CN105825045B - Phased mission systems can repair spare parts demand Forecasting Methodology - Google Patents
Phased mission systems can repair spare parts demand Forecasting Methodology Download PDFInfo
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Abstract
The invention discloses a kind of phased mission systems can repair spare parts demand Forecasting Methodology, for solving the technical problem of existing method spare parts demand prediction effect difference.Technical scheme is to analyze the equipment part composition required by different phase task first, determines part and system mode, including running status and failure state；Secondly, its failure mode modeling is carried out to each part (group), calculates its availability in all stage task.When having spare part and spare part can be repaiied, the different parts in system and its corresponding spare part are integrally considered as a parts group；Then, the BDD models of the phased mission systems are generated on the premise of certain ordering rule；Finally, Markov Chain based on the BDD models established and part (group), calculate the reliability of the stage fill under corresponding spare part quantity, compared with required task system reliability, so as to which that predicts the repairable system repaiies spare parts demand amount, it is accurate that spare parts demand prediction can be repaiied.
Description
Technical Field
The invention relates to a spare part demand prediction method, in particular to a repairable spare part demand prediction method of a multi-stage task system.
Background
The document "Chinese invention patent with publication number CN 101320455A" discloses a spare part demand prediction method based on in-service life evaluation ". Firstly, a statistical model is established by using the historical record of part replacement in the operation process of equipment, and the service life of the spare part under the current service condition is evaluated. Then, according to the estimated value of the service life and the actual service time, a spare part demand function is defined, and the total quantity of the spare part demand of the plurality of devices in a certain time range is further predicted. However, the task system reliability calculated by the method is that the component can only be replaced but not repaired under the condition that both the component and the spare part are not repairable, and the actual component and the spare part thereof are often repairable and cannot obtain a satisfactory spare part demand prediction result, so that the maintenance guarantee of the equipment system is influenced.
Disclosure of Invention
In order to overcome the defect that the spare part demand prediction effect of the existing method is poor, the invention provides a repairable spare part demand prediction method of a multi-stage task system. The method comprises the steps of firstly analyzing the components of equipment required by tasks at different stages, and determining the states of the components and the system, including an operation state and a failure state; secondly, each component (group) is modeled with its failure mode, and its availability for the task at the whole stage is calculated. When spare parts exist and can be repaired, different parts and corresponding spare parts in the system are taken into consideration as a whole part group; then, generating a BDD model of the multi-stage task system on the premise of a certain sequencing rule; and finally, calculating the reliability of the task system at the stage under the corresponding spare part quantity based on the established BDD model and the Markov chain of the component (group), and comparing the reliability with the required reliability of the task system so as to predict the repairable spare part demand of the repairable system.
The technical scheme adopted by the invention for solving the technical problems is as follows: a method for predicting the repairable part requirement of a multi-stage task system is characterized by comprising the following steps of:
step one, establishing a failure model of each part or part group in a task system carrying repairable parts in a multi-stage task by using a Markov chain, and taking different parts and corresponding spare parts in the equipment system as a whole of the part group.
The equipment system participating in the exercise task comprises three types of components c 1 ,c 2 ,c 3 The exercise task comprises two stages of target searching and target tracking, wherein the duration time of the two stages is T1 and T2 respectively. Component c is required when executing a target search task 1 Working and part c 2 ,c 3 At least one job, the task structure function being F 1 ＝c 1 c 2 +c 1 c 3 . When executing the target tracking task, the component c is required 1 ,c 2 ,c 3 Working simultaneously, the task structure function at this time is F 2 ＝c 1 c 2 c 3 . Reliability R of requiring equipment system to successfully complete the exercise task s =0.95, the system reliability of the successful task when carrying 0 spare parts is R 0 Carry c 1 ,c 2 ,c 3 The system reliability of the successful task is R under the condition of one spare part each 1 The amount of spare parts to be carried at this time is predicted.
