CN104572455A - Markov-chain-based component-based software reliability evaluation method - Google Patents

Abstract

The invention discloses a comprehensive and accurate Markov-chain-based component-based software reliability evaluation method. The method comprises the following steps: (1) analyzing a software structure to obtain a software state transition probability matrix, the number of function bodies contained in each component and the harm degree of each component; (2) testing software by using a use case number which is not less than a minimum test use case number, and counting the execution times, failure times and failure repair number of each component to obtain the execution probability of each component; (3) calculating the occupancy rate of each component according to an inter-component transition probability obtained in the step (1) and the execution probability of each component obtained in the step (2); (4) calculating the failure rate of each component; (5) calculating the complexity of each component according to the number of the function bodies contained in each component obtained in the step (1); (6) calculating the importance degree of each component according to the harm degree of each component obtained in the step (1); (7) evaluating the reliability of each component; (8) evaluating the reliability of the software.

Description

A kind of based on markovian component-based software reliability estimation method

Technical field

The invention belongs to field of software engineering, relate to software evaluation method, be specially a kind of based on markovian component-based software reliability estimation method.

Background technology

Large-scale real time software system is constantly expanding along with software size and function continuous renewal, and software configuration also becomes increasingly complex, also more and more higher to the requirement of its reliability, therefore needs to set up software reliability assessment model, carries out reliability evaluation to real-time software.Current large-scale real time software system is made up of many software components, and the frequency of utilization of each component and criticality different, therefore need to go to assess software reliability from the angle of software inhouse framework and software component.Typical component software appraisal procedure has three kinds: the method based on section, the method based on state and the method based on path.Based on the method for section mainly through analyzing different software input data sets, calculate the transition probability between probability and each component that different section occurs to assess software reliability; Based on the path that the method in path experiences mainly for test case, add up frequency of utilization and the probability of each component on this path, computational software system reliability; Operational process based on the method hypothesis software systems of state is the state migration procedure between each component, utilizes theory of random processes to assess software systems reliability.Wherein the random process model of main flow has two kinds: Markov Process Model (Markov Process Model) and nonhomogeneous Poisson process model (NonhomogeneousPoission Model).Markov Process Model, was proposed in 1972 by Jelinski and Moranda for representative with J-M model, and its main target is software estimation inefficacy moment, residual error number and fiduciary level.

The reliability estimation method of existing software systems mainly considers the single factor of component occupancy, thinks that the crash rate of the component that occupancy is high can produce material impact to the crash rate of software systems.But the complexity of component and importance degree also very important to the reliability of software systems, element structure is more complicated, and software systems reliability is lower; Although certain class important component utilization rate is not high, its inefficacy will produce devastating impact to software systems.Existingly all cannot carry out objective comprehensive evaluation to it, can not practical requirement.

Summary of the invention

For problems of the prior art, the invention provides one and can consider software occupancy, complexity and importance degree, carry out thoroughly evaluating based on markovian component-based software reliability estimation method.

The present invention is achieved through the following technical solutions:

The present invention is a kind of to be comprised the steps based on markovian component-based software reliability estimation method,

(1) by analysis software structure, obtain application state transition probability matrix, each component comprises function body quantity and each component extent of injury; Wherein, state transition probability matrix P is by the transition probability p between component _ijcomposition, p _ijrefer to the probability of component i to component j, the dimension of matrix is the sum of software component;

(2) use the use-case quantity being no less than minimum test case to test software, number is repaired in be performed number of times, failure number and the inefficacy of adding up each component, obtains each component and performs probability; Under the quantity of minimum test case, in software, all component actual figures are no less than 2 times; Failure member will be repaired in real time in test process, and obtains repairing the probability of success;

(3) each component occupancy is obtained according to transition probability between the component obtained in step (1) with according to each component execution probability calculation obtained in step (2);

(4) each component failure rate is calculated; The crash rate of component refers to the probability of component from normal condition to abnormality, and concrete steps are as follows:

Step 4.1: establish certain component to have normal and abnormal two states, forms component level Markov chain, sets up by g _i,jthe state transition probability matrix G of composition; g _i,jrepresent the probability being transferred to state j by state i, calculated by be performed number of times, failure number and inefficacy reparation number in step (2);

