CN109063264A - A kind of phased mission systems Reliability modeling and analysis method - Google Patents
A kind of phased mission systems Reliability modeling and analysis method Download PDFInfo
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Abstract
The invention belongs to system reliability technical fields, disclose a kind of phased mission systems Reliability modeling and analysis method, using the thought for individually modeling and solving by stage, each single phase reliability model in multiphase system is generated based on continuous time Markov chain, successively each stage model is solved, the reliability of terminal stage is the reliability of entire task；Single phase Markov model is pre-processed using sparse matrix compression and storage method；It is solved based on Markov reliability model of the Krylov subspace calculation method to generation；Using state mapping mechanism between the stage, successively solving by stage to single period model is realized.Using the thought individually modeled by stage, the problem that unified Modeling causes scale of model huge is avoided；Meanwhile based on sparse matrix compression storage and Krylov subspace method, the space storage efficiency and computational efficiency of model can be effectively improved.
Description
Technical field
The invention belongs to system reliability technical field more particularly to a kind of phased mission systems Reliability modeling and divide
Analysis method.
Background technique
As modern system becomes to become increasingly complex and intelligent, the operation of system is no longer single process, but including
The conversions of multiple functional sequences, or pass through the process that new system is formed to reconfiguration of cell in different phase.With space travel
For the aerial mission of device, including stages, each stage such as take off, rise, fly, land and land have different systems
Configuration, task success or failure standard and element failure parameter etc..Above-mentioned includes the multiple tasks stage, and system configuration and cell parameters with
Stage changed system is phased mission systems (phased-mission system, PMS).PMS is one kind in reality
Common system, such as cruise missile system in the engineer application of border, air defence weapon system and spaceflight TT&C system etc..
Reliability refers to the system interior ability for completing predetermined function of defined time under the defined conditions.From product design
Angle, reliability is divided into basic reliability and mission reliability.Mission reliability refers to that product is cutd open in a defined group task
The ability of predetermined function is completed in face.The mission reliability analysis of PMS is exactly that the system within given time that calculates executes all ranks
The probability of cross-talk task completion assignment of mission.
Currently, mainly having following three kinds of methods: built-up pattern for different models in the solution of PMS reliability
(combinatorial model) method, the method and system simulation method based on state space.(1) built-up pattern method: including reliable
Property block-scheme method and fault tree analysis method.Built-up pattern method has the characteristics that succinct intuitive, but is only suitable for modeling static PMS, generally
Assuming that the failure behaviour of each unit is mutually indepedent and is not maintainable；Meanwhile this method is not suitable for large scale system
Fail-safe analysis.With the increase of system scale, the complexity of this method is exponentially increased.Built-up pattern method further includes binary
Decision diagram (binary decision diagram, BDD) method, it is reliable that BDD method provides a kind of rapid solving static state PMS
Property mechanism, correlative study work, which is concentrated mainly on, solves multimodel failure, not totally wrong covering, common cause failure and BDD
Sort algorithm etc..It is typically only capable to handle static PMS using BDD method, and is mutual between each component in hypothesis system
It is independent；(2) based on the method for state space: including Markov method and Petri network method, being all based on the reason of random process
By being analyzed.Research of the Petri network in terms of the PMS reliability PMS based on Petri network of propositions such as including Mura is reliable
Property analysis model, the modeling tool DEEP based on Petri network method of exploitation, for the fail-safe analysis of PMS, to provide function strong
Big the integration environment.Compared to built-up pattern method application aspect limitation, based on the method for state space for various systems
Expression ability have very high flexibility, still, this flexibility and versatility need certain cost, it is adjoint and come the side of modeling
The more high complexity of method and model treatment；(3) system emulation method: when the unit correlation in system each stage is more complex, or
It is more difficult using the above method when system reliability, maintainability distribution are not exponential distributions, system emulation method can only be used.
The theoretical basis of emulation mode is the philosophy in probability theory --- law of great number.The advantages of emulation mode is suitable for any
Situation, but amount of calculation is its main bottleneck in application process greatly.The Raptor emulation tool of Murphy et al. exploitation
The achievable emulation to PMS reliability, but emulation mode used in the tool belongs to thick emulation, therefore simulation efficiency is lower.
Xu etc. proposes the PMS mission reliability emulation mode based on Force Law and failure offset, but this method is suitable for a large amount of
The system of redundant component and shorter task time.
About the solution of Markov reliability model, main problems faced is that state space is exponentially increased, specific body
To e in present solution proceduretQCalculating.etQReferred to as the calculation technique of matrix exponential function (matrix exponential), calculation method is big
Cause is divided into following a few classes: series expansion method, Ordinary Differential Equation Method, Polynomial Method, matrix disassembling method and matrix-split
Method.Correlative study point out there is no a kind of method in all cases can Efficient Solution, while existing most of method master
To be directed to middle and small scale matrix, be unsuitable for Large Scale Sparse Matrix Solving because a large amount of matrix power operation will lead to it is sparse
The disappearance of property, so that computational efficiency is low.Currently, for extensive environment exploitation method for solving include by Reibman and
The iterative solution thought that Rauzy is proposed converts matrix and vector operation for operation between matrix by deriving iterative formula；Li et al.
For the Calculation of Mission Reliability of TT & C Telecommunication, a kind of derivation algorithm based on Runge-Kutta, the party are proposed
Method computational accuracy with higher, but when matrix size is very big, algorithm time-consuming is still very big.