Component c 1 And the failure time of the spare part is subject to the parameter lambda (a) Is exponentially distributed, the repair time obeying parameter is mu (a) Is used as the index distribution of (1). Component c 2 And the failure time of the spare part is subject to the parameter lambda (b) Is exponentially distributed, the repair time obeying parameter is mu (b) The distribution of indices; component c 3 And the failure time compliance parameter of the spare part is lambda (c) Is exponentially distributed, the repair time obeying parameter is mu (c) Is used as the index distribution of (1).
Component c 1 And spare parts thereof as component group A, component c 2 And spare parts thereof as component group B, component c 3 And its spare parts as component group C. If with a 1 Representing the state of component set A at stage 1, the structural function of the two-stage task is written as:
Φ＝(a 1 b 1 +a 1 c 1 )·(a 2 b 2 c 2 )
wherein, b 1 ,c 1 ,a 2 ,b 2 And c 2 And a 1 Have the same meaning.
A state transition diagram of the system component groups is obtained according to the Markov process when each spare part exists in the delta t time. In the figure, 2 indicates that both the part and its spare part in the part set are normal, 1 indicates that one of the part and its spare part in the part set is normal, and 0 indicates that both the part and the spare part are failed. And meanwhile, obtaining a state transition diagram of the system component without the spare part.
And step two, establishing a BDD model of the task system by using a fault tree method according to the failure model of the task system.
The order rule of the generation sequence of the BDDs needs to be reasonably determined to establish the BDD model as follows:
sequencing according to the sequence of stage tasks, namely 1,2, …, n;
for the Boolean variables of the same-stage task, the variable with the small length L is at the front, and the variable with the large length L is at the back, wherein the length L of the variable is the minimum length of the product terms of the variable in the form of the sum of the product terms, and the lengths of the variables A, B, C, D and E in the structure function F = AB + AED + CEB + CD are respectively L (A) =2,L (B) =2,L (C) =2,L (D) =2,L (E) =2;
when the lengths L are equal, the Boolean variable which appears most frequently in the items of the stage task structure function is taken as the front variable.
And step three, listing paths from the root node to the terminal node 1 in the BDD model according to a depth-first search method to form an uncrossed path set.
Each path Π from the root node to the end node 1 is represented by the product of boolean variables. Searching the BDD of the generated task system for a path from the root node to the terminal node 1, wherein the total number is two: II type 1 ＝a 1 b 1 a 2 b 2 c 2 ，Where 1 is the boolean variable of the edge on this path.
And step four, calculating the probabilities of all the paths listed in the step three, calculating the joint probability of each group of related variables for the mutually related Boolean variables representing the same component/component group at different stages of each path, and multiplying the related probabilities of different components/component groups to obtain the probability of the corresponding path.
The number of the spare parts is different. Considering the component and its corresponding repairable part as a component group C, let Q (C) A generator matrix, P, representing a model of the availability of a group of components C (C) A generator matrix representing a reliability model of the component set C,representing the probability transition matrix of component set C at stage i,a probability transition matrix representing the state of normal operation of the component group C at stage i,a probability transition matrix representing that component set C is in a failed state at a time during stage i;
Q (C) write in split form:
wherein n represents the number of spare parts for the component, Q 11 Is an (n + 1) × (n + 1) matrix composed of transition rates of the operating state to the operating state. Q 21 ,Q 12 ,Q 22 The transition rates included are transition from the failure state to the operating state, transition from the operating state to the failure state, and transition from the failure state to the failure state, respectively.
P (C) Write in split form:
compared with Q (C) At P (C) There is no transition out of the failure state, i.e., C is not allowed to be repaired once it fails.
Will be provided withWriting into:
matrix arrayThe (j, k) element in (a) represents the probability that component group C is in state k at the end of stage i given that component group C is in state j at the beginning of stage i; matrix arrayThe (j, k) element in (a) represents the probability that the component C is in the state k at the end of phase i and remains in the running state throughout the phase i, given that the component group C is in the state j at the beginning of phase i; matrix arrayThe (j, k) element in (a) represents the probability that component set C is in state k at the end of stage i and has failed at some point in the process of stage i, given that component set C is in state j at the beginning of stage i. These index matrices are calculated using a normalization method.