Step 4.2: calculate the state probability vector B of this component at n all after date ⁽ⁿ⁾: B ⁽ⁿ⁾=B ⁽⁰⁾g ⁽ⁿ⁾, wherein, probability vector B ⁽⁰⁾=[01], G ⁽ⁿ⁾the n power of representing matrix G; According to steady markovian character, B ⁽ⁿ⁾fixed value will be converged to;

Step 4.3: the crash rate calculating this component;

The crash rate of current time equals state probability vector B ⁽⁰⁾second element, i.e. λ ⁽⁰⁾=g ₁₂;

The crash rate of n all after date equals the state probability vector B after restraining ⁽ⁿ⁾second element, namely ${\mathrm{\λ}}^{\left(n\right)}=g_{12}^{\left(n\right)}\text{;}$

(5) comprise function body quantity according to each component obtained in step (1) and calculate each component complexity;

(6) each component importance degree is calculated according to each component extent of injury obtained in step (1);

(7) each Member Reliability Analysis is assessed; Reliability assessment value and the predicted value of each component is obtained according to following Member Reliability Analysis model;

R(λ _j,γ _j,π _j,μ _j；t)＝exp(-γ _jλ _jπ _jμ _jt)，

Wherein, λ _jfor the crash rate of component j, γ _jfor the occupancy of component j, π _jfor the complexity of component j, μ _jfor the importance degree of component j, t is running software periodicity.

(8) software reliability is assessed; Obtain reliability assessment value and the predicted value of software according to each Member Reliability Analysis obtained in following software reliability model and step (7), the reliability of software is assessed;

$R({\mathrm{\λ}}_{1,...,n},{\mathrm{\π}}_{1,...,n},{\mathrm{\μ}}_{1,...,n};t)=\underset{j=1}{\overset{n}{\mathrm{\Π}}}\exp (-{\mathrm{\γ}}_j{\mathrm{\λ}}_j{\mathrm{\π}}_j{\mathrm{\μ}}_{j}t),$

Wherein, λ _jfor the crash rate of component j, γ _jfor the occupancy of component j, π _jfor the complexity of component j, μ _jfor the importance degree of component j, t is running software periodicity, and n is the number of components in software.

Preferably, in step (1), function body quantity obtains by Software Detailed Design instructions; The component extent of injury refers to the criticality of component to whole software, is divided into high, medium and low and without harm four grades;

When component failures will cause software crash, the extent of injury brought about great losses is for high;

When component failures lost efficacy causing the major function of software, but can not cause during the extent of injury of software crash is;

When component failures will cause software disabler, but the extent of injury not affecting the major function of software is low;

The extent of injury that can not affect the normal operation of software when component failures is nothing.

Further, in step (6), give the value of successively decreasing at random by four of the component extent of injury grades, high, medium and low and without corresponding to lev4 ~ lev1 respectively, then the importance degree of component j is:

Preferably, in step (3), occupancy refers to the ratio that component is performed in task, the occupancy of component j wherein, b _irepresent the probability performing component i, p _ijrepresent the probability being performed component j by component i.

Preferably, in step (5), the complexity of component j wherein, β _jfor the function body quantity that component j comprises, obtained by step (1).

Preferably, in step (7), the reliability assessment value obtaining current time component j as t=1 is R (λ _j, γ _j, π _j, μ _j; T)=exp (-γ _jλ _jπ _jμ _j); The reliability prediction value obtaining following k all after date component j as t=k is ) in formula: k is not equal to 1, ${\mathrm{\λ}}_j^{\left(k\right)}=g_{12}^{\left(k\right)}.$

Preferably, in step (8), the reliability assessment value obtaining current time software as t=1 is the reliability prediction value obtaining following k all after date software as t=k is $R({\mathrm{\λ}}_j^{\left(k\right)},{\mathrm{\γ}}_j^{\left(k\right)},{\mathrm{\π}}_{1,...,n},{\mathrm{\μ}}_{1,...,n};k)=\underset{j=1}{\overset{n}{\mathrm{\Π}}}\exp (-{\mathrm{\γ}}_j^{\left(k\right)}{\mathrm{\λ}}_j^{\left(k\right)}{\mathrm{\π}}_j{\mathrm{\μ}}_{j}k),$ In formula: k is not equal to 1, ${\mathrm{\γ}}_j^{\left(k\right)}=\underset{i=1}{\overset{n}{\mathrm{\Σ}}}{b}_i{p}_{\mathrm{ij}}^{\left(k\right)},{\mathrm{\λ}}_j^{\left(k\right)}=g_{12}^{\left(k\right)}\text{.}$

Compared with prior art, the present invention has following useful technique effect:

The present invention has taken into full account that software architecture and component feature are on the impact of software reliability, propose comprehensive assessment set of factors.Comprehensive assessment set of factors has not only paid close attention to Member Reliability Analysis that high-frequency calls to the impact of software systems reliability, have also contemplated that the complicacy of component and importance are on the impact of software systems reliability.Establishing with function body is the complexity measure model of scalar granularity, has taken into account the complexity of component and structural; Establish the importance measures model of priority and exponential model combination, avoid the subjectivity interference of artificial assignment.