In conclusion it is existing based on the multiphase system mission reliability analytical technology of Markov method there are the problem of
Be: one is the absence of the compression storage processing to model.Due to being sparse matrix based on the transition rates in Markov model,
In extensive Markov model solution procedure, transition rates are often the extensive dilute of tens of thousands of ranks even hundreds of thousands rank
Matrix is dredged, if solving sparse vectors with dense matrix method, amount of storage and calculation amount all will be very huge.Work as state
When amount of space is larger, using element in certain tactful condensation matrix, compression storage processing is carried out to model, it will save big
The memory space of amount；Second is that the dependence complexity between the stage causes state matching difficult.When using the independent side of modeling of each stage
When method, dependence of each unit across the stage in solution system is needed, i.e., is matched by state between the stage, by previous stage end system
The initial state of next stage system is obtained in each shape probability of state.But since the unit that different phase participates in work becomes
Change, cause in the same task period, difference, working time length are different at the time of each unit is due to beginning participating in work and connect
Continue or discontinuously appear in task phase, causes the state mapping process between two successive stages extremely complex.Currently, related text
Offer it is relatively simple for the research of state mapping mechanism between the stage, and do not have generality；Third is that existing reliability model
Derivation algorithm efficiency is lower.The solution of Markov reliability model is related to matrix exponetial operation etQ, when matrix size is larger
When, it calculates very difficult.Existing derivation algorithm is largely directed to middle and small scale matrix, when solving extensive Markov model
Inefficiency more especially includes the derivation algorithm of matrix power operation, and the sparsity of matrix fades away after power operation,
Its operation efficiency is more low.
Summary of the invention
In view of the problems of the existing technology, the present invention provides a kind of phased mission systems Reliability modelings and analysis
Method.
The invention is realized in this way a kind of phased mission systems Reliability modeling and analysis method, the multistage
Task system Reliability modeling, using individually modeling and solving by stage, is generated with analysis method based on continuous time Markov chain
Each single phase reliability model, successively solves each stage model in multiphase system, and the reliability of terminal stage is whole
The reliability of a task；Single phase Markov model is pre-processed using sparse matrix compression and storage method；It is based on
Krylov subspace calculation method solves the Markov reliability model of generation；Using state mapping mechanism between the stage,
Realize successively solving by stage to single period model.
Further, the phased mission systems Reliability modeling and analysis method the following steps are included:
Step 1 establishes the mission reliability model in system each stage based on continuous time Markov chain.Enable viTo be
It unites in the rate of transform of state i, pijThe probability of state j is transferred to from state i for system, then:
Wherein,
If vi=0, then state i is referred to as the absorbing state of system；
If vi=∞, then state i is referred to as the transient state of system；
If 0 < vi< ∞, then state i is referred to as the stable state of system；
qijIt is system from state i to the rate of transform of state j:
qii=-vi, following matrix is referred to as the transition rates of CTMC, or infinitely small generation:
And have
Qe=0；
Wherein, 1 e=1 ... 1TFor unit vector；
The Markov model transition rates Q of foundation:
Step 2: Markov single period model is pre-processed using sparse matrix compression and storage method, it is specific to wrap
It includes::
CRS stores matrix Q using three arrays: (1) value: storing the value of the nonzero element in Q, number line by line
It is floating type according to type；(2) colind: the column index of each non-zero entry in storage Q, data type is integer；(3) rowptr:
Index of first nonzero element of every row in array value is stored, data type is integer；
Step 3: for the Markov reliability model of generation, using based on Krylov subspace calculation method to model
It is solved, is successively calculated by stage, the reliability of final stage is the reliability of entire task, utilizes Krylov
Space-wise projects extensive Markov model into small-scale subspace, derives and calculates state using matrix after projection
The approximate formula of probability vector v (t)；
Step 4: the phased mission systems fail-safe analysis based on Markov process, generallys use and individually builds by stage
Mould simultaneously solves；It stage by stage in independent modeling method, is matched by state between the stage, each state is in by previous stage end system
Probability obtains the initial state of next stage system, is successively solved by stage based on state mapping mechanism realization between the stage.