To calculate the task success of the stage task system, the failure of a single component in the multi-stage task is first calculated.
Wherein the content of the first and second substances,indicating the state of the component group C at stage i.
In view of the above definitions, a formula for deriving association probabilities
If the probability vector of component group C at the beginning of stage i isThe probability vector for group C at the end of phase i is then:
wherein, T i Is the duration of phase i. The probability vector that component set C maintains the operating state during stage i is obtained from the reliability model of component set C:
wherein the content of the first and second substances,represents the probability that component set C remains in a running state throughout the task of phase i, and is in state k at the end of phase i. Probability of being in a failure state is determined by multiplying by a probability vectorAnd (4) obtaining. The reliability of the component group C at stage i is calculated by the equation (2):
wherein 1 is T Is a unit column vector of (n + 1) × 1.
According toAndobtaining the state probability vector of the component group C failing at a certain time in the stage i
Due to the memoryless nature of the homogeneous markov chain, the operating conditions of the set of components C at phase i depend only on their initial probability vector at the very beginning of phase i, and also on the probability vector at the end of phase i-1. Therefore, given s 1 ,s 2 ,…,s p The joint probability vector P for component group C is:
where the kth (1. Ltoreq. K. Ltoreq. N + 2) element of matrix p is the probability that component group C is in state k at the end of phase task p.Is defined as:
thenThe joint probability of (c) is:
the probability of occurrence of two non-intersecting paths in step three:
step five, adding the probabilities of all the non-intersecting paths in the decision graph corresponding to the number of different spare parts in the non-intersecting path set in the step four to obtain the reliability of the system for completing tasks under the corresponding number of the spare parts, and respectively calculating the system reliability R corresponding to 0 spare part 0 System reliability R for 1 spare part 1 System reliability R for 2 spare parts 2 System reliability R for 3 spare parts 3 . Given system requirement reliability R s Less than R 0 0 spare part, R, is required 0 <R s ≤R 1 1 spare part, R, is required 1 <R s ≤R 2 And 2 spare parts are needed to complete the demand prediction of repairable spare parts of the equipment system.
The invention has the beneficial effects that: firstly, analyzing the component of equipment required by tasks in different stages, and determining the states of the components and the system, including an operation state and a failure state; secondly, each component (group) is modeled with its failure mode, and its availability of tasks at all stages is calculated. When spare parts exist and can be repaired, different parts and corresponding spare parts in the system are taken into consideration as a whole part group; then, generating a BDD model of the multi-stage task system on the premise of a certain sequencing rule; and finally, calculating the reliability of the task system at the stage under the corresponding spare part quantity based on the established BDD model and the Markov chain of the component (group), and comparing the reliability with the required reliability of the task system so as to predict the repairable spare part demand of the repairable system. By adopting the BDD modeling analysis method, a multi-stage task system repairable part demand prediction model is established, the defect that only parts and spare parts are replaceable but not repairable in the existing repairable system spare part demand prediction is overcome, and the problem of prediction of task spare part demands in a repairable system under the condition that the parts and the spare parts are replaceable and the spare parts can be repaired is solved.
The present invention will be described in detail below with reference to the accompanying drawings and specific embodiments.
Drawings
FIG. 1 is a flow chart of a method for predicting the demand of a repairable part of a multi-stage mission system according to the present invention.
FIG. 2 is a diagram illustrating the state transition of system component groups when there is a spare part in the method for predicting the demand of repairable spare parts in a multi-stage mission system according to the present invention.
FIG. 3 is a state transition diagram of system components without spare parts in the method for predicting the repairable part requirement of a multi-stage mission system according to the present invention.
FIG. 4 is a BDD model generated in the multi-stage mission system repairable part demand prediction method of the present invention.