Further, utilize the model set up can carry out reliability assessment to software component, also can carry out reliability assessment to software systems; By the selected and assignment to the difference in cycle in model, can assess software current reliability, also can predict the reliability in software later stage, usable range is wider, this is to the global reliability improving software systems, and control software design quality is significant.

Accompanying drawing explanation

Fig. 1 is appraisal procedure process flow diagram of the present invention.

Fig. 2 is the component level Markov chain schematic diagram described in example of the present invention.

Embodiment

Below in conjunction with specific embodiment, the present invention is described in further detail, and the explanation of the invention is not limited.

In the middle of assessment method of the present invention being applied to software reliability during certain Large Real, by considering the impact on software reliability of component occupancy, complexity and importance degree, establish complexity model and importance degree model, use for reference the Reliability Evaluation Model that Markov chain thought creates component level and software systems level respectively, complete comprehensive reliability assessment is carried out to software.

As shown in Figure 1, concrete steps are as follows.

Step (1), software configuration during by analyzing this Large Real, obtains application state transition probability matrix, each component comprises function body quantity and each component extent of injury.Wherein, function body refer in component complete specific function can invoked code collection, be the chief component of component, can obtain by Software Detailed Design instructions.

State transition probability matrix P is by the transition probability p between component _i,jcomposition, p _i,jrefer to the probability of component i to component j, the dimension of matrix is 6; The component extent of injury refers to the criticality of component to whole software, is divided into four grades: high, medium and low, nothing harm.Statistics is as shown in table 1-1; And the division of the component extent of injury and corresponding sequence number are as shown in table 1-2.

$P=\left[\begin{array}{cccccc}0.2& 0.15& 0.15& 0.35& 0.1& 0.05\\ 0.15& 0.3& 0.05& 0.1& 0.2& 0.2\\ 0.05& 0.15& 0.3& 0.1& 0.1& 0.3\\ 0.1& 0.3& 0.1& 0.4& 0.05& 0.05\\ 0.05& 0.25& 0.2& 0.05& 0.3& 0.15\\ 0.1& 0.15& 0.25& 0.1& 0.1& 0.3\end{array}\right]$

This software structure analysis table of table 1-1

Component extent of injury sorted table in this software of table 1-2

Step (2), uses the test case of 1932 examples to test software, and that adds up each component is performed number of times, failure number, inefficacy reparation number.Wherein, failure number refers to the number of times that component makes a mistake in testing; By introducing inefficacy repair mechanism, failure member will be repaired in test process in real time, thus its reparation probability of success can be obtained, the number of times that after the reparation number that lost efficacy refers to component failure, success is repaired.Statistics is as shown in table 2.

Table 2 use-case test recording table

Step (3), calculates each component occupancy.Occupancy refers to the ratio that component is performed in task, the occupancy γ of component j _jcan be tried to achieve by following formula: wherein, b _irepresent the probability performing component i, p _ijrepresent the probability being performed component j by component i.

Therefore each component of current time occupancy is in systems in which γ ⁽⁰⁾=(0.11,0.22,0.19,0.15,0.13,0.2), following n each component of all after dates occupancy is in systems in which: γ ⁽ⁿ⁾=γ ⁽⁰⁾p ⁽ⁿ⁾=(0.1061,0.2233,0.1705,0.1706,0.1423,0.1871).