Further, reliability calculation method of the step 3 based on Krylov subspace includes:
For state probability vector v (t), enabling v (0) is the state probability vector of system initial time, then according to above
Description to Markov model, it is known that:
V (t)=v (0) eQt；
If enabling A=QΤ, ω (t)=v (t)T, π=ω (0) then has:
ω (t)=eAtπ t [0,T]；
ω=eAπ is then denoted as p to any m-1 rank multinomial expansion of ωm-1(A) π carries out approximate calculation, such as its m-1 rank
Taylor expansion isAll m-1 rank multinomial approximations of ω are all Krylov subspace KmElement in (A, π),
Wherein:
Km(A, π)=Span π, A π ..., Am-1π}；
Bm=[b1,b2,...,bm] it is Krylov subspace KmOne group of base in (A, π), BmBy Arnoldi procedure construction, ωoptFor
To the optimal value in the m-1 rank multinomial approximation of ω；According to principle of least square method, obtainBmFor
Orthonormal basis,
Base B is constructed using Arnoldi algorithmmWhen, if enable β=| | π | |2, takeπ is initial time state probability
Vector, eiIndicate i-th of column vector of unit matrix I, length is depending on the circumstances to be obtained:
π=β Bme1；
ωopt=Bmyopt, it obtains:
Following approximate formula is obtained according to Taylor expansion:
Then:
Time variable t is reintroduced back in formula, to arbitrary time t, is always had:
Further, state mapping mechanism between the step 4 stage are as follows:
The unit number in n expression system is enabled, then system state space is 2n；With SiI-th of state of expression system, by two
System string representation, Si=c1c2...cn, ck(1 k n) is the state of i-th of unit, if i-th unit is normal, ckIt is 1,
It otherwise is 0；
(1) it defines 1 state xor operation: using symbolIt indicates, there is two state SiAnd Sj, binary-coded character represented by state
The length of string is identical, and xor operation is the sum that corresponding position carries out XOR operation；
(2) it is equal that 2 states are defined: there are two state SiAnd SjIf its length is identical, andThen claim state SiWith
SjIt is equal；
The primitive rule that state maps between (3) two successive stages are as follows: by rejecting new addition equipment, being no longer participate in task
Equipment, or increase running equipment so that two stage status word string length having the same, then in the current generation state just
Beginning probability value is the corresponding probability value of previous stage end equivalent state, is enabledState in expression stage k-1,The expression stage
State in k, specifically includes:
1) there is new equipment addition in the current task stage, if having new equipment addition in stage k, as at the beginning of the equipment being newly added
Beginning state be it is normal, corresponding states value be 1；For any one state in stage kNew equipment is corresponded in status word string
State value be 0, thenIt is 0 in the probability of stage k；For the state that equipment is " 1 " is newly addedStage initial time
The corresponding probability of the state is unrelated with new equipment, and the correspondence position of new equipment in status word string is rejected, new state is obtainedThen previous stage stateWithLength having the same maps primitive rule pair according to stateWithIt is reflected
It penetrates, obtains the initial state probability vector of stage k；
2)In be not involved in the follow-up work stage equipment correspondence position reject, obtain new stateAccording to state
The primitive rule of mapping is to stateWithIt is mapped, obtains the initial state probability vector of stage k；
3) certain equipment do not execute task in the current generation, are constantly in though the running equipment of stage k is not carried out task
Operating status includes the description to running equipment in the Markov model of stage k.
Another object of the present invention is to provide a kind of phased mission systems Reliability modeling and analysis methods
Phased mission systems Calculation of Reliability system, the phased mission systems Calculation of Reliability system include:
The case resource file and assignment file of XML format are opened and read to file read module, obtains phased mission system
The equipment dependability data and task configuration information of system complete data preparation；
Calculation of Reliability module, it is raw based on continuous time Markov chain for calculating the reliability of phased mission systems
At reliability model, compression processing is carried out using sparse matrix compression and storage method to model, is based on Krylov subspace method
Model is calculated, and successively solving by stage for mission reliability is realized by the state mapping matching between the stage；
Result display module, for showing each session information and reliability results and total mission reliability meter
Calculate result；
Calculated result is saved as Excel file after the completion of calculating by preserving module.
Advantages of the present invention and good effect are illustrated in terms of model storage efficiency and Calculation of Reliability efficiency two:
One, (compressed row storage, CRS) method is stored to Markov reliability model using by row compression
Pretreated storage efficiency is carried out to compare.Foreign scholar Haque is concluded that by experiment in various sparse matrix storage sides
In case, fixed block stores the best performance of (Fixed-size Block Storage, FBS).FBS method can also be abbreviated as
FBSl, wherein l is the length of nonzero element block.Therefore, the storage efficiency of CRS and FBS2, FBS3 are compared.
Concrete case is as follows: note has phased mission system T1, which can be divided into three task phases P1, P2 and P3, hold
The continuous time is respectively 90s, 150s and 90s.The fault Tree difference in task T1 each stage is as shown in Figure 8, Figure 9.
Participating in the component count of task in task T1 in three phases is respectively 9,12 and 16, each stage matrix Q1、 Q2And Q3
Scale be respectively 512 × 512,4096 × 4096 and 65536 × 65536, nonzero element is respectively 950,11167 Hes
232713, the normal condition number in three phases system is respectively 95,859 and 13689.Matrix Q1、Q2And Q3Middle length is 2
Nonzero block number is 95,859 and 13689；The nonzero block number that length is 3 is 53,446 and 7037.
If real-coded GA needs 8 bytes, integer data needs 4 bytes, and character type data needs 8 bytes.Then
The amount of storage that can be obtained under CRS and FBSl scheme is
MCRS=8nnz+4 (nnz+N+1)
MFBSl=8nnz+4 (nnz+2N+2- (l-1) * βl)
Wherein, nnz is nonzero element number in matrix, N representing matrix dimension, βlThe nonzero block for being l for length in matrix
Number.To sum up, matrix Q can be obtained1、Q2And Q3CRS, FBS2, FBS3 and without compression when amount of storage it is as shown in table 3.
1 Q of table1、Q2And Q3Amount of storage needed under different storage modes compares (unit: byte)
| Q1 | Q2 | Q3 | |
| CRS | 13452 | 150392 | 3054704 |
| FBS2 | 15124 | 163344 | 3262096 |
| FBS3 | 15080 | 163212 | 3260556 |
| Without compression | 2097152 | 134217728 | 34359738368 |
As can be seen from Table 1, in the reliability model calculating process based on Markov process, using the effect of compression storage
Rate is apparently higher than the mode without compression storage, and in tri- kinds of compression schemes of CRS, FBS2 and FBS3, storage needed for CRS
It measures smaller.
