SUPERCAB V14 English
Contents
1. 1 l Reconfiguration 2 M 1 H A EN 1 element loss 3 Reconfiguration 4 2 element loss 5 Sytemloss 2NMy2 L T TITI r L 57 5 6 Global Redundancy This more generic model can take into account ON and OFF failure rates and Aorr probability of failure before using up y average reconfiguration duration in which the mission is stopped s and average non operational duration due to failure 1 Both durations can be modelled by exponential rates or Erlang law with k fictitious states Several politics of maintenance can be considered with 1 or n repair officers or with duration of reparation independent of the number of units in failure GLOBAL REDONDANCY y 1 Type 1 65 ks Type s Re Na Nb This parametrical function assesses the availability at t or or the reliability considering the loss of system as an absorbing state In case of the mission is not stopped during reconfiguration duration the markovian matrix with dimension N M 2 as shown in the following figure is used N M 1 N M 2 ne ep eee ome T pepe a M among N redundancy system available during reconfigurations In case of the mission is stopped during reconfiguration duration the markovian matrix shown in the following figure with dimension 2 N M 22 is used S 3 4 5 6 20 2 2 No failure a wem prem2. the use made for results he obtains CAB INNOVATION is liable for the software package conformity with his documentation The customer shall prove any possible non conformity CAB INNOVATION does not assume any whatsoever guarantee whether explicit or implicit relating to the software package manuals attached documentation or any supporting item or material provided and especially any guarantee for marketing of any products relating to software package or for using software package for a determined use any guarantee for absence of forgery etc Under no circumstances CAB INNOVATION could be held responsible for any whatsoever damage especially loss in performance data loss or any other financial loss resulting from the use or impossibility to use the SUPERCAB software package even if CAB INNOVATION was told about the possibility of such damage In the event where CAB INNOVATION liability is retained it is expressly agreed upon that the total amount of compensation to be paid by CAB INNOVATION all cases taken together could not in any way exceed the initial royalty price reduced by 25 per period of twelve months elapsed as from the mailing delivery date ARTICLE 8 DURATION This Agreement is entered into for an undetermined period of time as of the date set forth in Article 3 ARTICLE 9 TERMINATION Each party may terminate this Agreement by registered mail with acknowledgement of receipt forwarded to the other party for any
3. More complex examples of modelling are given in chapter 4 10 of this manual and are quoted for demonstration in online help menu Help 25 4 2 Entering an Markov Matrix Menu command matrix entry helps obtaining on a calculation sheet from the selected cell top left corner the following format of a blank matrix with dimension n MAT 1 2 3 4 5 1 _ 2 3 4 5 INIT 1 0 0 0 0 STATE 1 I 1 1 1 0 Then the user enters matrix s transition rates in hour 71 likely to be expressed either in the discrete form or in the form of formulas incorporating variable names terms of main diagonal are automatically computed during the processing and empty cells are replaced by 0 He enters vector INIT corresponding to probabilities to be in different states at T 0 and defines a combination of states to be assessed by entering 1 or 0 in corresponding cells of vector STATE The following utilities bar Matrixes allows to facilitate the entry It is displayed or disappears using menu command Utilities bar Meaning of buttons is as follows Defining names Initiating this button allows to display the following dialog box which helps define variable names in selected sheet 26 DEFINING ONE NAME 2 x Names used Name Refersto 0 01 to active cell contents C2 V with recopy of name on sheet add Such names may
4. 0 00001 0 01389 0 02083 2 0 0139 hr 1 MTTR 72 hours INIT 1 0 0 0 Availabity STATE 1 1 1 0 atT 2000 hours 0 9999833 P 2000hr 0 96479 0 00046 0 03473 1 70 05 32 4 4 Fictitious State Method Only the homogeneous Markovian processes in which transition rates are constant may be processed with a matrix calculation resolution of Chapman Kolmogorov equation dP dt 2 P M Now if a constant fault rate may have a physical meaning especially for certain electronic components it differs for a repair rate However the fictitious state method helps compensate for such limitation It consists in replacing in a Markovian model any transition between two states by a set of constant rate transitions between fictitious states Any type of transition may be modelled in this way and the sequence of transitions between fictitious states may be limited to Cox model shown below Relevant Markov matrix If transition rates of direct chain are identical Aa Ab Ac model is said generalized Erlang If moreover the sequence of transitions is limited to direct chain the resulting law is a simple Erlang law with parameter k corresponding to the convolution of exponential laws k Adding up k 1 fictitious states Close to a log normal law the latter is often used for modelling repair rates 33 The features of a simple Erlang law are as follows Simple Erlang law m 8