Detailed Description
Reference is made to fig. 1-4. The method for predicting the repairable part requirement of the multi-stage task system comprises the following specific steps:
1. establishing a failure model of each component or component group in a multi-stage task in a task system carrying repairable parts by using a Markov chain, and integrally considering different components and corresponding spare parts in an equipment system as a component group, wherein the specific mode is as follows:
the radar system participating in the exercise task comprises three types of components c 1 ,c 2 ,c 3 The exercise task comprises two stages of target searching and target tracking, wherein the duration time of the two stages is T1 and T2 respectively. Component c is required when executing a target search task 1 Working and part c 2 ,c 3 At least one job, the task structure function being F 1 ＝c 1 c 2 +c 1 c 3 . Component c is required when executing a target tracking task 1 ,c 2 ,c 3 Working simultaneously, the task structure function at this time is F 2 ＝c 1 c 2 c 3 . Reliability R of the radar system required to successfully complete the maneuver task s =0.95, when carrying 0 spare partsThe system reliability of the successful task is R 0 Carry a c 1 ,c 2 ,c 3 The system reliability for which the task is successful is R for each spare part 1 The amount of spare parts to be carried at that time is predicted.
Component c 1 And the failure time of the spare part is subject to the parameter lambda (a) Is exponentially distributed, the repair time obeying parameter is mu (a) Is used as the index distribution of (1). Similarly, part c 2 And the failure time of the spare part is subject to the parameter lambda (b) Is exponentially distributed, the repair time obeying parameter is mu (b) The distribution of indices; component c 3 And the failure time compliance parameter of the spare part is lambda (c) Is exponentially distributed, the repair time obeying parameter is mu (c) Is used as the index distribution of (1).
Component c 1 And spare parts thereof as a component group A, and a component c 2 And spare parts thereof as component group B, component c 3 And its spare parts as component group C. If with a 1 Representing the state of component group A at stage 1, the structural function of the two-stage task can be written as:
Φ＝(a 1 b 1 +a 1 c 1 )·(a 2 b 2 c 2 )
wherein b is 1 ,c 1 ,a 2 ,b 2 And c 2 And a 1 Have the same meaning.
According to the Markov process, the state transition diagram of the system component group when a spare part exists in the delta t time is obtained. Where 2 indicates that both the part and its spare part in the part set are normal, 1 indicates that one of the part and its spare part in the part set is normal, and 0 indicates that both the part and the spare part are failed. A state transition diagram of the system components without spare parts is similarly obtained.
2. And establishing a BDD model of the task system by using a fault tree method according to the failure model of the task system. The specific mode is as follows:
building a BDD model requires a reasonable determination of the order in which the BDDs are generated, which may grow exponentially if not selected properly. The ordering rule used in the prediction method is as follows:
(1) Sorting according to the sequence of stage tasks, namely 1,2, …, n;
(2) For boolean variables of the same phase task, the variable with a small length L is preceded and followed by a large length L, where the length L of the variable is the minimum length of the product term of the variable in the form of the sum of the individual product terms, such as: the length of each variable a, B, C, D, E in the structure function F = AB + AED + CEB + CD is L (a) =2,L (B) =2,L (C) =2,L (D) =2,L (E) =2;
(3) When the lengths L are equal, the boolean variable that appears most frequently in the terms of the phase task structure function is taken ahead.
And obtaining the BDD of the task system according to the sequencing rule.
3. The method for depth-first search enumerates paths from a root node to a terminal node 1 in the BDD model, and the paths form an disjoint path set, and the specific mode is as follows:
each path pi from the root node to the end node 1 is represented by the product of boolean variables (1-the boolean variable for the edge on this path). Searching the BDD of the generated task system for a path from the root node to the terminal node 1, wherein the total number is two: II type 1 ＝a 1 b 1 a 2 b 2 c 2 ，
4. Calculating the probabilities of all the paths listed in step 3, calculating the joint probability of each group of associated variables for the boolean variables which represent the same component/component group at different stages of each path and are associated with each other, and then multiplying the associated probabilities of different components/component groups to obtain the probability of the corresponding path, wherein the specific method is as follows:
the number of the standby devices is different, the state transition matrixes of the Boolean variables represented by the corresponding nodes in the decision diagram are different, and therefore the probability of each path obtained finally is different. Considering the component and its corresponding repairable part as a component group C, let Q (C) A generator matrix, P, representing a model of the availability of a group of components C (C) Represents a group of components CA generator matrix of the reliability model is generated,representing the probability transition matrix of component set C at stage i,a probability transition matrix representing the state of normal operation of the component group C at stage i,a probability transition matrix representing that component group C is in a failed state at a time during stage i;
Q (C) write in split form:
wherein n represents the number of spare parts for the component, Q 11 Is an (n + 1) × (n + 1) matrix composed of transition rates of the operating state to the operating state. Similarly, Q 21 ,Q 12 ,Q 22 The transition rates included are transition from the failure state to the operating state, transition from the operating state to the failure state, and transition from the failure state to the failure state, respectively.
