CN103744778A - Change point based ISQ-FDEFCE software reliability growth model - Google Patents

Abstract

The invention provides a change point based ISQ-FDEFCE software reliability growth model and belongs to the software reliability engineering field. The change point based ISQ-FDEFCE software reliability growth model aims at solving the problems of the existing change point based NHPP type software reliability growth model. According to the change point based ISQ-FDEFCE software reliability growth model, the formula of an ISQ-FDEFCE-CP model which comprises n change points is as follows and accordingly the software reliability growth model is a powerful tool for performing evaluation, analysis and prediction on the software reliability and the guidance is provided for the improvement of the software reliability.

Description

ISQ-FDEFCE software reliability growth model based on transfer point

Technical field

The present invention relates to the ISQ-FDEFCE software reliability growth model based on transfer point, belong to software reliability engineering field.

Background technology

Software fault adjusted rate is a very important parameter in fault makeover process, has reflected efficiency that fault is corrected and fault correction personnel's ability to work, for software developer and supvr judge whether to need to increase or reduce Resource Supply foundation.Most software reliability growth model all supposes that software fault adjusted rate obeys same distribution.In actual conditions, fault adjusted rate is subject to the impact of the factors such as fault correction personnel's skill, complexity, fault correction environment and the instrument of fault itself.Therefore software fault adjusted rate is neither constant neither be level and smooth, but changes at some point, exists transfer point.For Evaluation and Prediction software reliability exactly, there were in recent years some documents to propose application transfer point technology to come the variation tendency of analysis software fail data, the software reliability growth model of having set up the software failure process based on transfer point has good performance in experimental analysis.

For software fault adjusted rate, be subject to the impact of various factors, fault makeover process exists transfer point, and research is applied to transfer point technology in the modeling of software fault makeover process.

Transfer point (Change Point, be called for short CP) be a kind of analytical approach in statistics, refer to the point that certain or some parameters change, in a sequence or process, certain or some statistical characteristic values in certain τ moment sequence or process change, and this τ moment is exactly transfer point [85].Formalization representation is as follows:

(1) sequence of random variables or process X ₁, X ₂x _τ, X _τ+1, X _τ+2... X _nseparate;

(2) X ₁, X ₂x _τobey a certain probability distribution function G (X), X _τ+1, X _τ+2... X _nobey another probability distribution function F (X), and G (X) ≠ F (X);

(3) G (X) and F (X) can be arbitrariness probability distributing functions.

Claim that parameter τ is transfer point.

Transfer point technology has good application in a lot of fields, the aspects such as such as industry automatic control, navigation analysis, economy, meteorology, signal process, medical science, behaviouristics and computing machine.Tan Changchun utilize transfer point technical discussion the implied volatility of Hong Kong's Stock Market to the in the recent period forewarning function of fluctuation of stock market.There is recently scholar to propose transfer point to be applied in Software Reliability Modeling.The software reliability growth model of many classics all supposes that software fault detection rate is stably.In actual conditions, software fault detection rate depends on many factors.Generalized case, in the software test starting stage, a large amount of faults are detected, and software fault detection rate depends on other the factor such as fault discovery efficiency, fault density, test job amount, testing tool and running environment.At software test mid-term stage, fault detect rate depends on corresponding relation, code spreading factor and CPU schedule every day of implementation rate, software failure and the software fault of cpu instruction, therefore can calculate software fault detection rate.Utilize this fault detect rate can understand the progress of fault detect activity, the validity of Estimation Software test plan, and whether the fault detection method that assessment adopts is effective.In sum, once above-mentioned factor changes, cause the variation of software fault detection rate.

In recent years, the NHPP class software reliability growth model based on transfer point is set up in succession.The existing NHPP class software reliability growth model based on transfer point is set up for software failure detection behavior, seldom consider also likely because some influence factor changes, to produce transfer point in software fault makeover process, and more do not consider fault detect workload and the impact of fault correction workload on fault detect and fault makeover process, could not from the details of software fault detection and fault correction, carry out Reliability modeling better.

Summary of the invention

The present invention seeks to the problem existing in order to solve the existing NHPP class software reliability growth model based on transfer point, a kind of ISQ-FDEFCE software reliability growth model based on transfer point is provided.

ISQ-FDEFCE software reliability growth model based on transfer point of the present invention, the described ISQ-FDEFCE software reliability growth model based on transfer point is abbreviated as ISQ-FDEFCE-CP model, this model has n transfer point (CP) in software fault makeover process, and ISQ-FDEFCE-CP model hypothesis is as follows:

Software fault detection process is followed a NHPP;

Software systems inefficacy is at any time all caused by the remaining fault existing in software;

(t, t+ Δ t] number of faults that detected in the time interval is directly proportional to remaining number of faults in system and fault detect workload;

Separate between software fault;

Software fault makeover process cannot be left in the basket, and the number of faults of correction lags behind the fault sum detecting;

Cause that the fault that software systems lost efficacy finally will be corrected, software fault detection process and fault makeover process are executed in parallel at every turn, and fault makeover process can not have influence on process fault detection and fault correction is perfect;

Describe fault detect and fault correction activity with ISQ model, and model meets NHPP and arrive, obey general distribution service time;

In fault makeover process, fault adjusted rate changes on some time point;

ISQ-FDEFCE-CP model is:

$\begin{array}{c}{m}_c\left(t\right)=\underset{k=0}{\overset{n}{\mathrm{\Σ}}}\left\{{\∫}_{{\mathrm{\τ}}_k}^{{\mathrm{\τ}}_{k+1}}{m}_d^{\′}\left(x_{k+1}\right)\right[1-\exp (-{\mathrm{\ρ}}_{k+1}{W}_c\left({\mathrm{\τ}}_{k+1}\right)+{\mathrm{\ρ}}_{k+1}{W}_c\left(x_{k+1}\right))]{\mathrm{dx}}_{k+1}\\ +[m_d\left({\mathrm{\τ}}_{k+1}\right)-m_c\left({\mathrm{\τ}}_{k+1}\right)][1-\exp (-{\mathrm{\ρ}}_{k+1}{W}_c\left({\mathrm{\τ}}_{k+1}\right)+{\mathrm{\ρ}}_{k+1}{W}_c\left({\mathrm{\τ}}_k\right))]\}\end{array}$

Wherein:

M _c(t) be the mean value function with the fault makeover process of n transfer point;

M ' _d() is m _dthe first order derivative of (), m _d() represent till the mean value function of the process fault detection of t moment ISQ-FDEFCE-CP model,

$m_d\left(t\right)=a[1-\exp (-b{W}_d\left(t\right)+{\mathrm{bW}}_d\left(0\right))]=a[1-\exp (-b{W}_d^{*}\left(t\right))]$

Wherein: a represents that software test starts to close potential fault sum;

B represents the fault detect rate of per unit test job amount;

W _d(t) represent till t moment accumulation test job amount;

W (0) represents the test job amount in 0 moment;

W ^*(t)＝W(t)-W(0)；

ρ _k+1represent the fault adjusted rate of per unit fault correction workload; W _c() represents the fault correction workload of accumulating till the t moment.

Advantage of the present invention: in order to address these problems, the present invention combines and starts with from two aspects, sets up sophisticated software reliability growth model more.The software reliability growth model of the overwhelming majority all supposes that software fault adjusted rate is to obey same distribution.But in actual software fault makeover process, fault adjusted rate is subject to the impact of many factors, for example skill, fault correction environment and instrument of the complexity of fault itself, fault correction personnel etc.In software fault makeover process, can not guarantee the stability of these influence factors, once these influence factors change, fault adjusted rate also can change thereupon.Therefore fault adjusted rate neither constant also non-stationary may change at some point, transfer point problem in Here it is software fault makeover process.Practical experience shows [7]: in fault makeover process, fault correction personnel or software developer's correction skill or ability may increase along with the increase of time.For fear of waste resource or exceed the term of delivery and also can reduce or increase fault correction workload, i.e. fault correction workload changes and causes fault correction delay to change, and may in fault makeover process, produce transfer point.In addition,, in modern soft project, using different from the past, new debugging acid is also very rational phenomenon.These instruments can be guaranteed steadily improving of software fault detection and fault correction efficiency, also may be considered to a transfer point opportunity of therefore introducing new tool.Software correction rate is significantly improved.

