- T1, - mean time of non-failure operation;

- K g (t)- readiness function (probability that the system is operational at any arbitrary point in time t);

- K g= - readiness factor;

T- time between failures of the restored system.

Topological methods have the following features:

Simplicity of computational algorithms;

High clarity of procedures for determining the quantitative characteristics of reliability;

Possibility of approximate estimates;

No restrictions on the type of block diagram (systems, recoverable and non-recoverable, non-redundant and redundant with any type of redundancy and any multiplicity).

This chapter will discuss the limitations of topological methods:

The failure and recovery rates of the elements of a complex system are constant values”;

Time indicators of reliability, such as the probability of failure-free operation and the availability function, are determined in Laplace transforms;

Difficulties, in some cases insurmountable, in the analysis of the reliability of complex systems described by a multiply connected state graph.

The idea of topological methods is as follows.

The state graph is one of the ways to describe the functioning of the system. It determines the type of differential equations and their number. The intensities of transitions, which characterize the reliability of elements and their recoverability, determine the coefficients of differential equations. The initial conditions are chosen by coding the nodes of the graph.

The state graph contains all the information about the reliability of the system. And this is the reason to believe that reliability indicators can be calculated directly from the state graph.

5.4.1. Determining the probabilities of system states

Probability of finding the recoverable system in a state i at a fixed point in time t in the Laplace transform can be written in the following form:

where ∆(s)- the main determinant of the system of differential equations written in Laplace transformations; Δi(s) is a private determinant of the system.

It can be seen from expression (5.13) that Pi(s) will be determined if the degrees are found from the state graph type polynomials of the numerator and denominator, as well as the coefficients Bij (j = 0,1,2,..., m) and A i(i = 0,1, 2,..., n-1).

Let us first consider the method of determining Pi(s) the state graph of only such systems, in the state graph of which there are no transitions through states. These include all non-redundant systems, redundant systems with general redundancy with integer and fractional multiplicity, redundant systems of any structure with maintenance of failed devices in the reverse order of their receipt for repair. This class of systems also includes some redundant systems with equally reliable devices with different disciplines for their maintenance.

The functioning of the system is described by differential equations, the number of which is equal to the number of graph nodes. This means that the main determinant of the system ∆(s) in general will be a polynomial n th degree, where n is the number of state graph nodes. It is easy to show that the denominator polynomial does not contain an intercept. Indeed, since then the denominator of the function Pi(s) must contain s as a factor, otherwise the final probability Pi (∞) will be equal to zero. The exception is when the number of repairs is limited.

Degree of the numerator polynomial∆ i found from the expression:

m i \u003d n - 1 - l i,

where n- number of nodes of the state graph; l i- the number of transitions from the initial state of the system, determined by the initial conditions of its functioning, to the state i along the shortest path.

If the initial state of the system is the state when all devices are operational, then l i- state level number i, i.e. l i is equal to the minimum number of failed system devices in the state i. Thus, the degree of the probability numerator polynomial P i (s) stay of the system in i-th state depends on the state number i and from the initial conditions. Since the number of transitions l i maybe 0,1,2,..., n-1, then the degree of the polynomialΔi(s) based on (5.14) can also take the values m i = 0,1,2,..., n-1.

Lecture 9

Topic: Reliability assessment by the method of paths and sections. Logical-probabilistic methods for the analysis of complex systems

Plan

1. The method of minimal paths and sections for calculating the reliability indicators of systems with a branched structure.

2. Basic definitions and concepts of logical-probabilistic methods of analysis and evaluation of IS reliability.

3. The essence of the method of the shortest path of successful operation and the minimum section of failures.

4. Calculation of the health function and failure function for the bridge structure.

5. Areas of application of these methods. Statistical modeling to assess the reliability of IS.

Keywords

Reliability indicators, branched structure of IS, minimum path, section, logic-probabilistic method, bridge circuit, health function, shortest path of successful operation, minimum failure section, probability of failure-free operation, logic algebra function, structural diagram of reliability calculation.

There are structures and ways of organizing IS when redundancy takes place, but it cannot be represented by the scheme of serial and parallel inclusion of elements or subsystems. To analyze the reliability of such structures, the method of minimum paths and sections is used, which refers to approximate methods and allows you to determine the boundary estimates of reliability from above and below.

A path in a complex structure is a sequence of elements that ensure the functioning (operability) of the system.

A section is a set of elements whose failures lead to system failure.

The probability of failure-free operation of series-connected parallel circuits gives the upper estimate for the FBG of a system of this structure. The probability of failure-free operation of parallel-connected serial circuits of path elements gives a lower estimate for the FBG of a system of this structure. The actual value of the reliability indicator is between the upper and lower limits.

Consider a bridge circuit for connecting the elements of a system consisting of five elements (Fig. 1).

Rice. 1. Bridge circuit for connecting elements (subsystem)

Here, a set of elements forms a minimum path if excluding any element from the set causes the path to fail. It follows from this that within the limits of one path, the elements are in the main connection, and the paths themselves are connected in parallel. Set of minimum paths for bridging presented in fig. 2. Paths form element 1, 3; 2, 4; 1, 5, 4; 2, 5, 3.

Rice. 2. A set of minimal paths.

