WANG Tao,CAI Jinyan,MENG Yafeng,and ZHU Sai

Department of Electronic and Optical Engineering,Army Engineering University,Shijiazhuang 050003,China

Abstract: Due to the limitations of the existing fault detection methods in the embryonic cellular array (ECA), the fault detection coverage cannot reach 100%.In order to evaluate the reliability of the ECA more accurately, embryonic cell and its input and output(I/O) resources are considered as a whole, named functional unit(FU). The FU fault detection coverage parameter is introduced to ECA reliability analysis, and a new ECA reliability evaluation method based on the Markov status graph model is proposed.Simulation experiment results indicate that the proposed ECA reliability evaluation method can evaluate the ECA reliability more effectively and accurately. Based on the proposed reliability evaluation method, the influence of parameters change on the ECA reliability is studied, and simulation experiment results show that ECA reliability can be improved by increasing the FU fault detection coverage and reducing the FU failure rate. In addition, by increasing the scale of the ECA, the reliability increases to the maximum first,and then it will decrease continuously.ECA reliability variation rules can not only provide theoretical guidance for the ECA optimization design,but also point out the direction for further research.

Keywords:embryonic,Markov status graph model,reliability,fault detection,evaluation.

1.Introduction

Reliability refers to the ability that an item will perform its specified function under the specified conditions within the specified time [1], which can be used to evaluate the quality of a product.The embryonic cellular array(ECA)is a newly designed bio-inspired hardware by simulating the growth and development of multi-cellular organisms, which is with fault self-testing and self-repairing property[2].The ECA can improve the reliability and environmental adaptability of electronic circuits effectively,which provides a new method for the design of electronic circuits in the fields of aerospace,deep-sea exploration and complex electromagnetic environment.

With the development of the ECA technology, the research on the ECA reliability evaluation method has been enriched.Ortega and Tyrrell[3,4]analyzed the basic structure and self-repairing process of the two dimensional ECA,and ECA reliability evaluation models with different self-repairing strategies were proposed respectively based on the k-out-of-n system theory.Prodan et al.[5]proposed an original approach to ECA reliability analysis, by introducing the accuracy threshold measure borrowed from the fault-tolerant quantum computing theory, as one of the main parameters for our qualitative assessment.Zhang et al. [6] proposed a practical ECA reliability evaluation model based on the ideal reliability evaluation model,considering the implementation details in the embryonic cell,while the assistant hardware of extra register and routing channels in the switch box is extracted into the practical model. At the same time, cell granularity parameter and layout parameter were introduced to ECA reliability analysis [7,8], which makes the reliability evaluation model closer to the ECA actual working process.Zhu et al.[9]analyzed the ECA working process,a novel ECA reliability evaluation model based on the Markov non-repairable system theory was proposed.Wang et al.[10,11]proposed an ECA reliability model based on the multi-state system reliability theory,by dividing the ECA working process into multiple working states with different self-repairing strategies.

Based on the proposed ECA reliability evaluation methods, Ortega et al. [12] researched the effect on the ECA reliability with the changing of the electronic cell failure rate and the number of spare cells, which laid a foundation for the subsequent research.Prodan et al.[5]brought new perspectives on designing reliable embryonic memory structures at both the molecular and the cellular levels,and appropriate design principles were provided on both infor-mation encoding and storage.Lin et al.[13]researched the effect on the ECA reliability with different design methods in an n×n scale ECA, and ECA layout optimal design rules were given. Zhang et al. [6–8] researched the method of self-repairing strategy selection, cell granularity optimization selection and ECA layout optimization design of the ECA on reliability analysis.Zhu et al. [14,15]proposed a gene backup number selection method based on the ECA reliability and hardware consumption analysis.Wang et al.[11]studied the structure optimization design,performance analysis and preventive maintenance decision of the bus-based ECA. At the same time, based on the ECA reliability and hardware resource consumption analysis,the mathematical description method for the ECA,the method of electronic cells number selection in the ECA and the ECA layout optimization method are studied[16–19],which can be used to guide the optimization design of the ECA.