Accompanying drawing explanation

Fig. 1 is at x ₁the fault that detects of point (x, t] in the schematic diagram revised completely;

Fig. 2 is at x ₂the fault that detects of point (x2, t] in the schematic diagram that is corrected;

Fig. 3 is Laplce's trend analysis figure;

Fig. 4 is C figure.

Embodiment

Embodiment one: present embodiment is described below in conjunction with Fig. 1 and Fig. 2, ISQ-FDEFCE software reliability growth model based on transfer point described in present embodiment, the described ISQ-FDEFCE software reliability growth model based on transfer point is abbreviated as ISQ-FDEFCE-CP model, this model has 1 transfer point in software fault makeover process, and ISQ-FDEFCE-CP model hypothesis is as follows:

Software fault detection process is followed a NHPP;

Software systems inefficacy is at any time all caused by the remaining fault existing in software;

(t, t+ Δ t] number of faults that detected in the time interval is directly proportional to remaining number of faults in system and fault detect workload;

Separate between software fault;

Software fault makeover process cannot be left in the basket, and the number of faults of correction lags behind the fault sum detecting;

Cause that the fault that software systems lost efficacy finally will be corrected, software fault detection process and fault makeover process are executed in parallel at every turn, and fault makeover process can not have influence on process fault detection and fault correction is perfect;

Describe fault detect and fault correction activity with ISQ model, and model meets NHPP and arrive, obey general distribution service time;

In fault makeover process, fault adjusted rate changes on some time point;

ISQ-FDEFCE-CP model is:

$\begin{array}{c}{m}_c\left(t\right)=m_c(0,{\mathrm{\τ}}_1]+m_c({\mathrm{\τ}}_1,t]\\ ={\∫}_0^{{\mathrm{\τ}}_1}{m}_d^{\′}\left(x_1\right)G_1({\mathrm{\τ}}_1-x_1){\mathrm{dx}}_1\\ +{\∫}_{{\mathrm{\τ}}_1}^t{m}_d^{\′}\left(x_2\right)G_2(t-x_2){\mathrm{dx}}_2+[m_d\left({\mathrm{\τ}}_1\right)-m_c\left({\mathrm{\τ}}_1\right){]G}_2(t-{\mathrm{\τ}}_1)\end{array}$

Wherein:

M _c(t) be the mean value function with the fault makeover process of 1 transfer point;

M _c(0, τ ₁] be at (0, τ ₁] be detected the mean value function of fault at makeover process in the time interval;

M _c(τ ₁, t] and be (τ ₁, t] and in the time interval, be detected the mean value function of fault at makeover process;

M ' _d() is m _dthe first order derivative of (), m _d() represent till the mean value function of the process fault detection of t moment ISQ-FDEFCE-CP model,

$m_d\left(t\right)=a[1-\exp (-b{W}_d\left(t\right)+{\mathrm{bW}}_d\left(0\right))]=a[1-\exp (-b{W}_d^{*}\left(t\right))]$

Wherein: a represents that software test starts to close potential fault sum;

B represents the fault detect rate of per unit test job amount;

W _d(t) represent till t moment accumulation test job amount;

W (0) represents the test job amount in 0 moment;

W ^*(t)＝W(t)-W(0)；

G ₁(τ ₁-x ₁) expression (0, τ ₁] fault correction time cumulative distribution function in the time interval; Some trouble spots are at x ₁moment is detected, and at (x ₁τ ₁] revised completely in the time interval;

And G ₁(τ ₁-x ₁) press formula G ₁(τ ₁-x ₁)=1-exp[-ρ W _c(τ ₁)+ρ W _c(x ₁)] obtain;

Wherein: ρ represents the fault adjusted rate of per unit fault correction workload; W _c(τ ₁) represent to τ ₁the fault correction workload of accumulating till the moment; W _c(x ₁) represent to x ₁the fault correction workload of accumulating till the moment;

G ₂(t-x ₂) expression (τ ₁, t] and in the time interval, the cumulative distribution function of fault correction time;

And G ₂(t-x ₂) press formula G ₂(x ₁-t)=1-exp[-ρ W _c(x ₁)+ρ W _c(t)] obtain;

Wherein: ρ represents the fault adjusted rate of per unit fault correction workload; W _c(t) represent the fault correction workload of accumulating till the t moment.

Software reliability evaluation is one of main contents of the Research on Software Reliability Engineering.The reliability of Evaluation and Prediction software systems exactly, grasp reliability state and the Changing Pattern thereof of software systems, can not only helper applications supvr's prediction of test reach the residual number of faults of test duration, estimation software that the reliability objectives of appointment needs, the maintenance cost of software, the crash rate of software systems etc., more can provide foundation for software issue and software test resource distribution, software can be issued according to plan on time, can meet again user's reliability requirement.

Software reliability growth model is one of important means of software reliability evaluation, be in software reliability evaluation and forecasting research, receive much concern, achievement is maximum, a most active research field.Software reliability growth model is the math equation of describing the relation of relation, software failure and the operation profile of software failure and software fault.By software reliability growth model, can make quantitative assessment and prediction to the reliability characteristic of software, for example predict the reliability growth in performance history, assessment or the fiduciary level of forecasting software in schedule operating time there is mean value, software crash rate, the probability distribution in the software failure time interval and the delivery time of software expects at any time of Failure count etc. in forecasting software in official hour interval.Software reliability growth model is not only and is carried out that software reliability is assessed, the powerful tool of analysis and prediction, and provides guide for improving software reliability.

An important content of software reliability growth model investigation is exactly the mean value function of setting up the software accumulative total failure number in can accurate description test process.So the model obtaining should be process fault detection model m _dand fault makeover process model m (t) _c(t).Namely software test procedure is divided into process fault detection and fault makeover process.

In ISQ-FDEFCE-CP model, we had both considered software fault detection workload and fault correction workload, had considered that again fault adjusted rate changes before and after transfer point.ISQ-FDEFCE-CP model hypothesis is as follows:

(1) software fault detection process is followed a NHPP;

(2) software systems inefficacy is at any time all caused by the remaining fault existing in software;

(3) (t, t+ △ t] number of faults that detected in the time interval is directly proportional to remaining number of faults in system and fault detect workload;

(4) separate between software fault;

(5) software fault makeover process cannot be left in the basket, and the number of faults of correction lags behind the fault sum detecting;

(6) cause that the fault that software systems lost efficacy finally will be corrected, software fault detection process and fault makeover process are executed in parallel at every turn, and fault makeover process can not have influence on process fault detection and fault correction is perfect;

(7) with ISQ model, describe fault detect and fault correction activity, and model meets NHPP arrival, the general distribution of obedience service time;

(8) in fault makeover process, fault adjusted rate changes on some time point.

Suppose counting process Nd (t), t >=0} represents software fault detection process, stochastic variable Nd (t) represents till the software fault number that detected of t moment.Software failure process meets NHPP, therefore till the mean value function of the process fault detection of t moment ISQ-FDEFCE-CP model be:

$m_d\left(t\right)=a[1-\exp (-b{W}_d\left(t\right)+{\mathrm{bW}}_d\left(0\right))]=a[1-\exp (-b{W}_d^{*}\left(t\right))]---(3-8)$

Suppose counting process Nc (t), t >=0} represents software fault makeover process, stochastic variable Nc (t) represents till the number of faults revised completely of t moment.First study the situation that software fault makeover process has 1 CP, the mean value function of fault makeover process is:

m _c(t)＝m _c(0，τ ₁]+m _c(τ ₁，t] (3-9)

At (0, τ ₁] in the time interval, suppose p (0, τ ₁] represent that a certain fault is at x ₁moment is detected and at (x ₁, τ ₁] the interior probability of being revised completely of the time interval.As shown in Figure 1, according to multiplication formula, p (0, τ ₁] can be expressed as:

$\begin{array}{c}p(0,{\mathrm{\τ}}_1]={\∫}_0^{{\mathrm{\τ}}_1}P\{(T_d=x_1)\∩(T_c\≤{\mathrm{\τ}}_1-x_1)\}{\mathrm{dx}}_1\\ ={\∫}_0^{{\mathrm{\τ}}_1}P\{T_d=x_1\}P\{T_c\≤{\mathrm{\τ}}_1-x_1|T_d=x_1\}{\mathrm{dx}}_1\\ ={\∫}_0^{{\mathrm{\τ}}_1}P\{T_d=x_1\}{G}_1({\mathrm{\τ}}_1-x_1){\mathrm{dx}}_1\end{array}---(3-10)$

Td in formula---represent (0, τ ₁] in the time interval, the moment that fault is detected;

Tc---represent (0, τ ₁] in the time interval, the time period that fault is corrected;

G ₁(τ ₁-x ₁)---represent (0, τ ₁] in the time interval, fault correction time cumulative distribution function.