Two, Calculation of Reliability efficiency compares.Forward direction Euler algorithm (Forward EulerMethod, FEM) is due to process
Simply, it is easy to implement, therefore speed is better than the ordinary differentials side such as Runge-Kutta (Runge-Kutta) and Adam Mu Si (Adams)
Cheng Fangfa.Meanwhile foreign scholar A.Rauzy also turns out FEM method in most cases by experiment and can obtain preferably
Calculated result.Therefore, using FEM method compared with Krylov subspace method carries out computational efficiency.
Concrete case is as follows: using 3 Single-phase missions T2, T3 and T4, carries out Calculation of Mission Reliability.Assuming that participating in appointing
The number of devices of business is respectively 11,12 and 13, and the MTTR and MTBF of equipment are 20min and 30min, mission duration
It is 30min.To simplify description, it is assumed that equipment room is parallel relationship in each task.
Matrix size in task task4, task5, task6 is respectively 2048 × 2048,4096 × 4096 and 8192 ×
8192.Krylov method subspace scale takes 20 to be calculated, and is denoted as Krylov20；FEM algorithm step-size takes 0.0001, is denoted as
FEM0.00001.Each method calculated result and Riming time of algorithm difference are as shown in Table 1 and Table 2.
2 task of table4、task5And task6Reliability results comparison
The comparison of 3 Riming time of algorithm of table
It can be obtained by table 1, the calculating error of Krylov method and FEM method is respectively 4.46e-005,5.33e-005,
6.29e-005.To more massive problem it can be seen from Tables 1 and 2, Krylov method still is able to guarantee very high speed
Degree and precision, performance are substantially better than FEM method.It can be seen that Krylov method runing time and the direct phase of subspace scale
It closes, in the case that in subspace, scale is constant, the increase of matrix size influences Riming time of algorithm little in initial problem.
Detailed description of the invention
Fig. 1 is the flow chart of phased mission systems Reliability modeling and analysis method provided in an embodiment of the present invention；
The CTMC state transition diagram of Fig. 2 task phase provided in an embodiment of the present invention；
The algorithm flow of matrix and vector product operation is realized under Fig. 3 row compression and storage method provided in an embodiment of the present invention
Figure；
The state mapping process figure of the first situation when Fig. 4 new equipment provided in an embodiment of the present invention is added；
The state mapping process figure of second situation when Fig. 5 new equipment provided in an embodiment of the present invention is added；
The state mapping process figure of the third situation when Fig. 6 new equipment provided in an embodiment of the present invention is added；
The fault Tree of Fig. 7 stage P1 provided in an embodiment of the present invention；
The fault Tree of Fig. 8 stage P2 provided in an embodiment of the present invention；
The fault Tree of Fig. 9 stage P3 provided in an embodiment of the present invention.
Specific embodiment
In order to make the objectives, technical solutions, and advantages of the present invention clearer, with reference to embodiments, to the present invention
It is further elaborated.It should be appreciated that the specific embodiments described herein are merely illustrative of the present invention, it is not used to
Limit the present invention.
Application principle of the invention is explained in detail with reference to the accompanying drawing.
As shown in Figure 1, phased mission systems Reliability modeling provided in an embodiment of the present invention and analysis method include with
Lower step:
Step 1: the reliability model of phased mission systems is generated based on continuous time Markov chain, for consecutive hours
Between Markov chain, viIt is system in the rate of transform of state i, finds out the solution of the Markov model respectively by stage, so that it is determined that appoints
Be engaged in phase process in and at the end of, the working condition of each unit；If using the end-state of previous stage as at the beginning of the next stage
State, after the Task Reliability for finding out each stage, the Task Reliability of terminal stage is the Task Reliability of overall process；
Step 2: being pre-processed to Markov single period model using sparse matrix compression and storage method；
Step 3: for the Markov reliability model of generation, using based on Krylov subspace calculation method to model
It is solved, is successively calculated by stage, the reliability of final stage is the reliability of entire task；
Step 4: the phased mission systems fail-safe analysis based on Markov process, use individually models simultaneously by stage
The thought of solution；It stage by stage in independent modeling method, needs to match by state between the stage, is in each by previous stage end system
Shape probability of state obtains the initial state of next stage system, successively solves by stage to realize.
In step 2, sparse matrix compression and storage method the following steps are included:
The first step enables i representing matrix element be expert at, and i value is from 0 to m-1, then Rowptr [i] indicates the of the i-th row
Index value of one nonzero element in array Value, at the same know Rowptr [i+1] -1 indicate the i-th row the last one is non-
Index value of the neutral element in array Value；
Second step enables k=Rowptr [i], then knows that Value [k] is first nonzero element value of the i-th row, and
Colind [k] indicates the column index of the element；
Third step enables j=Colind [k], then knows that j is nonzero element qijColumn index, and qijLine index be i,
qij=Value [k]；According to the operation rule of matrix and multiplication of vectors, y [i]=y [i]+Value [k] * p [j] can be obtained；
4th step is incremented by k value, until k=Rowptr [i+1] -1, it is complete to have traversed operation for the nonzero element in the i-th row at this time
Finish；
5th step then takes i=i+1, carries out traversal operation to next line matrix non-zero element, is incremented by i value, until i=N-
1, result vector y can be obtained.