Summarizing the current studies,in the process of analyzing and evaluating the ECA reliability,it is assumed that the fault detection coverage of the electronic cell is 100%.However, in the actual working process, fault detection coverage cannot reach 100%. In addition, in the existing ECA reliability models, only the failure of the electronic cells is considered,and the failure of the input and output(I/O) resources among the electronic cells is not considered. At present, ECA reliability evaluation methods are mostly based on the k-out-of-n system reliability model[20–22],which can provide a time metric to evaluate the ECA reliability,and they are helpful to instructing the optimization design of the ECA.However,they cannot reflect the process of self-repairing and the ECA working states transition,which is helpful to researching the nature of the ECA reliability.

Analyzing the structural characteristics and working process of the ECA, in the paper, the electronic cell and its corresponding I/O resources are equivalent to a functional unit(FU),and the FU fault detection coverage is introduced to the ECA reliability evaluation method. With two kinds of self-repairing strategies,ECA reliability evaluation methods were proposed based on the Markov status graph model[23].With the ECA reliability evaluation method based on the k-out-of-n system theory model as a comparison object,the validity and rationality of the ECA reliability evaluation method based on the Markov state graph model have been verified. Based on the proposed ECA reliability evaluation method,the influence of the parameters change on the ECA reliability was studied, and the research results can not only provide theoretical guidance for the optimization design of the ECA,but also point out the direction for further research.

Section 2 briefly addresses the basic structure and selfrepairing strategies of the ECA. In Section 3, with two kinds of self-repairing strategies, ECA reliability evaluation methods are proposed based on the Markov status graph model respectively. Section 4 verifies the validity and rationality of the proposed ECA reliability evaluation method,and the influence of the parameters change on the ECA reliability is studied in detail.Finally,conclusions and suggestions for future work are presented in Section 5.

2.ECA and its self-repairing strategies

2.1 ECA

The ECA is a two-dimensional array of embryonic cells,which is shown in Fig. 1, and each cell in the ECA represents a processing element.Embryonic cells in the ECA can be divided into active cells and spare cells, and the structures of embryonic cells are the same.Active cells realize the function of the circuit,and spare cells are redundant units of active cells.

Fig.1 Structure of ECA and embryonic cell

The embryonic cell is mainly composed of an address generator,a gene memory and configuration block,a functional block,a control block,a fault detection block and an input output(I/O) router[24].The address generator will generate the co-ordinate of the cell, which will determine the function of the embryonic cell.The gene memory and configuration block stores the genes of the embryonic cell,which will be configured to all blocks in the embryonic cell.The functional block performs the function of the embryonic cell, which is determined by configuration genes.The control block will control all the blocks so as to ensure the normal and orderly working of the embryonic cell.The fault detection block realizes the fault self-testing of embryonic cells, which is the premise of fault self-repairing.The I/O router realizes the connection of embryonic cells.

2.2 ECA fault self-repairing strategies

The fault self-repairing strategy can effectively improve the reliability of the circuit, and the nature of fault selfrepairing is the shift and re-expression of address information. According to the structural characteristics of the ECA, there are a row (column)elimination self-repairing strategy and a cell elimination self-repairing strategy for the ECA[24].Row and column elimination self-repairing are equivalent cell replacement strategies. In the following discussion, only the row elimination self-repairing strategy will be analyzed; however, the same results apply for the column elimination self-repairing strategy.The self-repairing process of the row elimination self-repairing strategy is shown in Fig. 2, and the self-repairing process of the cell elimination self-repairing strategy is shown in Fig.3.

Fig.2 Process of row elimination self-repairing

Fig.3 Process of cell elimination self-repairing

In the working process of the row elimination selfrepairing strategy, the failing of one active cell provokes the elimination of the corresponding row, and cells are logically shifted upwards making use of a spare row.Active cells in the faulty cell row become transparent cells,which act as a conductor.Each embryonic cell is defined by two co-ordinates with enough memory to contain the configuration genes of the corresponding column.

The row elimination self-repairing strategy is far from optimal with respect to the use of spare resources,but the short time needed to recover from a failure makes it attractive to implement real-time fault-tolerant systems. In addition,as the array grows larger,the percentage of cells lost during reconfiguration decreases dramatically.

In order to improve the utilization of spare cells,the cell elimination self-repairing strategy is proposed. Each embryonic cell in the ECA is defined by two co-ordinates and contains the configuration genes of the entire array with enough memory.

Fig.3 shows the self-repairing process of the cell elimination self-repairing strategy. In the cell elimination selfrepairing strategy, spare cells replace faulty cells in two stages.First, when the number of the faulty cells in a row is less than the number of spare cells, and spares located in the same row replace the faulty cells; second,when the number of the faulty cells in a row surpasses the number of spare cells,row elimination self-repairing is performed.