5. 0 93194 0 90353 0 87635 0 84897 0 82035 0 78983 0 75712 0 72217 0 68521 0 6466 M1 Initial status 1 2 3 4 5 Final status 6 Reliability at T 2000 hours 0 952194 Equivalent elementary fault rate fits 400000 300000 200000 100000 0 0 1000 hours Reliability of element and redundancy eS 2 0 9 0 8 0 7 0 6 0 1000 2000 hours 38 186539 165912 154781 152713 158719 171472 189533 211530 236250 262686 290037 4 5 Evolutionary System SUPERCAB allows to process Markovian systems the state or the behaviour o determined times Such developments may correspond to changes in transition rate values or to forcing of states as shown by examples below f which is modified at Maintenance limited to working hours 1 0 95 09 s 0 85 lt 0 8 0 75 0 1 2 3 4 5 6 7 8 days Repair rates are zero out of working hours Reliability of a high speed train 1 0 998 0 996 0 994 0 992 e e e ce e e e e e e e N e 175 wo r e e kilometres Failure rates are modified during the journey high or low speed sections Performing a preventive maintenance action 0 2000 4000 6000 8000 10000 hours System condition is periodically modified by a preventive maintenance action 39 2 0 bili 0 6 with
6. TABLE4 ol x UNITS Nb Kind of TTF Operation Switching MTTR p redundancy E rate delay hour hour hour r hour 1250000 5000000 2000000 1000000 133333 33 455996 35 1333333 3 07777778 7 8 11363636 Serie E Reliability Block Diagram 1 Reliability availability Unte m7 Unte Unit 9 55965 hr 3 275000 hr ori irks y Mttr 1500h Reliability 04 Availability of x 02 Markov s Matrix Passive Passive 7 8 Serie C Boc T Unit 5 off if Unit 2 and Unit 3 nd c b a nd cnb ndnc b a d nc nb na nd nc nb na Architectures may consist of classical redundancies of type M among N active or passive with possible reconfiguration or repair times or more complex redundancies defined by user expressing their operating mode by logical expressions or using specific modelling Used with a posterior version of Excel to version 4 SUPERCAB allows to enter several tables in the same sheet corresponding for example each one to a subsystem and to obtain the block diagram in the form of drawing or image The various tables can be treated individually or collectively The operation of the system from that of subsystems as well as the operation of the latter from that of t
7. b in hot condition b b in cold condition b b 10 c in hot condition 2 c c in cold condition c c 10 Simultaneous activation of the minimum elements No fault Loss of aor a loss of bor b Loss of a and b ora and b Loss of all redundancies 4 0 b a 0 System loss 0 1 0 0 0 0 0 O G IN 9 8 08 hr 1 1 1E 07 8 6E 08 0 6 8E 08 b 7 8 08 hr 1 0 7 8 08 1 4 08 c 6 2 08 hr 1 9 8E 08 1 6E 08 0 0 Reliab t 10000 0 9999977 0 0 51 Example 12 Commands of redundancy mechanisms 3 among 4 M1 2 1 2 mechanisms E1 E2 E3 E4 E control electronics Lambda Lambda Lambda Lambda E in hot condition E E in cold condition E E 10 M in hot condition M M in cold condition 2 M 2 M 10 No fault 1 3 E M E Loss of 1 passive elements M 2 2 E M E Loss of 2 passive elements M 3 E M M E Loss of all redundancies 4 System loss 5 E 1 325E 06 hr 1 5 6465E 06 5 13E 07 hr 1 3 8085E 06 1 838E 06 2 0218 06 3 676 06 Reliab t 4000hr 0 999752 5 514E 06 INIT STATE 52 Example 13 Redundancy Calculators Example 10 OSR Supervision and Reconfiguration Component watchdog Calculator Eff Supervision Efficiency rate Lambda Calculator in hot condition Lbd Lambda Calculator in cold condition Lbdf Lam
8. t y o4 Ssrv e without 0 2 Menu command Evolutionary System helps processing systems submitted to such determinist developments After stating the different numbers of Markov matrixes and forcings required for describing system development the user defines the latter numbers on a program formatted calculation sheet see example 6 The user specifies the beginning and duration of validity phase of each matrix which can possibly be periodic If validity phases of different matrixes are superimposed the program warns about it and only considers the phase of matrix with superscript Similarly user specifies the occurrence time of each forcing as well as any possible periodicity If forcing sequences take place simultaneously the latter will be successively performed in their numbering order Probabilities P1 to Pn following a forcing of states may be defined either by numerical values or by expressions comprising such probabilities just prior to forcing sum is to be equal to 1 P1 Pl alpha P2 P3 P2 1 alpha P2 P3 0 P4 unchanged if empty box sum of probabilities after forcing is equal to Pi prior to forcing Finally the user defines vectors INIT and STATE duration of mission considered Tmax and the calculation sequence The latter should be lower than half the duration of matrix validity phases and lower than forcing periods Processing will be performed by initiating menu comma