At (0, τ ₁] in the time interval, fault is at x ₁to probability be:

$P\{T_d=x_1\}=\frac{m_d^{\′}\left(x_1\right)}{m_d\left({\mathrm{\τ}}_1\right)}---(3-11)$

Formula (3-11) substitution formula (3-10) is obtained:

$p(0,{\mathrm{\τ}}_1]={\∫}_0^{{\mathrm{\τ}}_1}\frac{m_d^{\′}\left(x_1\right)}{m_d\left({\mathrm{\τ}}_1\right)}{G}_1({\mathrm{\τ}}_1-x_1){\mathrm{dx}}_1---(3-12)$

At (0, τ ₁] fault that is detected in the time interval, the mean value function of its makeover process is expressed as:

$\begin{array}{c}{m}_c(0,{\mathrm{\τ}}_1]=m_d(0,{\mathrm{\τ}}_1]p(0,{\mathrm{\τ}}_1]\\ =m_d\left({\mathrm{\τ}}_1\right){\∫}_0^{{\mathrm{\τ}}_1}\frac{m_d^{\′}\left(x_1\right)}{m_d\left({\mathrm{\τ}}_1\right)}{G}_1({\mathrm{\τ}}_1-x_1){\mathrm{dx}}_1\\ ={\∫}_0^{{\mathrm{\τ}}_1}{m}_d^{\′}\left(x_1\right)G_1({\mathrm{\τ}}_1-x_1){\mathrm{dx}}_1\end{array}---(3-13)$

M _c(τ ₁, t] and comprise two parts: a part is at (0, τ ₁] be detected in the time interval and at (τ ₁, t] and the interior number of faults of being revised completely of the time interval; Another part is at (τ ₁, t] in the time interval, be detected and at (τ ₁, t] and the interior number of faults of being revised completely of the time interval.

At (τ ₁, t] and in the time interval, the number that residue fault is revised is completely:

[m _d(τ ₁)-m _c(τ ₁)]G ₂(t-τ ₁) (3-14)

G in formula ₂(t-τ ₁)---represent (τ ₁, t] and in the time interval, the cumulative distribution function of fault correction time.

At (τ ₁, t] and in the time interval, suppose q (τ ₁, t] and represent that a certain fault is at x ₂moment is detected and at (x ₂, t] and the interior probability of being revised completely of the time interval, as shown in Figure 2.

According to multiplication formula, q (τ ₁, t] can be expressed as:

$\begin{array}{c}q({\mathrm{\τ}}_1,t]={\∫}_{{\mathrm{\τ}}_1}^{t}P\{(T_d=x_2)\∩(T_c\≤t-x_2)\}{\mathrm{dx}}_2\\ ={\∫}_{{\mathrm{\τ}}_1}^{t}P\{T_d=x_2\}P\{T_c\≤t-x_2|T_d=x_2\}{\mathrm{dx}}_2\\ ={\∫}_{{\mathrm{\τ}}_1}^{t}P\{T_d=x_2\}{G}_2(t-x_2){\mathrm{dx}}_2\end{array}---(3-15)$

At (τ ₁, t] and in the time interval, fault is at x ₂moment, detected probability was:

$P\{T_d=x_2\}=\frac{m_d^{\′}\left(x_2\right)}{m_d\left(t\right)-m_d\left({\mathrm{\τ}}_1\right)}---(3-16)$

Formula (3-16) substitution formula (3-15) is obtained:

$q({\mathrm{\τ}}_1,t)={\∫}_{{\mathrm{\τ}}_1}^t\frac{m_d^{\′}}{m_d\left(t\right)-m_d\left({\mathrm{\τ}}_1\right)}{G}_2(t-x_2){\mathrm{dx}}_2---(3-17)$

At (τ ₁, t] and the fault that is detected in the time interval, the mean value function of its makeover process is expressed as:

$\begin{array}{c}{m}_d({\mathrm{\τ}}_1,t]q({\mathrm{\τ}}_1,t]=[m_d\left(t\right)-m_d({\mathrm{\τ}}_1)]{\∫}_{{\mathrm{\τ}}_1}^t\frac{m_d^{\′}\left(x_2\right)}{m_d\left(t\right)-m_d\left({\mathrm{\τ}}_1\right)}{G}_2(t-x_2){\mathrm{dx}}_2\\ ={\∫}_{{\mathrm{\τ}}_1}^t{m}_d^{\′}\left(x_2\right)G_2(t-x_2){\mathrm{dx}}_2\end{array}---(3-18)$

By formula (3-14) and formula (3-18), the mean value function that obtains fault makeover process is:

$m_c({\mathrm{\τ}}_1,t]=[m_d\left({\mathrm{\τ}}_1\right)-m_c\left({\mathrm{\τ}}_1\right)]G_2(t-{\mathrm{\τ}}_1)+{\∫}_{{\mathrm{\τ}}_1}^t{m}_d^{\′}\left(x_2\right)G_2(t-x_2){\mathrm{dx}}_2---(3-19)$

By formula (3-9), formula (3-13) and formula (3-19), the mean value function must with the fault makeover process of 1 transfer point is:

$\begin{array}{c}{m}_c\left(t\right)=m_c(0,{\mathrm{\τ}}_1]+m_c({\mathrm{\τ}}_1,t]\\ ={\∫}_0^{{\mathrm{\τ}}_1}{m}_d^{\′}\left(x_1\right)G_1({\mathrm{\τ}}_1-x_1){\mathrm{dx}}_1\\ +{\∫}_{{\mathrm{\τ}}_1}^t{m}_d^{\′}\left(x_2\right)G_2(t-x_2){\mathrm{dx}}_2+[m_d\left({\mathrm{\τ}}_1\right)-m_c\left({\mathrm{\τ}}_1\right)]G_2(t-{\mathrm{\τ}}_1)\end{array}---(3-20)$

Special τ ₁=0 o'clock, formula (3-20) can be write as:

$m_c\left(t\right)={\∫}_0^t{m}_d^{\′}\left(x_2\right)G_2(t-x_2){\mathrm{dx}}_2$

Embodiment two: the ISQ-FDEFCE software reliability growth model based on transfer point described in present embodiment, the described ISQ-FDEFCE software reliability growth model based on transfer point is abbreviated as ISQ-FDEFCE-CP model, this model has 2 transfer points in software fault makeover process, and ISQ-FDEFCE-CP model hypothesis is as follows:

Software fault detection process is followed a NHPP;

Software systems inefficacy is at any time all caused by the remaining fault existing in software;

(t, t+ Δ t] number of faults that detected in the time interval is directly proportional to remaining number of faults in system and fault detect workload;

Separate between software fault;

Software fault makeover process cannot be left in the basket, and the number of faults of correction lags behind the fault sum detecting;

Cause that the fault that software systems lost efficacy finally will be corrected, software fault detection process and fault makeover process are executed in parallel at every turn, and fault makeover process can not have influence on process fault detection and fault correction is perfect;

Describe fault detect and fault correction activity with ISQ model, and model meets NHPP and arrive, obey general distribution service time;

In fault makeover process, fault adjusted rate changes on some time point;

ISQ-FDEFCE-CP model is:

$\begin{array}{c}{m}_c\left(t\right)=m_c(0,{\mathrm{\τ}}_1]+m_c({\mathrm{\τ}}_1,{\mathrm{\τ}}_2]+m_c({\mathrm{\τ}}_2,t]\\ ={\∫}_0^{{\mathrm{\τ}}_1}{m}_d^{\′}\left(x_1\right)G_1({\mathrm{\τ}}_1-x_1){\mathrm{dx}}_1\\ +{\∫}_{{\mathrm{\τ}}_1}^{{\mathrm{\τ}}_2}{m}_d^{\′}\left(x_2\right)G_2({\mathrm{\τ}}_2-x_2){\mathrm{dx}}_2+[m_d\left({\mathrm{\τ}}_1\right)-m_c\left({\mathrm{\τ}}_1\right)]G_2({\mathrm{\τ}}_2-{\mathrm{\τ}}_1)\\ +{\∫}_{{\mathrm{\τ}}_2}^t{m}_d^{\′}\left(x_3\right)G_3(t-x_3){\mathrm{dx}}_3+[m_d\left({\mathrm{\τ}}_2\right)-m_c\left({\mathrm{\τ}}_2\right)]G_3(t-{\mathrm{\τ}}_2)\end{array}$

Wherein:

M _c(t) be the mean value function with the fault makeover process of 2 transfer points;

M _c(0, τ ₁] be at (0, τ ₁] be detected the mean value function of the makeover process of fault in the time interval;

M _c(τ ₁, τ ₂] be (τ ₁, τ ₂] be detected the mean value function of the makeover process of fault in the time interval;

M _c(τ ₂, t] and be (τ ₂, t] and in the time interval, be detected the mean value function of the makeover process of fault;

M ' _d() is m _dthe first order derivative of (), m _d() represent till the mean value function of the process fault detection of t moment ISQ-FDEFCE-CP model,

$m_d\left(t\right)=a[1-\exp (-b{W}_d\left(t\right)+{\mathrm{bW}}_d\left(0\right))]=a[1-\exp (-b{W}_d^{*}\left(t\right))]$

Wherein: a represents that software test starts to close potential fault sum;

B represents the fault detect rate of per unit test job amount;

W _d(t) represent till t moment accumulation test job amount;

W (0) represents the test job amount in 0 moment;

W ^*(t)＝W(t)-W(0)；

M _c(τ ₁) be τ ₁the number of faults of being revised completely in moment, m _c(τ ₂) be τ ₂the number of faults of being revised completely in moment;

0≤τ ₁≤τ ₂≤t，m _d(0)＝m _c(0)

G ₁(τ ₁-x ₁) expression (0, τ ₁] fault correction time cumulative distribution function in the time interval;

And G ₁(τ ₁-x ₁) press formula G ₁(τ ₁-x ₁)=1-exp[-ρ W _c(τ ₁)+ρ W _c(x ₁)] obtain;

Wherein: ρ represents the fault adjusted rate of per unit fault correction workload; W _c(τ ₁) represent to τ ₁the fault correction workload of accumulating till the moment; W _c(x ₁) represent to x ₁the fault correction workload of accumulating till the moment;

G ₂(τ ₂-x ₂) expression (τ ₁, τ ₂)] in the time interval, the cumulative distribution function of fault correction time;

And G ₂(τ ₂-x ₂) press formula G ₂(τ ₂-x ₂)=1-exp[-ρ W _c(τ ₂)+ρ W _c(x ₂)] obtain;

G ₃(t-x ₃) expression (τ ₂, t)] and in the time interval, the cumulative distribution function of fault correction time;

And press formula G ₃(t-x ₃)=1-exp[-ρ W _c(t)+ρ W _c(x ₃)] obtain;

Wherein: ρ represents the fault adjusted rate of per unit fault correction workload; W _c() represents the fault correction workload of accumulating till the t moment.

When fault makeover process has 2 transfer points, the mean value function of fault makeover process is:

m _c(t)＝m _c(0，τ ₁]+m _c(τ ₁，τ ₂]+m _c(τ ₂，t] (3-22)

At (0, τ ₁] in the time interval, fault makeover process all

$m_c(0,{\mathrm{\τ}}_1]={\∫}_0^{{\mathrm{\τ}}_1}{m}_d^{\′}\left(x_1\right)G_1({\mathrm{\τ}}_1-x_1){\mathrm{dx}}_1---(3-23)$

G in formula ₁(τ ₁-x ₁)---represent (0, τ ₁] in the time interval, the cumulative distribution function of fault correction time.

At (τ ₁, τ ₂] in the time interval, the mean value function of fault makeover process is:

$m_c({\mathrm{\τ}}_1,{\mathrm{\τ}}_2]={\∫}_{{\mathrm{\τ}}_1}^{{\mathrm{\τ}}_2}{m}_d^{\′}\left(x_2\right)G_2({\mathrm{\τ}}_2-x_2){\mathrm{dx}}_2+[m_d\left({\mathrm{\τ}}_1\right)-m_c\left({\mathrm{\τ}}_1\right)]G_2({\mathrm{\τ}}_2-{\mathrm{\τ}}_1)---(3-24)$

G in formula ₂(τ ₂-x ₂)---represent (τ ₁, τ ₂] in the time interval, the cumulative distribution function of fault correction time.

At (τ ₂, t] and in the time interval, the mean value function of fault makeover process is:

$m_c({\mathrm{\τ}}_2,t]={\∫}_{{\mathrm{\τ}}_2}^t{m}_d^{\′}\left(x_3\right)G_3(t-x_3){\mathrm{dx}}_3+[m_d\left({\mathrm{\τ}}_2\right)-m_c\left({\mathrm{\τ}}_2\right)]G_3(t-{\mathrm{\τ}}_2)---(3-25)$

G in formula ₃(t-x ₃)---represent (τ ₂, t] and in the time interval, the cumulative distribution function of fault correction time.

By formula (3-22), formula (3-23), formula (3-24) and formula (3-25), the mean value function can with the fault makeover process of 2 transfer points is:

$\begin{array}{c}{m}_c\left(t\right)=m_c(0,{\mathrm{\τ}}_1]+m_c({\mathrm{\τ}}_1,{\mathrm{\τ}}_2]+m_c({\mathrm{\τ}}_2,t]\\ ={\∫}_0^{{\mathrm{\τ}}_1}{m}_d^{\′}\left(x_1\right)G_1({\mathrm{\τ}}_1-x_1){\mathrm{dx}}_1\\ +{\∫}_{{\mathrm{\τ}}_1}^{{\mathrm{\τ}}_2}{m}_d^{\′}\left(x_2\right)G_2({\mathrm{\τ}}_2-x_2){\mathrm{dx}}_2+[m_d\left({\mathrm{\τ}}_1\right)-m_c\left({\mathrm{\τ}}_1\right)]G_2({\mathrm{\τ}}_2-{\mathrm{\τ}}_1)\\ +{\∫}_{{\mathrm{\τ}}_2}^t{m}_d^{\′}\left(x_3\right)G_3(t-x_3){\mathrm{dx}}_3+[m_d\left({\mathrm{\τ}}_2\right)-m_c\left({\mathrm{\τ}}_2\right)]G_3(t-{\mathrm{\τ}}_2)\end{array}---(3-26)$

Wherein 0≤τ ₁≤ τ ₂≤ t, m _d(0)=m _c(0)=0.