3.ECA reliability evaluation based on Markov status graph model

In the process of the ECA reliability evaluation,the electronic cell and its corresponding I/O resources are equivalent to FU, and it is the basic unit for evaluating the reliability of embryonic cellular row (ECR). In the process of ECR reliability evaluation, assume the FU fault detection coverage is p, and the FU failure process follows the exponential distribution. With two kinds of self-repairing strategies,the ECR reliability based on the Markov status graph model is analyzed.Based on the analysis of the ECR reliability, the reliability of the ECA will be evaluated by analyzing the dynamic characteristics of row elimination self-repairing.

3.1 Markov status graph model

The Markov status graph model is a typical Markov chain model [25,26], which is widely used in the modeling of stochastic decay processes in multi-state systems or multistate components.

Assume the discrete states space E = {e1,e2,...,en},and the discrete time parameter stochastic process{X(t)|X(t) ∈ E,t0}. For any moment 0the probability of tncan be defined by conditional probability,which is shown in(1).

If the stochastic process satisfies the requirements of(2),the stochastic process is called a Markov chain.As in the case of a general Markov process,(2)implies that the chain behavior in the future depends only on its present state and does not depend on its behavior in the past.

The Markov status graph model is suitable for both repairable and un-repairable systems[23,27].In the Markov status graph model,a system must meet the following conditions.

Firstly, the system must satisfy the Markov process.Note that only the Markov process where the state probabilities at a future instant does not depend on the states occupied in the past.

Secondly, the system and elements must be in the discrete state,that is, working or failure.In the working process, the system and elements only have two states. If the system and elements can operate normally,they are in the working state.When a fault occurs in the system and elements,they are in the failure state.

Thirdly,the system failure rate and the repair rate should be constant.The degradation process and repair process of the system obey the exponential distribution,so the failure rate and the repair rate are constant.

Finally,the state transition can take place at any instant of time.However,in a fairly small time interval Δt, there are no simultaneous state transitions of any different elements. In other words, there may be only one failure or one repair in a system at any instant t.

The general steps of the Markov status graph model are as follows[23,27,28].

(i) By analyzing the system working process, working states of the system are given.According to the states transition law of components in the system, the Markov state space graph of the system is given. The states transition rate usually refers to the failure rate or the maintenance rate.

(ii) According to system states transition space graph,the system states transition probability matrix T is given.The system states equation coefficient matrix A can be given by|T ?U|, which can be used to establish system states equation.(iii)Based on the established system states equation,the system states probabilitypi(t)can be calculated.In accordance with the definition ofpi(t),the reliability of the system can be calculated,and the calculation process is realized by Laplace transformation.

3.2 ECA reliability evaluation method

According to the ECA working process,the ECA is an unrepairable system. In the ECA, the working state at any moment is only related to that at the previous moment.Therefore,the ECA working process satisfies the Markov process.Embryonic cells in the ECA only have two kinds of states, that is, working or failure. The failure process of embryonic cells obeys the exponential distribution. In the ECA,there are no simultaneous state transitions of any different embryonic cells in a fairly small time interval Δt.

Above all,the ECA working process conforms to the requirements of the Markov status graph model.Therefore,the Markov status graph model can be used in the analysis and evaluation of the ECA reliability.

3.2.1 FU

The ECA is mainly composed of embryonic cells and I/O resources.In the existing ECA reliability evaluation model,only the failure of embryonic cells is considered, and the effect of I/O resources failure on the ECA reliability is not considered. In the ECA, The connection method among embryonic cells is shown in Fig.4.

Fig.4 Connection among embryonic cells

In the ECA, the failure of the embryonic cell will directly cause the failure of the circuit function. Similarly,the failure of I/O resources will also cause the failure of the circuit function.Therefore,the embryonic cell and its corresponding I/O resources can be equivalent to a series system,which is shown in Fig.5.

In a series system withNunits,assume the failure rate of units isλiand all units are independent of each other. Therefore,the series system failure rate is the sum of units failure rates,that is,

Fig.5 A series system

In order to facilitate the analysis of the ECA reliability,the embryonic cell and its corresponding I/O resources are equivalent to an FU.In an FU,assume the embryonic cell failure rate isλc,and its corresponding I/O resources failure rate isγc.Therefore,the FU failure rateλfisλc+γc.