In step 3, using Krylov subspace method, extensive Markov model is projected to small-scale subspace
In, matrix size is reduced, then derives the approximate formula for calculating state probability vector v (t) using matrix after projection.
Application principle of the invention is further described combined with specific embodiments below.
Phased mission systems Reliability modeling provided in an embodiment of the present invention and analysis method the following steps are included:
Step 1 generates the reliability model of phased mission systems based on continuous time Markov chain.For consecutive hours
Between Markov chain (continuous-time Markov chain, CTMC), viFor system state i the rate of transform
(transition rate).Remember pijThe probability of state j is transferred to from state i for system.Then
Wherein,
If vi=0, then state i is referred to as the absorbing state (absorbing state) of system；
If vi=∞, then state i is referred to as the transient state (instantaneous state) of system；
If 0 < vi< ∞, then state i is referred to as the stable state (stable state) of system.
Define qijThe rate of transform (transition rate of moving from for system from state i to state j
State i to state j):
It can prove that
qij=vipij
Define qii=-vi, following matrix be referred to as CTMC transition rates (transition rate matrix,
TRM), or infinitely small generation is sub (infinitesimal generator).
And have
Qe=0
Wherein, 1 e=1 ... 1TFor unit vector.
If sharing m in the equipment that stage k participates in task for a certain phased mission system, each equipment be divided into it is normal and
Failure two states, system mode are represented by Si=c1c2...cn, wherein ci=1 indicates that i-th of equipment is normal, otherwise indicates
I-th of equipment failure.Then in Markov model, the microstate space S of system is shared | S |=2mA state.
Remember that in the state vector of moment t system be X (t), if task phase initial time is 0.By system reliability model,
It can determine the state transition probability relationship of system.It generally can be written as Kolmogorov forward equation:
Above-mentioned solution of equations are as follows:
P (t)=P (0) eQt
According to the relationship of state probability vector in Markov model and transition probability matrix, to above-mentioned equation according to ordinary differential
Equation group theory can solve
V (t)=v (0) eQt
Above formula is the solution formula of state probability vector v (t) in Markov model.The state probability of system initial time to
It is commonly known to measure v (0).
Example: being equipped with a phased mission system, and mission profile is as shown in table 4 below:
Each phased mission section of 4 phased mission system T of table
For the task T shown in the table 4, the second stage of the TT&C task is taken to illustrate Single-phase mission reliability mould
The establishment process of type.Task phase p2(t1,t2) in system be made of two parallel units, then the microscopic indices of system share 4
A, if the service life of two units and repair process are exponential distribution, the corresponding failure rate of two units and repair rate are respectively
λi, μi, i=1,2, then it can establish the continuous time Markov chain model shown such as Fig. 3.
To sum up, it can establish following Markov model transition rates Q:
According to system reliability principle, since the initial stage, the solution of the Markov model can be found out respectively by stage,
So that it is determined that task phase during and at the end of, the working condition of each unit.If the end-state with previous stage is
The initial state of next stage, after the Task Reliability for finding out each stage, the Task Reliability of terminal stage is the task of overall process can
By degree.
Step 2: being pre-processed to above-mentioned Markov single period model using sparse matrix compression and storage method.By
Transition rates include therefore sparse matrix compression can be used in a large amount of neutral elements, as sparse matrix in Markov model
Storage method is pre-processed.Sparse matrix compression and storage method mainly includes by coordinate storage, by row compression storage, fixed block
It stores and by the methods of row compression piecemeal storage, by comparing, in the reliability model calculating based on Markov process, presses
The compression efficiency highest of row compression storage (compressed row storage, CRS) method.The specific implementation of this method is such as
Under:
CRS stores matrix Q using three arrays: (1) value: storing the value of the nonzero element in Q, number line by line
It is floating type according to type；(2) colind: the column index of each non-zero entry in storage Q, data type is integer；(3) rowptr:
Index of first nonzero element of every row in array value is stored, data type is integer.
Assuming that there is following sparse matrix Q:
Example matrix specific storage organization under CRS compression scheme is as shown in table 5.Wherein, the line index of matrix element
Initial value with column index is all since 0.
The storage organization of sparse matrix Q in 5 CRS scheme of table
| value | 0.0005 | -0.0805 | 0.04 | 0.04 | 0.0005 | -0.0805 | 0.04 | 0.04 |
| colind | 0 | 1 | 3 | 5 | 0 | 2 | 3 | 6 |
| rowptr | 0 | 0 | 4 | 8 | 12 | 16 | 20 | 24 |
| value | 0.0005 | 0.0005 | -0.041 | 0.04 | 0.0005 | -0.0805 | 0.04 | 0.04 |
| colind | 1 | 2 | 3 | 7 | 0 | 4 | 5 | 6 |
| rowptr | ||||||||
| value | 0.0005 | 0.0005 | -0.041 | 0.04 | 0.0005 | 0.0005 | -0.041 | 0.04 |
| colind | 1 | 4 | 5 | 7 | 2 | 4 | 6 | 7 |
| rowptr | 28 | |||||||
| value | 0.0005 | 0.0005 | 0.0005 | -0.0015 | ||||
| colind | 3 | 5 | 6 | 7 | ||||
| rowptr |
If enabling the non-zero entry prime number in nnz representing matrix Q, array it can be seen from the storage organization of CRS scheme
Element number in value and colind is equal to nnz, since the element type in value is floating type, the member in colind
Plain type is integer, and therefore, the total memory space of the two arrays is 8nnz+4nnz.Element number etc. in array rowptr
Add 1, i.e. N+1 in the line number of matrix Q.Again since its data type is integer, memory space needed for array rowptr is big
Small is 4 (N+1).To sum up, the amount of storage needed for can obtaining under CRS scheme are as follows:
M=8nnz+4 (nnz+N+1)
If enabling qijNonzero element in representing matrix, the result vector of y representing matrix Q and vector p product.Row compression storage
Algorithm flow chart such as Fig. 4 of matrix and vector product operation is realized under method.