P (C) Also written in the form of a partition:
compared with Q (C) At P (C) There is no transition out of the failure state, i.e., C is not allowed to be repaired once it fails.
Will be provided withWriting into:
matrix arrayThe (j, k) element in (a) represents the probability that component group C is in state k at the end of stage i given that component group C is in state j at the beginning of stage i; matrix arrayThe (j, k) element in (a) represents the probability that, given the condition that component group C is in state j at the beginning of phase i, component C is in state k at the end of phase i and remains in a running state throughout phase i; matrix arrayThe (j, k) element in (a) represents the probability that component set C is in state k at the end of stage i and has failed at some point in the process of stage i, given that component set C is in state j at the beginning of stage i. These index matrices are calculated using a normalization method.
To calculate the task success of this stage task system, the failure of a single component (group) in a multi-stage task is first discussed.
Wherein, the first and the second end of the pipe are connected with each other,indicating the state of the component group C at stage i.
In view of the above definitions, a formula for deriving association probabilities
Wherein s is i (1. Ltoreq. I.ltoreq.p) is 0,1 or x.
If the probability vector of component group C at the beginning of stage i is(Is the initial probability vector), the probability vector of component group C at the end of stage i is:
wherein T is i Is the duration of phase i. Similarly, the probability vector that component set C remains in the operating state during stage i is obtained from the reliability model of component set C:
whereinRepresents the probability that component set C remains in a running state throughout the task of phase i, and is in state k at the end of phase i. Probability of being in failure state is determined by multiplying probability vectorAnd (4) obtaining. The reliability of the component group C at stage i is calculated by the equation (2):
wherein 1 is T Is a unit column vector of (n + 1) × 1.
According toAndobtaining the state probability vector of the component group C failing at a certain time in the stage i
Due to the memoryless nature of the homogeneous markov chain, the operating conditions of the set of components C at phase i depend only on their initial probability vector at the very beginning of phase i, and also on the probability vector at the end of phase i-1. Therefore, given s 1 ,s 2 ,…,s p The joint probability vector P for component group C is:
where the kth (1. Ltoreq. K. Ltoreq. N + 2) element of the matrix p is the probability that the component group C is in state k at the end of the phase task p.Is defined as:
thenThe joint probability of (c) is:
TABLE 1 values of the parameters
| Parameter(s) | λ (a) | μ (a) | λ (b) | μ (b) | λ (c) | μ (c) | T 1 ＝T 2 |
| Value (h) -1 ) | 0.01 | 1 | 0.05 | 0.5 | 0.04 | 2 | 1h |
The time of the three phase tasks is assumed to be the same. Referring to table 1, the transfer matrix E, D, U of the component groups a, B, C at each stage is obtained as follows:
thereby obtaining the occurrence probability of two non-intersecting paths in the step 3:
5. adding the probabilities of all the non-intersecting paths in the decision graph corresponding to different spare parts in the non-intersecting path set in the step 4 to obtain the reliability of the system for completing tasks under the corresponding spare part number, and respectively calculating the system reliability R corresponding to 0 spare parts 0 System reliability R for 1 spare part 1 System reliability R for 2 spare parts 2 System reliability R for 3 spare parts 3 . Given system requirement reliability R s Less than R 0 0 spare part, R, is required 0 <R s ≤R 1 1 spare part, R, is required 1 <R s ≤R 2 And 2 spare parts are needed to complete the demand prediction of repairable spare parts of the equipment system.