Assume the scale of the ECA isR×C;the scale of the working ECA isr×c.Ris the row number of the ECA,Cis the column number of the ECA,ris the row number of the working ECA,cis the column number of working ECA, andRr,Cc. With different self-repairing strategies, the ECA is divided into the row elimination ECA and the cell elimination ECA,and the ECA reliability evaluation method is proposed based on the Markov status graph model.

3.2.2 Reliability evaluation method of cell elimination ECA

In the cell elimination ECA, the ECR is a basic unit for analyzing ECA reliability.In the ECR,if an FU fails,only the faulty FU will be eliminated.The ECR working process can be divided into multiple working states with different numbers of faulty FUs,which are shown in Table 1.

Table 1 Working states of cell elimination ECR

In Table 1, the stateC0to the stateare normal working states of the ECR,the stateC0means all FUs work normally in the ECR, the stateC1to the statemean one faulty FU fails in the ECR, the stateto the statemean two faulty FUs fail in the ECR,the stateto the statemeanC ?cfaulty FUs fail in the ECR.The stateF1to the stateare failure states of the ECR,which meanC ?c+1 faulty FUs fail in the ECR.For the FU fault detection coveragepless than 1,there is a stateFn,which means faults in the FU are not detected.Therefore,the working states set of the ECR

The reliability of the ECR is the probability that the ECR works in normal working states,that is,the probability that the ECR works in the stateC0to the stateIn the ECR, assume the failure rate of the FU isλf, and the FU fault detection coverage isp. Therefore, the state transition rate means the failure rate of the FU.According to the ECR working principle,the ECR states transition rate matrixTcan be given.In order to calculate the reliability of the ECR, only the ECR normal working states transition rate matrixTwis needed, and the ECR normal working states equation coefficient matrixSwequalsTw ?Uw,Uwis an identity matrix withdimensions.The Markov states differential equation is obtained by the Markov chain as follows.

According to the Markov states differential equation,the probability that the ECR works in normal working states can be calculated,and the ECR reliabilityR(t)is

In order to describe the solution process of the ECR reliability more clearly, assume the scale of ECAR ×Cis 4×4, and the scale of the working ECAr × cis 2×2. Therefore, the ECR working states setScis{C0,C1,...,C10,F1,F2,F3,F4,Fn}. Array is used to describe the working states of the ECR,andScis{0 000,1 000,0 100,0 010,0 001,1 100,1 010,1 001,0 110,0 101, 0 011, 1 110, 1 011, 1 101, 0 111,Fn}, in which 1 means the FU fails, 0 means the FU works normally.Because the ECR working states transition process is complicated, only the transition process of the stateC0is given,which is shown in Fig.6.

Fig.6 Transition process of state C0

In Fig. 6, the stateC0means no FU fails in ECR, the statesC1,C2,C3andC4mean one FU fails in the ECR.The transition rates from the stateC0to statesC1,C2,C3andC4are the same, which ispλf. Therefore, the probability that the ECR stays in the stateC0is 1?4λf,and the transition rate from the stateC0to the stateFnis 4λf(1?p). The transition process of other states can be obtained by this method,and the Markov status transition graph of all working states can be given.According to the obtained Markov status transition graph, the ECR normal working states transition rate matrixTwis

Therefore,the ECR normal working states equation coefficient matrix Swis Tw?Uw, and the Markov states differential equation is obtained by the Markov chain as follows.

According to the Markov states differential equation,the probability that the ECR works in normal working states can be calculated.In order to ensure the ECR works normally,the ECR must work in one state among the state C0to the state C10.Therefore,the probability that the ECR works in the state C0is

The probabilities that the ECR works in the states C1,C2,C3and C4are the same,which is

The probabilities that the ECR works in the state C5to the state C10are the same,which is

Therefore,in the above ECA,the reliability of one ECR is

In order to ensure the ECR works properly, the ECR must work in the the state C0to the statethe probability that the ECR works in the state C0is

The probability that the ECR works in the state C1to the stateis

The probability that the ECR works in the stateto the stateis

The probability that the ECR works in the stateto the state

The probability that the ECR works in the stateto the stateis

Therefore,in cell elimination ECA,the reliability of one ECR is

In the cell elimination ECA, when redundant FUs are used up in an ECR, row elimination self-repairing of the ECA will work. In the working process of row elimination self-repairing,the ECR is regarded as a whole,and the working principle of row elimination self-repairing in the ECA is the same as that of cell elimination self-repairing in the ECR.Based on the reliability evaluation method of the cell elimination ECR,the reliability of the cell elimination ECA is