For all circuit elements, FBGs are known R 1 , R 2 , R 3 , R 4 , R 5 and their corresponding failure probabilities of the "open" typeQ 1 hour Q 5 , it is necessary to determine the probability of the presence of a chain between points a and in. Since the same element is included in two parallel paths, the result of the calculation is an upper reliability estimate.

R in = 1- Q 13 ∙ Q 24 ∙ Q 154 ∙ Q 253 = 1- (1-R 1 R 3)(1-R 2 R 4)(1-R 1 R 5 R 4)(1-R 2 R 5 R 3)

When determining the minimum cross sections, the selection of the minimum number of elements is carried out, the transfer of which from an operable state to an inoperable one causes a system failure.

With the correct selection of the section elements, the return of any of the elements to a working state restores the working state of the system.

Since the failure of each of the sections causes a system failure, the first ones are connected in series. In the limits of each section, the elements are connected in parallel, since for the system to work, it is sufficient to have an operable state of any of the section elements.

The diagram of the minimum cross sections for the bridge circuit is shown in fig. 3. Since the same element is included in two sections, the resulting estimate is a lower estimate.

Pn = P 12 ∙ P 34 ∙ P 154 ∙ P 253 = (1- q 1 q 2 )∙ (1- q 3 q 4 )∙ (1- q 1 q 5 q 4 )∙ (1- q 2 q 5 q 3 )

Rice. 3. Set of minimum sections

System uptime probability R s is then estimated by the double inequality

R n ≤R with ≤R in

Thus, this method makes it possible to represent a system with an arbitrary structure in the form of parallel and series circuits. (When compiling the minimum paths and sections, any system is transformed into a structure with a parallel-serial or series-parallel connection of elements). The method is simple, but requires precise definition of all paths and sections. It has been widely used in calculating the reliability of APCS subsystems, especially in relation to protection and logic control systems. It is used in reactor power control systems, which provide for the possibility of switching from one faulty control circuit to another, which is in a standby state.

Logical and probabilistic methods for analyzing the reliability of systems

The system can only be in two states: in a state of full operability ( at= 1) and in a state of complete failure ( at= 0). It is assumed that the action of the system is deterministically dependent on the action of its elements, i.e. at is a function X 1 , X 2 , … , x i, … , x n. Items can be also in only two incompatible states: full operability (x i = 1) and complete failure (x i = 0).

A function of the algebra of logic that relates the state of elements to the state of the system at (X 1 , X 2 ,…, x n) are called health function systemsF(y) = 1.

To assess the operable states of the system, two concepts are used:

where i- belongs to multiple numbers corresponding to this
l-mu way.

where means the set of numbers corresponding to the given section.

In other words, the MCO of the system describes one of the possible ways to disrupt the system with the help of a minimum set of failed elements.

Every redundant system has a finite number of shortest paths (l= 1, 2,…, m ) and minimum cross sections (j= 1, 2,…, m).

Using these concepts, we can write down the conditions for the system to work.

1) in the form of a disjunction of all available shortest paths for successful functioning.

;

2) in the form of a conjunction of negations of all MCOs

;

For example, for the bridge structure of the IC, the system health function using the KPUF will be written as follows:

;

the operability function of the same system through the MCO can be written in the following form:

With a small number of elements (no more than 20), a tabular method for calculating reliability can be used, which is based on the use of the addition theorem for the probabilities of joint events.

The probability of failure-free operation of the system can be calculated by the formula (through a probabilistic function of the form):

For the reliability of a computer system with a complex redundancy structure, a statistical modeling method can be used.

The idea of the method is to generate boolean variablesx i c given probability pi occurrence of a unit, which are substituted into the logical structural function of the simulated system in an arbitrary form, and then the result is calculated.

Aggregate X 1 , X 2 ,…, X nindependent random events that form a complete group is characterized by the probabilities of occurrence of each of the eventsp(x i), and .

To simulate this set of random events, a random number generator is used, uniformly distributed in the interval

Meaning pi is chosen equal to the probability of failure-free operationith subsystem. In this case, the calculation process is repeatedN 0 times with new, independent random argument valuesx i(this counts the numberN(t) single values of the logical structural function). AttitudeN(t)/ N 0 is a statistical estimate of the probability of uptime

where N(t) - the number of faultlessly working up to the point in timetobjects, with their original number.

Generating Random Boolean Variablesx iwith a given probability of occurrence of one R iis carried out on the basis of random variables uniformly distributed in the interval, obtained using standard programs included in the mathematical software of all modern computers.

Control questions and tasks

1. What is the method for assessing the reliability of IS, where the probability of failure-free operation of the system is defined as R n ≤R with ≤R in.

2. To calculate the reliability of which systems, the method of paths and sections is used?

3. What method can be used to evaluate the reliability of bridge-type devices?

4. What methods for determining the reliability indicators of recoverable systems are known?

5. Structurally represent a bridge circuit as a set of minimal paths and sections.

6. Define the minimum path and the minimum section.

7. Write a health function for a branched device?

8. What is a performance function?

9. What is the shortest path to successful operation (KPUF). Write down the working conditions in the form of KPUF.

10. Where is the logical-probabilistic method of reliability assessment used?

Literature: 1, 2, 3, 5, 6, 8.