Step 3: for the Markov reliability model of generation, using based on Krylov subspace calculation method to model
It is solved, is successively calculated by stage, the reliability of final stage is the reliability of entire task.Based on Krylov
The phased mission system fail-safe analysis in space, be exactly using Krylov subspace method, by extensive Markov model project to
In small-scale subspace, matrix size is reduced, then derives and calculates state probability vector v (t) using matrix after projection
Approximate formula.Therefore, by utilizing Krylov subspace method, problem scale can be reduced, computation complexity is reduced.
Calculation of Reliability process based on Krylov subspace is as follows:
For state probability vector v (t), if enabling v (0) is the state probability vector of system initial time, according to above
In description to Markov model, it is known that
V (t)=v (0) eQt
If enabling A=QΤ, ω (t)=v (t)T, π=ω (0) then has
ω (t)=eAt π t [0,T]
It puts aside time t, enables ω=eAπ then can be denoted as p to any m-1 rank multinomial expansion of ωm-1(A) π into
Row approximate calculation, as its m-1 rank Taylor expansion isAll m-1 rank multinomial approximations of ω are all Krylov
Space KmElement in (A, π), wherein
Km(A, π)=Span π, A π ..., Am-1π}
If Bm=[b1,b2,...,bm] it is Krylov subspace KmOne group of base in (A, π), BmBy Arnoldi process structure
It makes, ωoptFor the optimal value in the m-1 rank multinomial approximation to ω.According to principle of least square method, can obtainDue to B obtained by Arnoldi processmFor orthonormal basis, therefore,Thus may be used
Know
Base B is constructed using Arnoldi algorithmmWhen, if enable β=| | π | |2, take(π be initial time state probability to
Amount), and set eiIndicate i-th of column vector of unit matrix I, length is depending on the circumstances, then can obtain
π=β Bme1
In summary formula, simultaneously because ωopt=Bmyopt, can obtain
It is still extremely complex for calculate with above-mentioned formula (5.13).If being approximately AB by formula (5.6)m≈
BmHm, then haveI.e.Following approximate formula can be obtained according to Taylor expansion:
It can then obtain:
Time variable t is reintroduced back in formula, to arbitrary time t, is always hadIt can obtain:
It also include matrix exponetial operation in above-mentioned formulaBut due to matrix HmOrder m be less than original matrix A rank
Number n, it can thus be assumed that converting initial extensive fabric problem to the meter of small-scale matrix by Krylov-subspace projection
It calculates, to reduce computation complexity.
Step 4: the phased mission systems fail-safe analysis based on Markov process, generallys use and individually builds by stage
Mould and the thought solved.Stage by stage in independent modeling method, need to match by state between the stage, by previous stage end system
The initial state of next stage system is obtained in each shape probability of state, is successively solved by stage to realize.But due to different phase
The equipment and its quantity of middle participation task would generally change, and cause the state probability vector in two stages cannot be direct
Match.In existing calculation method, the realization mapped state the stage is all fairly simple, does not establish the rank of a system high efficiency
State mapping mechanism between section.
State matches between implementation phase, establishes state mapping mechanism between the following stage.For ease of description, the system of definition
State and the concept of state intersection.The unit number in n expression system is enabled, then system state space is 2n.With SiExpression system
I-th of state, is indicated by string of binary characters, Si=c1c2...cn, ck(1 k n) is the state of i-th of unit, if i-th
Unit is normal, ckIt is 1, is otherwise 0.
It defines 1 state xor operation: using symbolIt indicates, is equipped with two state SiAnd Sj, string of binary characters represented by state
Length it is identical, xor operation is the sum that corresponding position carries out XOR operation.For example, S0=1001, S1=0011, then
It is equal to define 2 states: being equipped with two state SiAnd SjIf its length is identical, andThen claim state SiWith
SjIt is equal.
According to the equal concept of state, state maps between two successive stages primitive rule are as follows: pass through and reject new be added
Equipment is no longer participate in task device, or increases running equipment, so that two stage status word string length having the same, then
The probability values of state are the corresponding probability value of previous stage end equivalent state in current generation.Due to observing and controlling in different phase
The variation of the communication resource is more complicated, is discussed according to three kinds of situations described above between mapping mechanism the stage, enablesTable
Show the state in stage k-1,State in expression stage k.It particularly may be divided into following three kinds of situations:
Situation one: there is new equipment addition in the current task stage.If having new equipment addition in stage k, due to what is be newly added
Equipment original state be it is normal, i.e., corresponding states value be 1.Therefore for any one state in stage kIf its status word
The state value that new equipment is corresponded in symbol string is 0, thenIt is 0 in the probability of stage k.For the shape that equipment is " 1 " is newly added
StateThe corresponding probability of the stage initial time state is unrelated with new equipment, therefore by the correspondence of new equipment in status word string
Position is rejected, and new state is obtainedThen previous stage stateWithLength having the same can map base according to state
This rule pairWithIt is mapped, obtains the initial state probability vector of stage k.