And the mean time to failure(MTTF)of the cell elimination ECA is

3.2.3 Reliability evaluation of row elimination ECA

According to the reliability evaluation method in the cell elimination ECA, the reliability of the row elimination ECA can be evaluated similarly, so the concrete analysis process will not be repeated here. In the row elimination ECA, any working FU’s failing will result in the failure of the ECR, and the faulty ECR will be eliminated. In the ECR, the working process can be divided into multiple working states with different numbers of faulty working FUs and faulty redundant FUs,and working states are shown in Table 2.

Table 2 Working states of row elimination ECR

In Table 2,the state R0to the stateare normal working states of the ECR,the state R0means no FU fails in the ECR, the state R1to the state RC?cmean one redundant FU fails in the ECR, the state RC?c+1to the statemean two redundant FUs fail in the ECR,the statemeans all redundant FUs fail in the ECR.The state F1to the stateare failure states of the ECR,the state F1to statemean one working FU fails in the ECR,the stateto the statemean one working FU and one redundant FU fail in the ECR, the statethe statemean one working FU and two redundant FUs fail in the ECR,the stateto the statemean one working FU and C ?c redundant FUs fail in the ECR. For the FU fault detection coverage p less than 1, there is a state Fn. The state Fnmeans the faulty FU is not detected. Therefore,the working states set of the row elimination ECR Sris

The probability that the ECR works in the state R0is

The probability that the ECR works in the state R0to state RC?cis

The probability that the ECR works in the state RC?c+1to the stateis

The probability that the ECR works in the stateto the state

The probability that the ECR works in the state

Therefore,the reliability of the row elimination ECR is

Above all,the reliability of the row elimination ECA is

The MTTF of the row elimination ECA is

4.Experimental verification and parameters analysis

At present,the research on the ECA technology is still in the theoretical stage,and there is lack of mature specialized

hardware.Therefore,it is difficult to obtain the experimental data of the ECA through the accelerated degradation experiment so as to verify the validity and rationality of the proposed ECA reliability evaluation method. At present,thek-out-of-nreliability theory is widely applied to analyzing the ECA reliability, and the ECA reliability evaluation method based on thek-out-of-nreliability theory is chosen as a comparison object. By comparing ECA reliability data from two kinds of reliability evaluation methods,the validity of the proposed ECA reliability evaluation method based on the Markov status graph model is verified.Analyzing the influence of the parameters change on the ECA reliability, the rationality of the proposed ECA reliability evaluation method is verified.

4.1 Comparative analysis

Six kinds of ECAs are chosen as experimental objects,and the scales of ECAs are shown in Table 3.The reliability of ECAs is analyzed based on two kinds of ECA reliability evaluation methods respectively.

Table 3 ECAs with six kinds of scales

In the ECA with a determined scale, the evaluation of the ECA reliability based on thek-out-of-nsystem theory is only related to the FU failure rateλf,and the evaluation of the ECA reliability based on the Markov status graph model is related to both the FU failure rateλfand the FU fault detection coveragep.In order to verify the validity of the proposed ECA reliability evaluation method based on the Markov status graph model,assume the FU fault detection coveragep=1,λf=6E-6 h.With two kinds of ECA reliability evaluation methods,the reliability curves of six kinds of ECAs are shown in Fig.7,and the MTTF values of six kinds of ECAs are shown in Table 4.

Fig.7 Reliability curves of six kinds of ECAs

Table 4 MTTF values of six kinds of ECAs h

In Fig. 7, “n/k” means the ECA reliability evaluation method based on thek-out-of-nsystem reliability model,“MSG”means ECA reliability evaluation method based on the Markov status graph model,“ECA1”meansR=C=5 andr=c= 3,“ECA2”to “ECA6”are shown in Table 3.Fig. 7(a) shows the reliability curves of six kinds of cell elimination ECAs, which are calculated by two kinds of ECA reliability evaluation methods above.ECA reliability curves of six kinds of ECAs are completely coincident,and the MTTF values of six kinds of scales are the same respectively.