System reliability R of successful task when the radar system components each have 1 spare part 1 Comprises the following steps:
under the condition of different spare parts, the transfer matrixes E, D, U of each stage of each part (group) in the step 4 are different, but the decision diagram in the step 3 is the same as the set of the non-intersection paths in the step 4, and the system reliability under the condition of different spare parts can be conveniently obtained only by calculating the step 4. System reliability R of successful task when 0 spare part is obtained through calculation 0 Comprises the following steps:
due to R 0 <R s <R 1 Therefore, each key part can achieve the task only by needing one spare partAnd achieving the required system reliability, thereby completing the demand prediction of the repairable part.
Claims (1)
1. A method for predicting the demand of repairable parts of a multi-stage task system is characterized by comprising the following steps:
establishing a failure model of each component or component group in a task system carrying repairable parts in a multi-stage task by using a Markov chain, and taking different components and corresponding spare parts in the equipment system as a whole component group;
the equipment system participating in the exercise task comprises three types of components c 1 ,c 2 ,c 3 The exercise task comprises two stages of target searching and target tracking, wherein the duration time of the two stages is T1 and T2 respectively; component c is required when executing a target search task 1 Working and part c 2 ,c 3 At least one job, the task structure function being F 1 ＝c 1 c 2 +c 1 c 3 (ii) a Component c is required when executing a target tracking task 1 ,c 2 ,c 3 Working simultaneously, the task structure function at this time is F 2 ＝c 1 c 2 c 3 (ii) a Reliability R of requiring the equipment system to successfully complete the exercise task s =0.95, the system reliability of success of the task when carrying 0 spare parts is R 0 Carry c 1 ,c 2 ,c 3 The system reliability of the successful task is R under the condition of one spare part each 1 Predicting the quantity of spare parts to be carried at the moment;
component c 1 And the failure time of the spare part is subject to the parameter lambda (a) Is exponentially distributed, the repair time obeying parameter is mu (a) The distribution of indices; component c 2 And the failure time of the spare part is subject to the parameter lambda (b) Is exponentially distributed, the repair time obeying parameter is mu (b) The exponential distribution of (a); component c 3 And the failure time compliance parameter of the spare part is lambda (c) Is exponentially distributed, the repair time obeying parameter is mu (c) The distribution of indices;
component c 1 And spare parts thereof as component group A, component c 2 And spare parts thereof as component group B, component c 3 And spare parts thereof as a component group C; if with a 1 Representing the state of component set A at stage 1, the structural function of the two-stage task is written as:
Φ＝(a 1 b 1 +a 1 c 1 )·(a 2 b 2 c 2 )
wherein, b 1 ,c 1 ,a 2 ,b 2 And c 2 And a 1 Have the same meaning;
obtaining a state transition diagram of the system component group when each spare part exists in delta t time according to a Markov process; in the figure, 2 indicates that both the part and its spare part in the part group are normal, 1 indicates that one of the part and its spare part in the part group is normal, and 0 indicates that both the part and the spare part are failed; meanwhile, a state transition diagram of the system component without the spare part is obtained;
step two, establishing a BDD model of the task system by using a fault tree method according to the failure model of the task system;
the order rule of the generation sequence of the BDDs needs to be reasonably determined to establish the BDD model as follows:
sequencing according to the sequence of stage tasks, namely 1,2, …, n;
for the Boolean variables of the same-stage task, a variable with a small length L is in front of the Boolean variable and a variable with a large length L is behind the Boolean variable, wherein the length L of the variable is the minimum length of product terms of the variable in the form of the sum of the product terms, and the lengths of variables F, G, H, D and E in the structural function F = KG + KED + HEG + HD are respectively L (K) =2,L (G) =2,L (H) =2,L (D) =2,L (E) =2;
when the lengths L are equal, taking the Boolean variable with the most occurrence times in the items of the stage task structure function as the front;
step three, enumerating paths from a root node to a terminal node 1 in the BDD model according to a depth-first search method to form an uncrossed path set;
each path pi from the root node to the terminal node 1 is represented by the product of boolean variables; searching for slave root nodes in a BDD of a generated task systemThe path to end node 1, there are two in total: II type 1 ＝a 1 b 1 a 2 b 2 c 2 ，Where 1 is the boolean variable of the edge on this path;