Step (4), calculates each component failure rate.The crash rate of component refers to the probability of component from normal condition to abnormality, and concrete solution procedure is:

Step 4.1: establish component to have normal and abnormal two states, forms component level Markov chain, as shown in Figure 2.Set up by g _i,jthe state transition probability matrix G of composition, g _i,jrepresent the probability being transferred to state j by state i, can by step (2) be performed number of times, failure number, inefficacy repair number and calculate.The state transition probability matrix of 6 components is:

$G_1=\left[\begin{array}{cc}0.8704& 0.1296\\ 0.9286& 0.0714\end{array}\right],G_2=\left[\begin{array}{cc}0.9299& 0.0701\\ 1& 0\end{array}\right],G_3=\left[\begin{array}{cc}0.9515& 0.0485\\ 1& 0\end{array}\right];$

$G_4=\left[\begin{array}{cc}0.9620& 0.0380\\ 1& 0\end{array}\right],G_5=\left[\begin{array}{cc}0.9118& 0.0882\\ 0.9167& 0.0833\end{array}\right],G_6=\left[\begin{array}{cc}0.8811& 0.1189\\ 1& 0\end{array}\right];$

Step 4.2: calculate the state probability vector B of component at n all after date ⁽ⁿ⁾, its method is: B ⁽ⁿ⁾=B ⁽⁰⁾g ⁽ⁿ⁾, wherein, probability vector B ⁽⁰⁾=[01], G ⁽ⁿ⁾the n power of representing matrix G.The state probability vector of 6 component n all after dates is:

$B_1^{\left(n\right)}=\left[\begin{array}{cc}0.8775& 0.1225\end{array}\right],B_2^{\left(n\right)}=\left[\begin{array}{cc}0.9345& 0.0655\end{array}\right],B_3^{\left(n\right)}=\left[\begin{array}{cc}0.9538& 0.0462\end{array}\right];$

$B_4^{\left(n\right)}=\left[\begin{array}{cc}0.9634& 0.0366\end{array}\right],B_5^{\left(n\right)}=\left[\begin{array}{cc}0.9122& 0.0878\end{array}\right],B_6^{\left(n\right)}=\left[\begin{array}{cc}0.8937& 0.1063\end{array}\right];$

Step 4.3: the crash rate calculating component.Crash rate equals the state probability vector B after restraining ⁽ⁿ⁾second element, i.e. λ=g ₁₂.The then crash rate vector of each component n all after date crash rate composition ${\mathrm{\λ}}^{\left(n\right)}=({\mathrm{\λ}}_1^{\left(n\right)},{\mathrm{\λ}}_2^{\left(n\right)},...,{\mathrm{\λ}}_6^{\left(n\right)})=(\mathrm{0.1225,0.0655},\mathrm{0.0462,0.0366},0.0878,0.1063),$ The crash rate vector λ=(λ of each component current time crash rate composition ₁, λ ₂..., λ ₆)=(0.1296,0.0701,0.0485,0.0380,0.0882,0.1189).

Step (5), calculates the complexity of each component.The complexity π of component j _jcomputing method be: wherein, β _jfor the function body quantity that component j comprises, can be obtained by table 1.Therefore the complexity of each component is: π={ π _j; J=1...6}=(0.0857,0.2000,0.1429,0.2214,0.1929,0.1571).

Step (6), calculates the importance degree of each component.Component importance degree is relevant with the component extent of injury, and give the value of successively decreasing at random by four of the component extent of injury grades, be respectively lev4 ~ lev1, the importance degree of assignment and correspondence is as shown in table 3, then the importance degree computing formula of component j is:

Table 3 component importance degree allocation table

Therefore the vector of the importance degree of each component composition is: μ={ μ _j; J=1...n}=(0.6412,0.5134,0.4111,0.5134,0.3679,0.4111).

Step (7), assesses each Member Reliability Analysis.Value in model is tried to achieve in (6) in step (3) all, gets the reliability assessment that t=1 can obtain current time component j: R (λ _j, γ _j, π _j, μ _j; T)=exp (-γ _jλ _jπ _jμ _j), assessment result is as shown in table 4:

Table 4 Member Reliability Analysis evaluation form

Get the reliability prediction that t=k can obtain following k all after date component j: $R({\mathrm{\λ}}_j,{\mathrm{\γ}}_j,{\mathrm{\π}}_j,{\mathrm{\μ}}_j;k)=\exp \left({-\mathrm{\λ}}_j^{\left(k\right)}{\mathrm{\γ}}_j^{\left(k\right)}{\mathrm{\π}}_j{\mathrm{\μ}}_{j}k\right),$ In formula: ${\mathrm{\γ}}_j^{\left(k\right)}=\underset{i=1}{\overset{n}{\mathrm{\Σ}}}{b}_i{p}_{\mathrm{ij}}^{\left(k\right)},{\mathrm{\λ}}_j^{\left(k\right)}=g_{12}^{\left(k\right)}\text{.}$