In a similar way,Fig.7(b)shows the reliability curves of six kinds of row elimination ECAs, which are calculated by two kinds of ECA reliability evaluation methods above.ECA reliability curves of six kinds of ECAs are completely coincident,and MTTF values of six kinds of ECAs are the same respectively.

In order to further demonstrate the validity and accuracy of the proposed method, assume the working ECA scales of six kinds of ECAs are shown in Table 3,RandCincrease fromrandcto 4rand 4crespectively,and the growth rates are the same. With two kinds of evaluation methods above,the MTTF values of six kinds of row elimination ECAs are shown in Fig.8,and MTTF values of six kinds of row elimination ECAs are shown in Fig.9.

Fig.8 MTTF curves of six kinds of cell elimination ECAs

Fig.9 MTTF curves of six kinds of row elimination ECAs

In Fig. 8 and Fig. 9, “n/k” means the ECA reliability evaluation method based on thek-out-of-nsystem reliability model, “MSG” means the ECA reliability evaluation method based on the Markov status graph model,and the MTTF of ECA is evaluated by the above two kinds of methods respectively.In ECAs,the scale of working ECAs is determined.WhenRandCincrease fromrandcto 4rand 4crespectively,the MTTF curves of six kinds of cell elimination ECAs are completely coincident.Similarly,the MTTF curves of six kinds of row elimination ECAs are completely coincident. Therefore, the proposed ECA reliability evaluation method can effectively and accurately evaluate the reliability of the ECA.

In order to verify the rationality of the proposed ECA reliability method, assumep= 1, 0.95, 0.9, 0.85, 0.8,λf=6E–6 h,and ECA scales are shown in Table 3.When the cell elimination self-repairing strategy is adopted, the MTTF values of six kinds of ECAs are shown in Table 5.Similarly, when the row elimination self-repairing strategy is adopted,the MTTF values of six kinds of ECAs are shown in Table 6.

Table 5 MTTF values of six kinds of cell elimination ECAs with different p h

Table 6 MTTF values of six kinds of row elimination ECAs with different p h

As shown in Table 5 and Table 6, when the ECA scale and the self-repairing strategy are determined, the higherpis,the larger the MTTF of ECA is.In the process thatpincreases,the higherpis, the larger growth rate of MTTF is.In addition,in the process thatpincreases,the larger the ECA scale is, the larger the growth rate of ECA is. In the ECA,fault detection is the premise of fault self-repairing,if the fault cannot be detected,the ECA will fail immediately.Therefore,the higherpis, the lower the failure rate of the ECA is,that is,the larger MTTF is.The larger ECA scale means more embryonic cells;therefore,the probability that faults occur is higher;therefore,the higherpis,the larger the growth rate of MTTF is.

Above all, the proposed ECA reliability evaluation method based on the Markov status graph model can evaluate ECA reliability effectively and more accurately.

4.2 Analysis of parameters in ECA

Based on the analysis in Section 3.2.2,the ECA reliability is related to the number of ECA rowsRandr,the number of ECA columnsCandc,the FU fault detection coveragepand the FU failure rateλf. When the functional circuit is mapped to the ECA, the working scale of the ECA is determined, that is,candrare determined. In order to optimize the structural design of the ECA, the effects on reliability with the changing of parameters in the ECA are studied. Selecting different values ofp,λf,CandRin the ECA,the ECA reliability is evaluated by the proposed method respectively.In order to increase the credibility of the experiment,six kinds of ECAs with different scales are selected,which are shown in Table 7.

Table 7 Six kinds of ECAs

4.2.1 Effect on ECA reliability with the changing ofp

The changing of the FU fault detection coveragepwill directly affect the transition rates among ECA working states so as to affect the probability that the ECA works in working states, and it will finally affect the ECA reliability.In order to analyze the effect on the ECA reliability with the changing ofp, assumeλf=6E–6 h, and the scales of the ECA are shown in Table 7. During the process thatpincreases from 0.5 to 1, the MTTF curves of six kinds of ECAs are shown in Fig.10.

In Fig.10,“ECA1–ECA6”mean six kinds of ECAs in Table 7 respectively.Fig.10(a)shows the MTTF curves of six kinds of cell elimination ECAs with differentp, and Fig. 10(b) shows the MTTF curves of six kinds of row elimination ECAs with differentp. As shown in Fig. 10,in the process of increasingpfrom 0.5 to 1,MTTF values of six kinds of ECAs are increasing continuously,and the growth rates of MTTF are also increasing.