step four, calculating the probabilities of all the paths listed in the step three, calculating the joint probability of each group of related variables for the mutually related Boolean variables representing the same component/component group at different stages of each path, and then multiplying the related probabilities of different components/component groups to obtain the probability of the corresponding path;
the number of prepared pieces is different; considering the component and its corresponding repairable part as a component group C, let Q (C) A generator matrix, P, representing a model of the availability of a group of components C (C) A generator matrix representing a reliability model for the group of components C,representing the probability transition matrix of component set C at stage i,a probability transition matrix representing the state of normal operation of the component group C at stage i,a probability transition matrix representing that component set C is in a failed state at a time during stage i;
Q (C) write to the form of segmentation:
wherein n represents the number of spare parts for the component, Q 11 Is a (n + 1) × (n + 1) matrix composed of transition rates of the operating state to the operating state; q 21 ,Q 12 ,Q 22 Including the transfer rate ofThe method comprises the steps of transferring from a failure state to an operating state, transferring from the operating state to the failure state and transferring from the failure state to the failure state;
P (C) write in split form:
compared with Q (C) At P (C) There is no transition out of the failure state, i.e., C is not allowed to be repaired once it fails;
will be provided withWriting into:
matrix arrayThe (j, k) element in (a) represents the probability that component group C is in state k at the end of stage i given that component group C is in state j at the beginning of stage i; matrix arrayThe (j, k) element in (a) represents the probability that, given the condition that component group C is in state j at the beginning of phase i, component C is in state k at the end of phase i and remains in a running state throughout phase i; matrix ofThe (j, k) element in (a) represents the probability that, given the condition that component group C is in state j at the beginning of phase i, component group C is in state k at the end of phase i and has failed at some point in the course of phase i; these index matrices are calculated using a normalization method;
in order to calculate the task success of the stage task system, firstly calculating the failure condition of a single component in the multi-stage task;
wherein the content of the first and second substances,representing the state of the component group C at stage i;
in view of the above definitions, a formula for deriving association probabilities
If the probability vector of component group C at the beginning of stage i isThe probability vector for group C at the end of phase i is then:
wherein, T i Is the duration of phase i; the probability vector that component set C maintains the operating state during stage i is obtained from the reliability model of component set C:
wherein the content of the first and second substances,represents the probability that the component group C remains in the operating state throughout the task of phase i and is in state k at the end of phase i; probability of being in failure state is determined by multiplying probability vectorObtaining; the reliability of the component group C at stage i is calculated by the equation (2):
wherein 1 is T Is a unit column vector of (n + 1) × 1;
according toAndobtaining the state probability vector of the component group C failing at a certain time in the stage i
Due to the memoryless nature of the homogeneous markov chain, the operating conditions of the group of components C at phase i depend only on its initial probability vector at the very beginning of phase i, which is also the probability vector at the end of phase i-1; therefore, given s 1 ,s 2 ,…,s p The joint probability vector P for component group C is:
wherein, the k (1 ≦ k ≦ n + 2) of the matrix p) The element is the probability that component group C is in state k at the end time of stage task p;is defined as:
thenThe joint probability of (c) is:
the probability of occurrence of two non-intersecting paths in step three:
step five, adding the probabilities of all the non-intersecting paths in the decision graph corresponding to the number of different spare parts in the non-intersecting path set in the step four to obtain the reliability of the system for completing tasks under the corresponding number of the spare parts, and respectively calculating the system reliability R corresponding to 0 spare part 0 System reliability R for 1 spare part 1 System reliability R for 2 spare parts 2 System reliability R for 3 spare parts 3 (ii) a Given system requirement reliability R s Less than R 0 0 spare part, R, is required 0 ＜R s ≤R 1 1 spare part, R, is required 1 ＜R s ≤R 2 And 2 spare parts are needed to complete the demand prediction of repairable spare parts of the equipment system.
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