Citing, if the equipment for executing task in stage k-1 is r1、r2, the equipment in stage k is r1、r2With new equipment r3.Newly
State mapping process when equipment is added is as shown in Figure 5.
Situation two: certain equipment of previous stage are all no longer participate in task execution in the later stage.In this case,
It willIn be not involved in the follow-up work stage equipment correspondence position reject, obtain new stateThen it is reflected according to state
The primitive rule penetrated is to stateWithIt is mapped, obtains the initial state probability vector of stage k.Citing, if stage k-1
In equipment be r1、r2And r3, the equipment in stage k is r1And r2, equipment r3It is not all used in later task phase, this
There may be the merging of state in the case of kind, state mapping process is as shown in Figure 6.
Situation three: certain equipment of previous task phase do not execute task in the current generation, but can task rank afterwards
It is reused in section, these equipment is known as to the running equipment in a certain stage.In this case, though the running equipment of stage k not
Execution task but it is constantly in operating status, it is believed that these equipment carry out normal state transfer, i.e. stage k in stage k
Markov model in include description to running equipment, only running equipment will not have an impact task success or failure.Therefore,
State in stage kIt include the state description to running equipment.Citing, if the equipment in stage k-1 is r1、r2And r3, rank
Equipment in section k is r1, equipment r2And r3It is not involved in task in stage k, but will continue in the later stage to use then its shape
State mapping process is as shown in Figure 7.
The above are the detailed descriptions of state mapping mechanism between the stage, and various possible mapping modes are summarized as three classes, real
State matching in border between two successive stages may include a variety of situations simultaneously, then can in summary method progress state reflect
It penetrates.State mapping process is also related with the Success criteria of entire task.For example, if MISSION SUCCESS CRITERIA is that all stages are successful
It executes, then the absorbing state in the stage at this time will not generate contribution to entire mission reliability.Therefore, when carrying out state mapping,
The state equal with the absorbing state of previous stage in current generation needs its initial value at this stage being set as 0.
For phased mission systems Reliability modeling proposed by the present invention and analysis method, author has been developed accordingly
Computational software system.The system by the reading of case (scenario) file, Calculation of Reliability, as the result is shown with save etc. modules form.
Major function includes: opening and case (scenario) resource file and assignment file for reading XML format, obtains phased mission system system
The equipment dependability data and task configuration information of system complete data preparation；Calculate the reliability of phased mission systems, algorithm
It is based primarily upon continuous time Markov chain and generates reliability model, model is pressed using sparse matrix compression and storage method
Contracting processing, calculates model based on Krylov subspace method, and passes through the state mapping matching realization task between the stage
Reliability successively solves by stage；After the completion of calculating, each session information and reliability results, and total times can be shown
Business reliability results, and calculated result is saved as into Excel file.
Phased mission systems Reliability modeling provided in an embodiment of the present invention and the reliability model of analysis method are main
It is using the Reliability modeling side based on continuous time Markov chain (continuous time Markov chains, CTMC)
Method, derivation algorithm are mainly based upon Krylov subspace method and state mapping mechanism between the stage, realize to single period model
Successively solve by stage, and realize to reliability model by row compression storage (compressed row strorage,
CRS) the compression processing under mode.
The foregoing is merely illustrative of the preferred embodiments of the present invention, is not intended to limit the invention, all in essence of the invention
Made any modifications, equivalent replacements, and improvements etc., should all be included in the protection scope of the present invention within mind and principle.
Claims (6)
1. a kind of phased mission systems Reliability modeling and analysis method, which is characterized in that the phased mission systems can
Multiphase system is generated based on continuous time Markov chain using individually modeling and solving by stage with analysis method by property modeling
In each single phase reliability model, successively each stage model is solved, the reliability of terminal stage be entire task can
By property；Single phase Markov model is pre-processed using sparse matrix compression and storage method；Based on Krylov subspace
Calculation method solves the Markov reliability model of generation；Using state mapping mechanism between the stage, realize to single phase mould
Type successively solves by stage.
2. phased mission systems Reliability modeling as described in claim 1 and analysis method, which is characterized in that described multistage
Section task system Reliability modeling and analysis method the following steps are included:
Step 1 establishes the mission reliability model in system each stage, v based on continuous time Markov chainiIt is system in state
The rate of transform of i, pijThe probability of state j is transferred to from state i for system, then:
Wherein,
If vi=0, then state i is referred to as the absorbing state of system；
If vi=∞, then state i is referred to as the transient state of system；
If 0 < vi< ∞, then state i is referred to as the stable state of system；
qijIt is system from state i to the rate of transform of state j:
qii=-vi, following matrix is referred to as the transition rates of CTMC, or infinitely small generation:
And have
Qe=0；
Wherein, 1 e=1 ... 1TFor unit vector；
The Markov model transition rates Q of foundation:
Step 2: being pre-processed to Markov single period model using sparse matrix compression and storage method, specifically include:
CRS stores matrix Q using three arrays: (1) value: storing the value of the nonzero element in Q, data class line by line
Type is floating type；(2) colind: the column index of each non-zero entry in storage Q, data type is integer；(3) rowptr: storage
Index of first nonzero element of every row in array value, data type is integer；
Step 3: being carried out using based on Krylov subspace calculation method to model for the Markov reliability model of generation
It solves, is successively calculated by stage, the reliability of final stage is the reliability of entire task, utilizes Krylov subspace
Method projects extensive Markov model into small-scale subspace, derives and calculates state probability using matrix after projection
The approximate formula of vector v (t)；
Step 4: the phased mission systems fail-safe analysis based on Markov process, generallys use by stage individually modeling simultaneously
It solves；It stage by stage in independent modeling method, is matched by state between the stage, each shape probability of state is in by previous stage end system
The initial state of next stage system is obtained, is successively solved by stage based on state mapping mechanism realization between the stage.