9. A B C D 0 96904943 Fault Tree Return to the preceding menu 2 5 4 Method of the fictitious states Fictifs Method of the fictitious states Replacement of a complexe transition by transitions with constant rates between fictitious states Cox model Generalized Erlang a Simple Erlang 1 2 3 Probability density of Simple Erlang law Adjustment generalized Erlang 0002 00015 Failure rate 0001 Equivalent 010005 S C 20 24 25 32 P k 2 number of fictitious states Time hr Used to model duration of repair Modeling of youth and wear phenomena Example Return to the preceding menu 13 2 5 5 Treatment by phases E Phases Treatment by phases Use of Markovian models specific to given phases Availability Contribution of a maintenance preventive action 2000 4000 5000 8000 10000 hours Repair only during the business hours Vector of probability modified between two phases Example Return to the preceding menu 2 6 Formulas of redundancy DIDAC 6 m u as of red u nda n Passive redundancy M N 5 M M among N M elements necessary among N Active the N elements function simultaneously Passive the N M elements in redundancy are used if necessary 5 9 Hot Cold Energy State of the elements redundancy or OFF Time of reconfiguration Time of commutation on one of the elements in re
10. MDT TAT Operational Cost ON redundancy Cost hour hour availability Euros hour Euros LNA chanel PCG or PCD Down converterchanel PCG or PCD Time link 1 Electric box GLOBAL STATION 0 9609 143500 Availability gt Objective 0 96 Bm Optimization of a stock of spares by using GENCAB tool Criterion of cost with respect of a constraint of availability 61 Active redundancy 1 2 3 N M N M 1 N M 2 N M 3 N M 4 N M 5 N M 2 2 1 N M 2 2 N M 2 3 2 4 N M 2 S 1 1 N M 2 S 1 OK 1 Loss 1 unit 2 N 1 Aon S Aorr Loss 2 units 3 1 MDT 4 Loss N M 1 units N M M 1 Aon S Xorr Loss N M units N M 1 M Aon 2 1 MDT 4 S AorF Indisponible N M 2 s 1 MDT 4 1 spare N M 3 S 1 oFF Loss 1 unit 1 spare N M 4 N 1 Aon 1 MDT S 1 Aoer Loss 2 units 1 spare N M 5 1 MDT 4 Loss N M 1 units 1 spare N M 2 2 2 M 1 Aon Loss N M units 1 spare N M 2 2 1 M Aon 2 Indisponible 1 spare N M 2 2 2 spares N M 2 3 Loss 1 unit 2 spares N M 2 4 Loss N M 1 units S spares N M 2 S 1 2 M 1 A on Loss N M units S spares N M 2 S 1 1 Loss Syst me N M 2 S 1 1 TAT MDT 3 M 2 1 Reliability Availability 1 Final Loss of the system so less M units active Active M N N 4 2 Reliability Availability 2 or 3 Temporary unavailability so less M units act
11. Rate of transition DIDAC 2 Rate of transition f natataneods pata haud Att lim x xo P KO with teat OK with t At Reliability R tedt Rit Rit a t dt Rit dt RW dt R6 2 A c dc and R e jf is constant in h or fits 10 h MD MTTF R dt 1 2 0 Service life Early failures Errors of design Occasional failures Failures of wear production deterioration corrosion Ht Rate of repair H constant practical but without physical significance Example Return to the menu 2 3 Markovian process EYE Markovian process Modeling of dynamic systems per graph or matrix Active redundancy 2 2 a Regrouping of 2 equivalent states Graph of Markov o7rct7 s DOD Matrix of Markov Markovian processes stochastic process without memory 2 constant Example Return to the menu E D Processus DAR Step of modeling 1 Identification of the states with possible regrouping of equivalent states 2 Characterization of the transitions T failure rate 0 1 41 0 00001 Aon 10 z 42 0 00002 on the request pl 0 05 p2 0 02 Cold Redundancy commutated by a switch Availability 0 999981009 States 1 2 3 4 5 1 2 3 4 5 OK switch on 1 1 Marr 0 000009 0000002 0 000001 switch 2 2 Morr 1 D 42 2 B 0 000001 0 000018 0 000002 Lo
12. S Aorr M Aon 1 1 MDT 4 S Agrr M Aon 1 1 MDT 4 5 1 S 1 Aorr N M 1 Aorr 1 MDT 8 1 1 Treconf 1 MDT 4 1 Treconf M oN oFF i ore 1 Treconf M Aon 2 M Xoy 1 M Aoy 2 M on 1 z M on N M Aorr 1 Treconf 1 Treconf M oN oFF M on ore 1 Treconf M oN M on 1 TAT 3 65 OPERATING LICENCE AGREEMENT OF SUPERCAB SOFTWARE PACKAGE ARTICLE 1 SUBJECT The purpose of this Agreement is to define the conditions in which the CAB INNOVATION Company grants the customer with a non transferable non exclusive and personal right to use the software package referred to as SUPERCAB and whose features are specified in user s manual ARTICLE 2 SCOPE OF THE OPERATING RIGHT The customer may use the software package on one single computer and on a second one provided that the second computer does not operate at the same time as the first one The customer can only have one software package copy maintained in a safe place as a backup copy If this license is regarding a performance on site the customer may install the package software on a server while scrupulously complying with purchase conditions stated on specific conditions especially defining the maximum number of users authorized to use the software package from their terminal and the maximum number of users authorized to use it simultaneously The customer is therefo