Embodiment three: present embodiment is described below in conjunction with Fig. 3 and Fig. 4, ISQ-FDEFCE software reliability growth model based on transfer point described in present embodiment, the described ISQ-FDEFCE software reliability growth model based on transfer point is abbreviated as ISQ-FDEFCE-CP model, this model has n transfer point in software fault makeover process, and ISQ-FDEFCE-CP model hypothesis is as follows:

Software fault detection process is followed a NHPP;

Software systems inefficacy is at any time all caused by the remaining fault existing in software;

(t, t+ Δ t] number of faults that detected in the time interval is directly proportional to remaining number of faults in system and fault detect workload;

Separate between software fault;

Software fault makeover process cannot be left in the basket, and the number of faults of correction lags behind the fault sum detecting;

Cause that the fault that software systems lost efficacy finally will be corrected, software fault detection process and fault makeover process are executed in parallel at every turn, and fault makeover process can not have influence on process fault detection and fault correction is perfect;

Describe fault detect and fault correction activity with ISQ model, and model meets NHPP and arrive, obey general distribution service time;

In fault makeover process, fault adjusted rate changes on some time point;

ISQ-FDEFCE-CP model is:

$\begin{array}{c}{m}_c\left(t\right)=\underset{k=0}{\overset{n}{\mathrm{\Σ}}}\left\{{\∫}_{{\mathrm{\τ}}_k}^{{\mathrm{\τ}}_{k+1}}{m}_d^{\′}\left(x_{k+1}\right)\right[1-\exp (-{\mathrm{\ρ}}_{k+1}{W}_c\left({\mathrm{\τ}}_{k+1}\right)+{\mathrm{\ρ}}_{k+1}{W}_c\left(x_{k+1}\right))]{\mathrm{dx}}_{k+1}\\ +[m_d\left({\mathrm{\τ}}_{k+1}\right)-m_c\left({\mathrm{\τ}}_{k+1}\right)][1-\exp (-{\mathrm{\ρ}}_{k+1}{W}_c\left({\mathrm{\τ}}_{k+1}\right)+{\mathrm{\ρ}}_{k+1}{W}_c\left({\mathrm{\τ}}_k\right))]\}\end{array}$

Wherein:

M _c(t) be the mean value function with the fault makeover process of n transfer point;

M ' _d() is m _dthe first order derivative of (), m _d() represent till the mean value function of the process fault detection of t moment ISQ-FDEFCE-CP model,

$m_d\left(t\right)=a[1-\exp (-b{W}_d\left(t\right)+{\mathrm{bW}}_d\left(0\right))]=a[1-\exp (-b{W}_d^{*}\left(t\right))]$

Wherein: a represents that software test starts to close potential fault sum;

B represents the fault detect rate of per unit test job amount;

W _d(t) represent till t moment accumulation test job amount;

W (0) represents the test job amount in 0 moment;

W ^*(t)＝W(t)-W(0)；

ρ _k+1represent the fault adjusted rate of per unit fault correction workload; W _c() represents the fault correction workload of accumulating till the t moment.

According to embodiment one and two, the rest may be inferred, and when software fault makeover process has n transfer point, the mean value function of fault makeover process is:

$\begin{array}{c}{m}_c\left(t\right)=\underset{k=0}{\overset{n}{\mathrm{\Σ}}}{m}_c({\mathrm{\τ}}_k,{\mathrm{\τ}}_{k+1}]\\ =\underset{k=0}{\overset{n}{\mathrm{\Σ}}}[{\∫}_{{\mathrm{\τ}}_k}^{{\mathrm{\τ}}_{k+1}}{m}_d^{\′}\left(x_{k+1}\right)G_{k+1}({\mathrm{\τ}}_{k+1}-x_{k+1}){\mathrm{dx}}_{k+1}+[m_d\left({\mathrm{\τ}}_k\right)-m_c\left({\mathrm{\τ}}_k\right)\left]G_{k+1}({\mathrm{\τ}}_{k+1}-{\mathrm{\τ}}_k)\right]\end{array}---(3-27)$

G in formula _k+1(τ _k+1-x _k+1)---represent (τ _k, τ _k+1] in the time interval, the cumulative distribution function of fault correction time.Wherein 0≤τ _k≤ τ _k+1≤ t, τ ₀=0, τ _k+1=t, m _d(0)=m _c(0)=0.

Fault detect workload and fault correction workload temporal evolution situation have significant impact to software reliability growth curve, and fault correction workload is determining the result [1] of fault correction.For Evaluation and Prediction software fault makeover process exactly, consider fault correction workload to join in software fault makeover process, identical with 2.4.1 joint, establish the fault correction time and obey function and be: G _k+1(x _k+1)=1-exp[-ρ _k+1w _c(τ _k+1)+ρ _k+1w _c(τ _k+1-x _k+1)], wherein 0≤k≤n, 0≤τ _k≤ τ _k+1≤ t, τ ₀=0, τ _k+1=t, is distinguished substitution formula (3-20), formula (3-26) and formula (3-27), and the mean value function can with the fault makeover process of the ISQ-FDEFCE-CP model of 1 transfer point is:

$\begin{array}{c}{m}_c\left(t\right)={\∫}_0^{{\mathrm{\τ}}_1}{m}_d^{\′}\left(x_1\right)[1-\exp (-{\mathrm{\ρ}}_1{W}_c\left({\mathrm{\τ}}_1\right)+{\mathrm{\ρ}}_1{W}_c\left(x_1\right))]{\mathrm{dx}}_1\\ +{\∫}_{{\mathrm{\τ}}_1}^t{m}_d^{\′}\left(x_2\right)[1-\exp (-{\mathrm{\ρ}}_2{W}_c\left(t\right)+{\mathrm{\ρ}}_2{W}_c\left(x_2\right))]{\mathrm{dx}}_2\\ +[m_d\left({\mathrm{\τ}}_1\right)-m_c\left({\mathrm{\τ}}_1\right)][1-\exp (-{\mathrm{\ρ}}_2{W}_c\left(t\right)+{\mathrm{\ρ}}_2{W}_c\left({\mathrm{\τ}}_1\right))]\end{array}---(3-28)$

The mean value function with the fault makeover process of the ISQ-FDEFCE-CP model of 2 transfer points is:

$\begin{array}{c}{m}_c\left(t\right)={\∫}_0^{{\mathrm{\τ}}_1}{m}_d^{\′}\left(x_1\right)[1-\exp (-{\mathrm{\ρ}}_1{W}_c\left({\mathrm{\τ}}_1\right)+{\mathrm{\ρ}}_1{W}_c\left(x_1\right))]{\mathrm{dx}}_1\\ +{\∫}_{{\mathrm{\τ}}_1}^{{\mathrm{\τ}}_2}{m}_d^{\′}\left(x_2\right)[1-\exp (-{\mathrm{\ρ}}_2{W}_c\left({\mathrm{\τ}}_2\right)+{\mathrm{\ρ}}_2{W}_c\left(x_2\right))]{\mathrm{dx}}_2\\ +[m_d\left({\mathrm{\τ}}_1\right)-m_c\left({\mathrm{\τ}}_1\right)][1-\exp (-{\mathrm{\ρ}}_2{W}_c\left({\mathrm{\τ}}_2\right)+{\mathrm{\ρ}}_2{W}_c\left({\mathrm{\τ}}_1\right))]\\ +{\∫}_{{\mathrm{\τ}}_2}^t{m}_d^{\′}\left(x_3\right)[1-\exp (-{\mathrm{\ρ}}_3{W}_c\left(t\right)+{\mathrm{\ρ}}_3{W}_c\left(x_3\right))]{\mathrm{dx}}_3\\ +[m_d\left({\mathrm{\τ}}_2\right){-m_c\left({\mathrm{\τ}}_2\right)\left]\right[1-\exp (-{\mathrm{\ρ}}_3{W}_c\left(t\right)+{\mathrm{\ρ}}_3{W}_c\left({\mathrm{\τ}}_2\right))]}_{}\end{array}---(3-29)$

The mean value function with the fault makeover process of the ISQ-FDEFCE-CP model of n transfer point is:

$\begin{array}{c}{m}_c\left(t\right)=\underset{k=0}{\overset{n}{\mathrm{\Σ}}}\left\{{\∫}_{{\mathrm{\τ}}_k}^{{\mathrm{\τ}}_{k+1}}{m}_d^{\′}\left(x_{k+1}\right)\right[1-\exp (-{\mathrm{\ρ}}_{k+1}{W}_c\left({\mathrm{\τ}}_{k+1}\right)+{\mathrm{\ρ}}_{k+1}{W}_c\left(x_{k+1}\right))]{\mathrm{dx}}_{k+1}\\ +[m_d\left({\mathrm{\τ}}_{k+1}\right)-m_c\left({\mathrm{\τ}}_{k+1}\right)][1-\exp (-{\mathrm{\ρ}}_{k+1}{W}_c\left({\mathrm{\τ}}_{k+1}\right)+{\mathrm{\ρ}}_{k+1}{W}_c\left({\mathrm{\τ}}_k\right))]\}\end{array}---(3-30)$

Present embodiment model is carried out to Laplce's test:

In order to detect the fitting effect of ISQ-FDEFCE-CP model, we have selected one group of actual software failure dates set of publishing, and the fitting effect of this model is analyzed.Actual software failure dates set comes from the Rome air force development centre system T1 of (Rome Air Development Center is called for short RADC).System T1 is a real-time command control application system, comprises 21700 instructions, 9 programmers, and through 21 weeks, cost 25.3CPU hour, detected 136 faults, revises 136 faults.Concrete software failure data is as shown in table 1.