Step (8), assessment software systems reliability.Value in model is tried to achieve in (6) in step (3) all, gets the reliability assessment value that t=1 can obtain current time software systems: $R({\mathrm{\λ}}_{1,...,n},{\mathrm{\π}}_{1,...,n},{\mathrm{\μ}}_{1,...,n};t)=\underset{j=1}{\overset{n}{\mathrm{\Π}}}\exp (-{\mathrm{\γ}}_j{\mathrm{\λ}}_j{\mathrm{\π}}_j{\mathrm{\μ}}_j)=0.9942;$ Get the reliability prediction that t=k can obtain following k all after date software systems:

$R({\mathrm{\λ}}_j^{\left(k\right)},{\mathrm{\γ}}_j^{\left(k\right)},{\mathrm{\π}}_{1,...,n},{\mathrm{\μ}}_{1,...,n};k)=\underset{j=1}{\overset{n}{\mathrm{\Π}}}\exp (-{\mathrm{\γ}}_j^{\left(k\right)}{\mathrm{\λ}}_j^{\left(k\right)}{\mathrm{\π}}_j{\mathrm{\μ}}_{j}k),$ In formula: ${\mathrm{\γ}}_j^{\left(k\right)}=\underset{i=1}{\overset{n}{\mathrm{\Σ}}}{b}_i{p}_{\mathrm{ij}}^{\left(k\right)},{\mathrm{\λ}}_j^{\left(k\right)}=g_{12}^{\left(k\right)}\text{.}$

Claims (8)

1., based on a markovian component-based software reliability estimation method, it is characterized in that, comprise the steps,

(1) by analysis software structure, obtain application state transition probability matrix, each component comprises function body quantity and each component extent of injury; Wherein, state transition probability matrix P is by the transition probability p between component _ijcomposition, p _ijrefer to the probability of component i to component j, the dimension of matrix is the sum of software component;

(2) use the use-case quantity being no less than minimum test case to test software, number is repaired in be performed number of times, failure number and the inefficacy of adding up each component, obtains each component and performs probability; Under the quantity of minimum test case, in software, all component actual figures are no less than 2 times; Failure member will be repaired in real time in test process, and obtains repairing the probability of success;

(3) each component occupancy is obtained according to transition probability between the component obtained in step (1) with according to each component execution probability calculation obtained in step (2);

(4) each component failure rate is calculated; The crash rate of component refers to the probability of component from normal condition to abnormality, and concrete steps are as follows:

Step 4.1: establish certain component to have normal and abnormal two states, forms component level Markov chain, sets up by g _i,jthe state transition probability matrix G of composition; g _i,jrepresent the probability being transferred to state j by state i, calculated by be performed number of times, failure number and inefficacy reparation number in step (2);

Step 4.2: calculate the state probability vector B of this component at n all after date ⁽ⁿ⁾: B ⁽ⁿ⁾=B ⁽⁰⁾g ⁽ⁿ⁾, wherein, probability vector B ⁽⁰⁾=[01], G ⁽ⁿ⁾the n power of representing matrix G; According to steady markovian character, B ⁽ⁿ⁾fixed value will be converged to;

Step 4.3: the crash rate calculating this component;

The crash rate of current time equals state probability vector B ⁽⁰⁾second element, i.e. λ ⁽⁰⁾=g ₁₂;

The crash rate of n all after date equals the state probability vector B after restraining ⁽ⁿ⁾second element, namely ${\mathrm{\λ}}^{\left(n\right)}=g_{12}^{\left(n\right)};$

(5) comprise function body quantity according to each component obtained in step (1) and calculate each component complexity;

(6) each component importance degree is calculated according to each component extent of injury obtained in step (1);

(7) each Member Reliability Analysis is assessed; Reliability assessment value and the predicted value of each component is obtained according to following Member Reliability Analysis model;

R(λ _j,γ _j,π _j,μ _j；t)＝exp(-γ _jλ _jπ _jμ _jt)，

Wherein, λ _jfor the crash rate of component j, γ _jfor the occupancy of component j, π _jfor the complexity of component j, μ _jfor the importance degree of component j, t is running software periodicity;

(8) software reliability is assessed; Obtain reliability assessment value and the predicted value of software according to each Member Reliability Analysis obtained in following software reliability model and step (7), the reliability of software is assessed;

$R({\mathrm{\λ}}_{1,...,n},{\mathrm{\π}}_1,...,n,{\mathrm{\μ}}_{1,...,n};t)=\underset{j=1}{\overset{n}{\mathrm{\Π}}}\exp (-{\mathrm{\γ}}_j{\mathrm{\λ}}_j{\mathrm{\π}}_j{\mathrm{\μ}}_{j}t),$

Wherein, λ _jfor the crash rate of component j, γ _jfor the occupancy of component j, π _jfor the complexity of component j, μ _jfor the importance degree of component j, t is running software periodicity, and n is the number of components in software.