Fig.10 MTTF curves of six kinds of ECAs with different p

In the ECA, fault detection is the premise of fault selfrepairing. If the fault in the ECA cannot be detected in time,the ECA will fail immediately.Therefore,the largerpis,the lower the ECA failure rate is,that is,the MTTF values of ECA will increase with the increase ofp.As shown in (17) and (25), the reliability of ECA is positively corrected withp,and the largerpis,the larger the growth rate of MTTF is.

4.2.2 Effect on ECA reliability with the changing ofλf

The changing of the FU failure rateλfwill directly affect the transition rates among ECA working states so as to affect the probability that the ECA works in working states,and it will finally affect the reliability of ECA.In order to analyze the effect on the ECA reliability with the changing ofλf, assumep= 0.96,andC,R,candrare shown in Table 7.In the process thatλfdecreases from 1E–5 h to 1E–6 h,the MTTF curves of six kinds of ECAs are shown in Fig.11.

Fig.11 MTTF curves of six kinds of ECAs with different λf

In Fig.11,“ECA1–ECA6”mean six kinds of ECAs in Table 7 respectively. Fig. 11(a) shows the MTTF curves of six kinds of cell elimination ECAs with differentλf,and Fig.11(b)shows the MTTF curves of six kinds of row elimination ECAs with differentλf.As shown in Fig.11,in the process of reducingλffrom 1E–5 h to 1E–6 h,MTTF values of six kinds of ECAs are increasing continuously.

In the ECA, when the scale is determined, the smallerλfis, the lower the failure rate of the ECA is. Therefore,the MTTF values of the ECA will increase with the decreasing ofλf.In the process of reducingλffrom 1E –5 h to 1E–6 h,the growth rates of MTTF values are increasing in six kinds of row elimination ECAs. In six kinds of cell elimination ECAs,the growth rates of MTTF values are increasing first,then they will decrease in “ECA1–ECA3”.

4.2.3 Effect on ECA reliability with the changing ofR

As shown in (17) and (25), the changing ofRwill directly affect the reliability of ECA.In order to analyze the effect on the ECA reliability with differentR, assumingλf=6E–6 h,p= 0.96,C,randcare shown in Table 7.WhenRincreases fromrto 120,the MTTF curves of six kinds of ECAs are shown in Fig.12.

Fig. 12(a) shows the MTTF curves of six kinds of cell elimination ECAs with differentR. In six kinds of cell elimination ECAs, whenRincreases fromrto 120, the MTTF values of ECAs increase to the maximum first,and then the MTTF values of ECAs are decreasing continuously. From ECA1 to ECA6, whenRequals 20, 36, 45,72, 83 and 94 respectively,the MTTF values of six kinds of cell elimination ECAs reach the maximum.

Fig.12 MTTF curves of six kinds of ECAs with different R

In the cell elimination ECA, the theoretical maximum number of self-repairing isR(C ?c).Therefore,with the increasing ofR,the theoretical maximum number of selfrepairing is increasing, and the MTTF of the ECA is increasing. With the further increasing of ECA scales, the number of electronic cells is increasing, and the failure rate of the ECA is also increasing.Therefore,in six kinds of cell elimination ECAs, when the MTTF values reach the maximum,the MTTF values will decrease with the increasing ofR.

As shown in Fig. 12(a), in the increasing process of MTTF, the smaller the scale is, the larger the growth rate of MTTF is.In the ECA,the smaller the scale is,the lower the failure rate of ECA is. WhenR=r, the numbers of electronic cells in ECA1 to ECA6 are 50,320,690,2 040,3 440 and 5 200 respectively.At the same time,Rincreases by 1,the number of self-repairing increases byC ?c,that is, the numbers of self-repairing increase by 4, 8, 12, 22,44 and 52 respectively.

Fig. 12(b)shows the MTTF curves of six kinds of row elimination ECAs with differentR. In six kinds of row elimination ECAs, whenRincreases fromrto 120, the MTTF values of ECAs increase to the maximum first,and then the MTTF values of ECAs are decreasing continuously. From ECA1 to ECA6, whenRequals 23, 43, 54,69, 81 and 93 respectively,the MTTF values of six kinds of row elimination ECAs reach the maximum.