The system T1 of table 1 Rome air force development centre project

In software test procedure, it is to increase or reduce that often employing trend of researchist tests to analyze reliability.The trend method of testing kind of publishing is numerous, probably can be divided into: graphics test method and the large class of analysis test method two.Graphics test method is the common method that judges roughly data trend in Practical Project.It directly draws trend curve according to the actual data that detect, detected data are the time interval of software failure or the software failure number of time per unit normally.The advantage of graphics test method is intuitive, from figure, can directly observe the trend of reliability, and shortcoming is non-formalization.Graphics test method relatively, analysis test method is stricter test, they are that raw data draws the trend factor and relevant information after treatment from the viewpoint of statistics.

Laplce's method of testing is a kind of most important and the most frequently used method by analytical test method check fail data trend, and the fail data being detected can be that time between failures can be also failure intensity (failure number in the unit interval) or accumulative total failure number.The principle of work of Laplce's method of testing is to realize trend analysis by calculating Laplce factor u (j).Suppose the time interval (0, t] be divided into j isometric unit interval, and with n (i), represent the fail data observing in i unit interval, according to the method for existing document, the expression of Laplce's factor is provided by following formula:

$u\left(j\right)=\frac{\underset{i=1}{\overset{j}{\mathrm{\Σ}}}(i-1)n\left(i\right)-\frac{(j-1)}{2}\underset{i=1}{\overset{j}{\mathrm{\Σ}}}n\left(i\right)}{\sqrt{\frac{(j^2-1)}{12}\underset{i=1}{\overset{j}{\mathrm{\Σ}}}n\left(i\right)}}---(3-31)$

When the fail data observing is accumulative total during failure number, suppose with N (j) expression till the accumulative total failure number that moment j occurs,

formula (3-31) can be write as:

$u\left(j\right)=\frac{\underset{i=1}{\overset{j}{\mathrm{\Σ}}}N\left(i\right)-\frac{(j+1)}{2}N\left(j\right)}{\sqrt{\frac{(j^2-1)}{12}N\left(j\right)}}---(3-32)$

Below Laplce's factor u (j) is discussed:

(1) Laplce's factor u (j) is if negative value represents that failure intensity reduces, and software reliability increases;

(2) Laplce's factor u (j) if on the occasion of, represent failure intensity increase, software reliability reduce;

(3) Laplce's factor u (j) if in-2 and+change between 2, represent that software reliability is stable.

Software reliability is an important property of software quality, implements software reliability engineering and must, take the quality management of software and Quality Assurance as support, become an ingredient of software quality management and Quality Assurance.Control chart is a kind of important means and the instrument of quality management, and it is that procedure quality characteristic value is measured, records, assessed and the whether a kind of statistical method design drawing in state of a control of supervision process.Application controls figure can predict, analyze, monitor, judge and improve operation process status, realizes the process quality management putting prevention first.Control chart is divided into two classes: the control chart of the control chart of monitoring use and analysis use, most research at present all concentrates on the control chart of monitoring use.

C figure is one of the most frequently used instrument of monitoring of software process, is the one of the frequent control chart using in statistical process control technology.It is applicable to the number of faults (or per unit number of faults) of the enumeration feature of following Poisson distribution.Use C figure effectively product process and quality are analyzed, controlled and improve, guarantee that product process is under statistics is controlled, for process control provides, quantize basis.C figure principle of work is that line, the very big positive and negative deviation territory that may occur take process are upper control limit (Upper Control Limit is called for short UCL) and lower control limit (Lower Control Limit, abbreviation LCL) centered by the expectation value of process status.After controlling limit and being estimated, as fruit dot drops on, control among limit, illustrate it is a little controlled, otherwise, illustrate out of control.Calculate C figure trigger formula as follows:

${\mathrm{UCL}}_C=\stackrel{\‾}{C}+3\sqrt{\stackrel{\‾}{C}}---(3-34)$

${\mathrm{LCL}}_C=\stackrel{\‾}{C}-3\sqrt{\stackrel{\‾}{C}}---(3-35)$

Utilize Laplce's method of testing and C figure to determine number and the position of transfer point in software fault makeover process herein.On system T1 software failure data collection, calculate respectively Laplce's factor and C graph parameter, result of calculation is as shown in Figure 3 and Figure 4.From Fig. 3, observe, before the 11st week, value in-2 and+2 between, this shows that software reliability is stable.From Fig. 4, observe, before the 11st week, mean value is all under center limit, and after 11 weeks, curve fast rise also exceedes upper control limit.Judging 11 weeks is a transfer point.On the other hand, from Fig. 3 and Fig. 4, also can be observed, after the 18th week, curve fast-descending.Judging 18 weeks is also a transfer point.Researchist once advised the segmentation transfer point number [86,87,90] of avoiding undue.Therefore select a transfer point herein, 18 weeks.

In order to evaluate the fitting effect of ISQ-FDEFCE-CP model, on system T1 software failure data collection, the SRGM-TE model of the ISQFDEFCE-CP model of the ISQ-FDEFCE-CP model to application GMW function, application Logistic function, application GMW function and the SRGM-TE model of application Logistic function compare respectively.Selecting 18 weeks is a transfer point, and the mean value function of the software reliability growth model of contrast is shown in Table 2.

The mean value function of the software reliability growth model of table 2 matching comparison

Application least square method solves the parameter of all software reliability growth models in table 2, and estimates of parameters is in Table 3.According to the parameter value in table 3, calculate the standard of comparison value of all models on system T1 data set: SSEc, the RMSEc of process fault detection SSEd, RMS Ed and fault makeover process, the definition of these two standards of comparison is referring to formula (2-30) and formula (2-31), and fitting result is in Table 3.

The software reliability growth model parameter estimation value of table 3 based on system T1 data set

The software reliability growth models fitting effect comparison of table 4 based on system T1 data set

As can be seen from Table 4, the SSEd value of the ISQ-FDEFCE-CP model of the ISQ-FDEFCE-CP model of application GMW function and application Logistic function, SSEc value, RMSEd value and RMSEc value are less than the SRGM-TE model of application GMW function and the SRGM-TE model of application Logistic function, the particularly SSEc value of fault makeover process and RMSEc value, the ISQ-FDEFCE-CP model of the ISQ-FDEFCE-CP model of application GMW function and application Logistic function can be applied to system T1 software failure data collection preferably.As can also be seen from Table 4, the SSEd value of the ISQ-FDEFCE-CP model of application Logistic function, SSEc value, RMSEd value and RMSEc value are also less than the ISQFDEFCE-CP model of application GM function, and the fitting effect of this ISQ-FDEFCE-CP model that application Logistic function has been described on system T1 data set is better.

Embodiment four: present embodiment is considered the ISQ-FDEFCE-CP model of imperfect misarrangement.

Utilize the result of study of embodiment three, from the hypothesis of the perfect misarrangement of fault of revising ISQ-FDEFCE-CP model, start with, set up the ISQ-FDEFCE-CP model of considering the imperfect misarrangement of fault, the ISQ-FDEFCE-CP model of revising.