2. one according to claim 1 is based on markovian component-based software reliability estimation method, it is characterized in that, in step (1), function body quantity obtains by Software Detailed Design instructions.

3. one according to claim 1 is based on markovian component-based software reliability estimation method, it is characterized in that, in step (1), the component extent of injury refers to the criticality of component to whole software, is divided into high, medium and low and without harm four grades;

When component failures will cause software crash, the extent of injury brought about great losses is for high;

When component failures lost efficacy causing the major function of software, but can not cause during the extent of injury of software crash is;

When component failures will cause software disabler, but the extent of injury not affecting the major function of software is low;

The extent of injury that can not affect the normal operation of software when component failures is nothing.

4. one according to claim 3 is based on markovian component-based software reliability estimation method, it is characterized in that, in step (6), the value of successively decreasing is given at random by four of the component extent of injury grades, high, medium and low and without corresponding to lev4 ~ lev1 respectively, then the importance degree of component j is: ${\mathrm{\μ}}_j=\exp (\frac{{\mathrm{lev}}_j}{\underset{i=1}{\overset{4}{\mathrm{\Σ}}}{\mathrm{lev}}_i}-1).$

5. one according to claim 1 is based on markovian component-based software reliability estimation method, it is characterized in that, in step (3), occupancy refers to the ratio that component is performed in task, the occupancy of component j wherein, b _irepresent the probability performing component i, p _ijrepresent the probability being performed component j by component i.

6. one according to claim 1 is based on markovian component-based software reliability estimation method, it is characterized in that, in step (5), and the complexity of component j wherein, β _jfor the function body quantity that component j comprises, obtained by step (1).

7. one according to claim 1 is based on markovian component-based software reliability estimation method, it is characterized in that, in step (7), the reliability assessment value obtaining current time component j as t=1 is R (λ _j, γ _j, π _j, μ _j; T)=exp (-γ _jλ _jπ _jμ _j); The reliability prediction value obtaining following k all after date component j as t=k is $R({\mathrm{\λ}}_j,{\mathrm{\γ}}_j,{\mathrm{\π}}_j,{\mathrm{\μ}}_j;k)=\exp (-{\mathrm{\λ}}_j^{\left(k\right)}{\mathrm{\γ}}_j^{\left(k\right)}{\mathrm{\π}}_j{\mathrm{\μ}}_{j}k),$ In formula: k is not equal to 1, ${\mathrm{\γ}}_j^{\left(k\right)}=\underset{i=1}{\overset{n}{\mathrm{\Σ}}}{b}_i{p}_{\mathrm{ij}}^{\left(k\right)},$ ${\mathrm{\λ}}_j^{\left(k\right)}=g_{12}^{\left(k\right)}.$

8. one according to claim 1 is based on markovian component-based software reliability estimation method, it is characterized in that, in step (8), the reliability assessment value obtaining current time software as t=1 is the reliability prediction value obtaining following k all after date software as t=k is $R({\mathrm{\λ}}_j^{\left(k\right)},{\mathrm{\γ}}_j^{\left(k\right)},{\mathrm{\π}}_{1,...,n},{\mathrm{\μ}}_{1,...,n};k)=\underset{j=1}{\overset{n}{\mathrm{\Π}}}\exp (-{\mathrm{\γ}}_j^{\left(k\right)}{\mathrm{\λ}}_j^{\left(k\right)}{\mathrm{\π}}_j{\mathrm{\μ}}_{j}k),$ In formula: k is not equal to 1, ${\mathrm{\γ}}_j^{\left(k\right)}=\underset{i=1}{\overset{n}{\mathrm{\Σ}}}{b}_i{p}_{\mathrm{ij}}^{\left(k\right)},{\mathrm{\λ}}_j^{\left(k\right)}=g_{12}^{\left(k\right)}.$