3. phased mission systems Reliability modeling as claimed in claim 2 and analysis method, which is characterized in that the step
Three reliability calculation methods based on Krylov subspace include:
For state probability vector v (t), enabling v (0) is the state probability vector of system initial time, then according to above right
The description of Markov model, it is known that:
V (t)=v (0) eQt；
If enabling A=QΤ, ω (t)=v (t)T, π=ω (0) then has:
ω (t)=eAtπ t [0,T]；
ω=eAπ is then denoted as p to any m-1 rank multinomial expansion of ωm-1(A) π carries out approximate calculation, such as its m-1 rank Taylor
Expansion isAll m-1 rank multinomial approximations of ω are all Krylov subspace KmElement in (A, π), in which:
Km(A, π)=Span π, A π ..., Am-1π}；
Bm=[b1,b2,...,bm] it is Krylov subspace KmOne group of base in (A, π), BmBy Arnoldi procedure construction, ωopt
For the optimal value in the m-1 rank multinomial approximation to ω；According to principle of least square method, obtain
BmFor orthonormal basis,
Base B is constructed using Arnoldi algorithmmWhen, if enable β=| | π | |2, takeπ is initial time state probability vector,
eiIndicate i-th of column vector of unit matrix I, length is depending on the circumstances to be obtained:
π=β Bme1；
ωopt=Bmyopt, it obtains:
Following approximate formula is obtained according to Taylor expansion:
Then:
Time variable t is reintroduced back in formula, to arbitrary time t, is always had:
4. phased mission systems Reliability modeling as claimed in claim 2 and analysis method, which is characterized in that the step
State mapping mechanism between four stages are as follows:
The unit number in n expression system is enabled, then system state space is 2n；With SiI-th of state of expression system, by binary system
String representation, Si=c1c2...cn, ck(1 k n) is the state of i-th of unit, if i-th of unit is normal, ckIt is 1, otherwise
It is 0；
(1) it defines 1 state xor operation: using symbolIt indicates, there is two state SiAnd Sj, string of binary characters represented by state
Length is identical, and xor operation is the sum that corresponding position carries out XOR operation；
(2) it is equal that 2 states are defined: there are two state SiAnd SjIf its length is identical, andThen claim state SiWith SjPhase
Deng；
The primitive rule that state maps between (3) two successive stages are as follows: by rejecting new addition equipment, the task of being no longer participate in is set
It is standby, or increase running equipment so that two stage status word string length having the same, then in the current generation state it is initial
Probability value is the corresponding probability value of previous stage end equivalent state, is enabledState in expression stage k-1,Expression stage k
In state, specifically include:
1) there is new equipment addition in the current task stage, if there is new equipment addition in stage k, due to the equipment initial shape being newly added
State be it is normal, corresponding states value be 1；For any one state in stage kThe shape of new equipment is corresponded in status word string
State value is 0, thenIt is 0 in the probability of stage k；For the state that equipment is " 1 " is newly addedThe stage initial time shape
The corresponding probability of state is unrelated with new equipment, and the correspondence position of new equipment in status word string is rejected, new state is obtainedThen
Previous stage stateWithLength having the same maps primitive rule pair according to stateWithIt is mapped, is obtained
To the initial state probability vector of stage k；
2)In be not involved in the follow-up work stage equipment correspondence position reject, obtain new stateIt is reflected according to state
The primitive rule penetrated is to stateWithIt is mapped, obtains the initial state probability vector of stage k；
3) certain equipment do not execute task in the current generation, though the running equipment of stage k is not carried out task and is constantly in operation
State includes the description to running equipment in the Markov model of stage k.
5. a kind of phased mission systems Reliability modeling as described in claim 1 and the phased mission systems of analysis method can
By property computing system, which is characterized in that the phased mission systems Calculation of Reliability system includes:
The case resource file and assignment file of XML format are opened and read to file read module, obtains phased mission systems
Equipment dependability data and task configuration information, complete data preparation；
Calculation of Reliability module can based on continuous time Markov chain generation for calculating the reliability of phased mission systems
By property model, compression processing is carried out using sparse matrix compression and storage method to model, based on Krylov subspace method to mould
Type is calculated, and realizes successively solving by stage for mission reliability by the state mapping matching between the stage；
Result display module, for showing each session information and reliability results and total Calculation of Mission Reliability knot
Fruit；
Calculated result is saved as Excel file after the completion of calculating by preserving module.
6. a kind of multistage using phased mission systems reliability calculation method described in 4 any one of Claims 1 to 4
Task system.
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