The ISQ-FDEFCE-CP model of revising:

For assessment and forecasting software reliability more accurately, when Software Reliability Modeling, should consider to introduce new fault because of imperfect misarrangement.Therefore the perfect misarrangement assumed condition of the ISQ-FDEFCE-CP model to embodiment three is revised.The basic assumption of the ISQ-FDEFCE-CP model of revising: describe fault detect and fault recovery actions and process fault detection with ISQ model and meet NHPP, at (t, t+ Δ t] expectation value of the software fault number that detected in the time interval is directly proportional to expectation value and the fault detect workload of remaining fault in software systems, may introduce new fault and fault introducing rate and be constant beta (its function changing with the test duration using as from now on the object of research) when fault is excluded.All the other assumed conditions are identical with embodiment three.

According to assumed condition, set up differential equation group below:

$\left\{\begin{array}{c}\frac{{\mathrm{dm}}_d\left(t\right)}{\mathrm{dt}}={\mathrm{bw}}_d\left(t\right)[a\left(t\right)-m_d\left(t\right)]\\ a\left(t\right)=a+{\mathrm{km}}_d\left(t\right)\end{array}\right.---(3-36)$

By starting condition md (0)=0, the mean value function of the process fault detection of the ISQ-FDEFCE-CP model that the solution differential equation (3-36) can be revised is:

$m_d\left(t\right)=\frac{a}{1-k}[1-\exp (-b(1-k)W_d^{*}\left(t\right))]---(3-37)$

The fault sum function relevant to time t is:

$a\left(t\right)=\frac{a}{1-k}[1-\mathrm{kexp}(-b(1-k)W_d^{*}\left(t\right))]---(3-38)$

The mean value function that formula (3-37) substitution formula (3-28) is drawn to the fault makeover process of the ISQ-FDEFCE-CP model of the correction with 1 transfer point is:

$\begin{array}{c}{m}_c\left(t\right)={\∫}_0^{{\mathrm{\τ}}_1}{\mathrm{abw}}_d\left(x_1\right)\exp [-b(1-k)W_d^{*}\left(x_1\right)][1-\exp (-{\mathrm{\ρ}}_1{W}_c\left(x_1\right))]{\mathrm{dx}}_1\\ +{\∫}_{{\mathrm{\τ}}_1}^t{\mathrm{abw}}_d\left(x_2\right)\exp [-b(1-k)W_d^{*}\left(x_2\right)][1-\exp (-{\mathrm{\ρ}}_2{W}_c\left(t\right)+{\mathrm{\ρ}}_2{W}_c\left(x_2\right))]{\mathrm{dx}}_2\\ +[m_d\left({\mathrm{\τ}}_1\right)-m_c\left({\mathrm{\τ}}_1\right)][1-\exp {\left(}_{-\mathrm{\ρ}2W_c\left(t\right)+{\mathrm{\ρ}}_2{W}_c\left({\mathrm{\τ}}_1\right)}]\end{array}---(3-39)$

The ISQ-FDEFCE-CP model of this correction had both been considered fault detect workload and fault correction workload, considered again that fault adjusted rate changes before and after transfer point and fault makeover process in imperfect misarrangement, software test procedure described in detail preferably.

Experimental analysis:

For the fitting effect of the ISQ-FDEFCE-CP model of examination and correction, on system T1 software failure data collection (system T1 software failure data is referring to table 5), the ISQ-FDEFCE-CP model of the ISQ-FDEFCE-CP model to application GMW function and application Logistic function compares respectively.Table 6 has been listed the mean value function of the software reliability growth model comparing.Use SSEd, SSEc, RMSEd and RMSEc are as the standard of comparison of model, and the definition of these two standards of comparison is referring to formula (2-30) and formula (2-31), and fitting result is as shown in table 6.

Other test job flow function of table 5

The mean value function of the software reliability growth model of table 6 matching comparison

Table 6 continued

The software reliability growth models fitting result comparison of table 7 based on system T1 data set

As can be drawn from Table 7, for system T1 software failure data collection, the overall fitting result of the ISQ-FDEFCE-CP model of revising on process fault detection and fault makeover process is better than the ISQ-FDEFCE-CP model that embodiment three proposes, and illustrates that the ISQ-FDEFCE-CP model of revising can effectively be applied to system T1 software failure data collection.Also illustrate and when fault is got rid of, have imperfect misarrangement, may introduce new fault simultaneously.By drawing in table 7, on process fault detection, the SSEd of ISQ-FDEFCE-CP model of the correction of application Logistic function and the value of RMSEd are less than the ISQ-FDEFCE-CP model of the correction of application GMW function.This has illustrated the data set for system T1, the fitting effect of the ISQ-FDEFCE-CP model of application Logistic function on process fault detection is better, and on fault makeover process, the ISQ-FDEFCE-CP model SSEc of correction and the value of RMSEc of the ISQ-FDEFCE-CP model of the correction of application Logistic function and application GMW function are more approaching.

Claims (3)

1. the ISQ-FDEFCE software reliability growth model based on transfer point, the described ISQ-FDEFCE software reliability growth model based on transfer point is abbreviated as ISQ-FDEFCE-CP model, it is characterized in that, this model has 1 transfer point in software fault makeover process, and ISQ-FDEFCE-CP model hypothesis is as follows:

Software fault detection process is followed a NHPP;

Software systems inefficacy is at any time all caused by the remaining fault existing in software;

(t, t+ Δ t] number of faults that detected in the time interval is directly proportional to remaining number of faults in system and fault detect workload;

Separate between software fault;

Software fault makeover process cannot be left in the basket, and the number of faults of correction lags behind the fault sum detecting;

Cause that the fault that software systems lost efficacy finally will be corrected, software fault detection process and fault makeover process are executed in parallel at every turn, and fault makeover process can not have influence on process fault detection and fault correction is perfect;

Describe fault detect and fault correction activity with ISQ model, and model meets NHPP and arrive, obey general distribution service time;

In fault makeover process, fault adjusted rate changes on some time point;

ISQ-FDEFCE-CP model is:

$\begin{array}{c}{m}_c\left(t\right)=m_c(0,{\mathrm{\τ}}_1]+m_c({\mathrm{\τ}}_1,t]\\ ={\∫}_0^{{\mathrm{\τ}}_1}{m}_d^{\′}\left(x_1\right)G_1({\mathrm{\τ}}_1-x_1){\mathrm{dx}}_1\\ +{\∫}_{{\mathrm{\τ}}_1}^t{m}_d^{\′}\left(x_2\right)G_2(t-x_2){\mathrm{dx}}_2+[m_d\left({\mathrm{\τ}}_1\right)-m_c\left({\mathrm{\τ}}_1\right){]G}_2(t-{\mathrm{\τ}}_1)\end{array}$

Wherein:

M _c(t) be the mean value function with the fault makeover process of 1 transfer point;

M _c(0, τ ₁] be at (0, τ ₁] be detected the mean value function of fault at makeover process in the time interval;

M _c(τ ₁, t] and be (τ ₁, t] and in the time interval, be detected the mean value function of fault at makeover process;

M ' _d() is m _dthe first order derivative of (), m _d() represent till the mean value function of the process fault detection of t moment ISQ-FDEFCE-CP model,

$m_d\left(t\right)=a[1-\exp (-b{W}_d\left(t\right)+{\mathrm{bW}}_d\left(0\right))]=a[1-\exp (-b{W}_d^{*}\left(t\right))]$

Wherein: a represents that software test starts to close potential fault sum;

B represents the fault detect rate of per unit test job amount;

Wd ₍t) represent till t moment accumulation test job amount;

W (0) represents the test job amount in 0 moment;

|W ^*(t)＝W(t)-W(0)；

G ₁(τ ₁-x ₁) expression (0, τ ₁] fault correction time cumulative distribution function in the time interval; Some trouble spots are at x ₁moment is detected, and at (x ₁τ ₁] revised completely in the time interval;

And G ₁(τ ₁-x ₁) press formula G ₁(τ ₁-x ₁)=1-exp[-ρ W _c(τ ₁)+ρ W _c(x ₁)] obtain;

Wherein: ρ represents the fault adjusted rate of per unit fault correction workload; W _c(τ ₁) represent to τ ₁the fault correction workload of accumulating till the moment; W _c(x ₁) represent to x ₁the fault correction workload of accumulating till the moment;

G ₂(t-x ₂) expression (τ ₁, t] and in the time interval, the cumulative distribution function of fault correction time;

And G ₂(t-x ₂) press formula G ₂(x ₁-t)=1-exp[-ρ W _c(x ₁)+ρ W _c(t)] obtain;

Wherein: ρ represents the fault adjusted rate of per unit fault correction workload; W _c(t) represent the fault correction workload of accumulating till the t moment.