In the row elimination ECA, the theoretical maximum number of self-repairing isR ?r. Therefore, with the increase ofR, the theoretical maximum number of selfrepairing is increasing,and the MTTF of ECA is increasing. In the increasing process ofR, the number of electronic cells is increasing, and the failure rate of the ECA is also increasing. In six kinds of row elimination ECAs,when the MTTF values reach the maximum, the MTTF values will decrease with the increasing ofR.

As shown in Fig. 12(b), in the increasing process of MTTF, the smaller the scale is, the larger the growth rate of MTTF is.In the ECA,the smaller the scale is,the lower the failure rate of ECA is. WhenR=r, the numbers of electronic cells in ECA1 to ECA6 are 50,320,690,2 040,3 440, 5 200 respectively.At the same time,Rincreases by 1,the number of self-repairing increases by 1 too.

4.2.4 Effect on ECA reliability with the changing ofC

In order to analyze the effect on the ECA reliability with differentC, assumeλf= 6E–6 h,p= 0.96, andR,randcare shown in Table 7.In the process thatCincreases fromcto 100,the MTTF curves of six kinds of ECAs are shown in Fig.13.

Fig.13 MTTF curves of six kinds of ECAs with different C

Fig. 13(a) shows the MTTF curves of six kinds of cell elimination ECAs with different C.In six kinds of ECAs,when C increases from c to 100,the MTTF values of ECAs increase to the maximum first, and then the MTTF values of ECAs are decreasing continuously.From ECA1 to ECA6, when C equals 22, 28, 38, 48, 55 and 67 respectively, the MTTF values of six kinds of cell elimination ECAs reach the maximum.

In the cell elimination ECA, the theoretical maximum number of self-repairing is R(C ?c).Therefore,with the increase of C, the theoretical maximum number of selfrepairing is increasing,and the MTTF of the ECA is also increasing. In addition, with the increase of C, the number of electronic cells is increasing,and the failure rate of the ECA is also increasing.Therefore,in six kinds of cell elimination ECAs,when the MTTF values reach the maximum,the MTTF values will decrease with the increasing of C.

In the increasing process of MTTF,the smaller the ECA scale is,the larger the growth rate of MTTF is.In the ECA,the smaller the ECA scale is, the lower the failure rate of the ECA is.When C = c, the numbers of electronic cells in ECA1 to ECA6 are 72,336,936,2 428,3 600 and 5 760 respectively.At the same time,C increases by 1,the number of self-repairing increases by R, that is, the numbers of self-repairing increase by 12, 28, 52, 86, 100 and 120 respectively.

Fig. 13(b)shows the MTTF curves of six kinds of row elimination ECAs with different C.In six kinds of ECAs,the MTTF values do not change with the increase of C.In the row elimination ECA,the failing of one active cell provokes the elimination of the corresponding ECR. Therefore, the number of C does not affect the MTTF of the ECA.

Above all, the MTTF of the ECA can be improved by increasing p and reducing λf.In the ECA,the higher p is,the higher the MTTF is.At the same time,the lower λfis,the higher the MTTF of the ECA is.In the increasing process of R,the MTTF of the ECA increases to the maximum first,and then it will decline continuously.Similarly,in the increasing process of C, the MTTF of the ECA increases to the maximum first,and then it will decline continuously in the cell elimination ECA.

5.Conclusions

This paper analyzes the structural characteristics of the ECA,and the embryonic cell and its corresponding I/O resources can be equivalent to an FU. Introducing the FU fault detection coverage p to ECA reliability evaluation,a new ECA reliability evaluation method is proposed based on the Markov status graph model.

Selecting the most widely used ECA reliability evaluation method based on the k-out-of-n system reliability model as the comparison object, and experimental results indicate that the proposed ECA reliability evaluation model based on the Markov status graph model can evaluate the reliability of ECA effectively and more accurately.Based on the verification of the proposed method,parameters in the ECA are analyzed,and the ECA reliability variation law is got.In the ECA,the reliability of the ECA can be improved effectively by increasing the FU fault detection coverage rate and reducing the FU failure rate.In the process of increasing the scale of the ECA, the reliability will increase first, then it will reach the maximum,and after that it will decrease continuously.The ECA reliability variation law is able to guide the optimization design of the ECA from the theoretical point of view.

Based on the analysis of parameters in the ECA, in the future, more effective fault detection methods will be researched so as to improve the ECA reliability.At the same time,the optimization selection method of R and C in the ECA will be studied so as to guide the optimization design of the ECA layout.