2. the ISQ-FDEFCE software reliability growth model based on transfer point, the described ISQ-FDEFCE software reliability growth model based on transfer point is abbreviated as ISQ-FDEFCE-CP model, it is characterized in that, this model has 2 transfer points in software fault makeover process, and ISQ-FDEFCE-CP model hypothesis is as follows:

Software fault detection process is followed a NHPP;

Software systems inefficacy is at any time all caused by the remaining fault existing in software;

(t, t+ Δ t] number of faults that detected in the time interval is directly proportional to remaining number of faults in system and fault detect workload;

Separate between software fault;

Software fault makeover process cannot be left in the basket, and the number of faults of correction lags behind the fault sum detecting;

Cause that the fault that software systems lost efficacy finally will be corrected, software fault detection process and fault makeover process are executed in parallel at every turn, and fault makeover process can not have influence on process fault detection and fault correction is perfect;

Describe fault detect and fault correction activity with ISQ model, and model meets NHPP and arrive, obey general distribution service time;

In fault makeover process, fault adjusted rate changes on some time point;

ISQ-FDEFCE-CP model is:

$\begin{array}{c}{m}_c\left(t\right)=m_c(0,{\mathrm{\τ}}_1]+m_c({\mathrm{\τ}}_1,{\mathrm{\τ}}_2]+m_c({\mathrm{\τ}}_2,t]\\ ={\∫}_0^{{\mathrm{\τ}}_1}{m}_d^{\′}\left(x_1\right)G_1({\mathrm{\τ}}_1-x_1){\mathrm{dx}}_1\\ +{\∫}_{{\mathrm{\τ}}_1}^{{\mathrm{\τ}}_2}{m}_d^{\′}\left(x_2\right)G_2({\mathrm{\τ}}_2-x_2){\mathrm{dx}}_2+[m_d\left({\mathrm{\τ}}_1\right)-m_c\left({\mathrm{\τ}}_1\right)]G_2({\mathrm{\τ}}_2-{\mathrm{\τ}}_1)\\ +{\∫}_{{\mathrm{\τ}}_2}^t{m}_d^{\′}\left(x_3\right)G_3(t-x_3){\mathrm{dx}}_3+[m_d\left({\mathrm{\τ}}_2\right)-m_c\left({\mathrm{\τ}}_2\right)]G_3(t-{\mathrm{\τ}}_2)\end{array}$

Wherein:

M _c(t) be the mean value function with the fault makeover process of 2 transfer points;

M _c(0, τ ₁] be at (0, τ ₁] be detected the mean value function of the makeover process of fault in the time interval;

M _c(τ ₁, τ ₂] be (τ ₁, τ ₂] be detected the mean value function of the makeover process of fault in the time interval;

M _c(τ ₂, t] and be (τ ₂, t] and in the time interval, be detected the mean value function of the makeover process of fault;

M ' _d() is m _dthe first order derivative of (), m _d() represent till the mean value function of the process fault detection of t moment ISQ-FDEFCE-CP model,

$m_d\left(t\right)=a[1-\exp (-b{W}_d\left(t\right)+{\mathrm{bW}}_d\left(0\right))]=a[1-\exp (-b{W}_d^{*}\left(t\right))]$

Wherein: a represents that software test starts to close potential fault sum;

B represents the fault detect rate of per unit test job amount;

W _d(t) represent till t moment accumulation test job amount;

W (0) represents the test job amount in 0 moment;

W ^*(t)=W(t)-W(0)；

M _c(τ ₁) be τ ₁the number of faults of being revised completely in moment, m _c(τ ₂) be τ ₂the number of faults of being revised completely in moment;

0≤τ ₁≤τ ₂≤t，m _d(0)＝m _c(0)

G ₁(τ ₁-x ₁) expression (0, τ ₁] fault correction time cumulative distribution function in the time interval;

And G ₁(τ ₁-x ₁) press formula G ₁(τ ₁-x ₁)=1-exp[-ρ W _c(τ ₁)+ρ W _c(x ₁)] obtain;

Wherein: ρ represents the fault adjusted rate of per unit fault correction workload; W _c(τ ₁) represent to τ ₁the fault correction workload of accumulating till the moment; W _c(x ₁) represent to x ₁the fault correction workload of accumulating till the moment;

G ₂(τ ₂-x ₂) expression (τ ₁, τ ₂)] in the time interval, the cumulative distribution function of fault correction time;

And G ₂(τ ₂-x ₂) press formula G ₂(τ ₂-x ₂)=1-exp[-ρ W _c(τ ₂)+ρ W _c(x ₂)] obtain;

G ₃(t-x ₃) expression (τ ₂, t)] and in the time interval, the cumulative distribution function of fault correction time;

And press formula G ₃(t-x ₃)=1-exp[-ρ W _c(t)+ρ W _c(x ₃)] obtain;

Wherein: ρ represents the fault adjusted rate of per unit fault correction workload; W _c() represents the fault correction workload of accumulating till the t moment.

3. the ISQ-FDEFCE software reliability growth model based on transfer point, the described ISQ-FDEFCE software reliability growth model based on transfer point is abbreviated as ISQ-FDEFCE-CP model, it is characterized in that, this model has n transfer point in software fault makeover process, and ISQ-FDEFCE-CP model hypothesis is as follows:

Software fault detection process is followed a NHPP;

Software systems inefficacy is at any time all caused by the remaining fault existing in software;

(t, t+ Δ t] number of faults that detected in the time interval is directly proportional to remaining number of faults in system and fault detect workload;

Separate between software fault;

Software fault makeover process cannot be left in the basket, and the number of faults of correction lags behind the fault sum detecting;

Cause that the fault that software systems lost efficacy finally will be corrected, software fault detection process and fault makeover process are executed in parallel at every turn, and fault makeover process can not have influence on process fault detection and fault correction is perfect;

Describe fault detect and fault correction activity with ISQ model, and model meets NHPP and arrive, obey general distribution service time;

In fault makeover process, fault adjusted rate changes on some time point;

ISQ-FDEFCE-CP model is:

$\begin{array}{c}{m}_c\left(t\right)=\underset{k=0}{\overset{n}{\mathrm{\Σ}}}\left\{{\∫}_{{\mathrm{\τ}}_k}^{{\mathrm{\τ}}_{k+1}}{m}_d^{\′}\left(x_{k+1}\right)\right[1-\exp (-{\mathrm{\ρ}}_{k+1}{W}_c\left({\mathrm{\τ}}_{k+1}\right)+{\mathrm{\ρ}}_{k+1}{W}_c\left(x_{k+1}\right))]{\mathrm{dx}}_{k+1}\\ +[m_d\left({\mathrm{\τ}}_{k+1}\right)-m_c\left({\mathrm{\τ}}_{k+1}\right)][1-\exp (-{\mathrm{\ρ}}_{k+1}{W}_c\left({\mathrm{\τ}}_{k+1}\right)+{\mathrm{\ρ}}_{k+1}{W}_c\left({\mathrm{\τ}}_k\right))]\}\end{array}$

Wherein:

M _c(t) be the mean value function with the fault makeover process of n transfer point;

M ' _d() is m _dthe first order derivative of (), m _d() represent till the mean value function of the process fault detection of t moment ISQ-FDEFCE-CP model,

$m_d\left(t\right)=a[1-\exp (-b{W}_d\left(t\right)+{\mathrm{bW}}_d\left(0\right))]=a[1-\exp (-b{W}_d^{*}\left(t\right))]$

Wherein: a represents that software test starts to close potential fault sum;

B represents the fault detect rate of per unit test job amount;

W _d(t) represent till t moment accumulation test job amount;

W (0) represents the test job amount in 0 moment;

W ^*(t)＝W(t)-W(0)；

ρ _k+1represent the fault adjusted rate of per unit fault correction workload; W _c() represents the fault correction workload of accumulating till the t moment.

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