13. So the user will have to gather manually the features of some block elements to reduce the formula size 23 Rates entered by the program in the line defining the type of block redundancy second and fifth column correspond to rates at active and passive condition of relevant chains For evaluation of architecture the tool uses formulas of redundancy and Markovian treatments whose limitations are specified in the corresponding chapters For the treatment of the logical expressions of level higher than that of the block table and N 1 level it uses the following formula of probability in the shape of macro function of the spreadsheet PROBA A A B C Probability of A Probability of B Probability of C The first argument is the logical expression between quotation marks and the following corresponds to the value of the various probabilities in the alphabetical order By defect the results displayed at bottom of columns correspond to products of intermediate results shown in relevant columns 3 3 Generating Block Diagram Automatically The program automatically generates the Block Diagram as soon as the user initiates the menu command Block Diagram Such function may be achieved prior or after the reliability availability calculation The Block Diagram may be in line plotted or cut depending on printing format Portrait or Landscape and be subject to a reduction or enlargement rate Used with a posterior version of Exce
14. breach by such party of its obligations despite a notice remaining unresponsive for 15 days and this occurring with no prejudice to damages it could claim and provided that the last paragraph of Article 7 above be enforced At end of this Agreement or in case of termination for whatsoever reason the customer will have to stop using SUPERCAB software package pay all sums remaining due on date of termination and return all elements composing the software package computer programs documentation etc without maintaining any copy of it ARTICLE 10 ROYALTY As a payment for the operating right concession the customer pays CAB INNOVATION an initial royalty the amount of which is determined in specific conditions ARTICLE 11 PROHIBITED TRANFER The customer refrains from transferring the software package operating right granted personally to him by these provisions The customer also abstains from making documentation and supporting material CD even free of charge available to a person not expressly set forth in second paragraph of Article 2 ARTICLE 12 ADDITIONAL SERVICES Any additional services will be subject to an amendment of these provisions possibly through an exchange of letters so as to specify the contents modalities of achievement and the price ARTICLE 13 CORRECTIVE AND PREVENTIVE MAINTENANCE 67 The corrective and preventive maintenance may be subject upon customer s request to a separate Agreement at
15. hb Non pab bs D_Regroup Regrouping of equivalent states Exploitation of symmetries Reliability 0 99600892 2 3 Non failure 1 hathaort Abbo Loss of a or a 2 thCott Loss of b or b 3 Ab c a Loss of a and b or a and b 4 a b 2 c old _ Cold Loss of any redundancy 5 Loss of the system 6 000055 00033 0 0003 0 0085 Shared resource 3 0 0005 0 0083 0 0168 power supply 0 0088 Initial State Available States Return to the preceding menu 2 5 3 Coupling between Markovian treatments and Faults Trees Ell Couplage Coupling between Markovian treatments and Faults Trees Combination static of independent subsets dynamic Availability OR C AND D AND B Treatment of the Markov matrices then logical calculation starting from the results Example Return to the preceding menu 12 D Couplage aag Coupling between Markovian treatments and Faults Trees Mechanism model of wear 1 elo du OK CSI ESSE Up 121 1 3 1 Failure 2 S Trecont MDT Availability AL 1 1 2 10000 1000 15 70 0 99850185 Markov B 1 3 2 3000 20000 30 650 0 98892623 Markov c 2 3 1 200 2000 10 20 50 0 9137931 Markov D Generalized Erlanglaw 100 7 0 96918929 Markov
16. k 4 0 12 k E Mean value s Li 4 8 0 08 un m k Standard deviation T 5 M Jk 004 k kA 4t e uv At e 8 902 Density f x kal SoN O A e e NANG RS Time hr The fictitious state method is implemented on SUPERCAB using different programs we are about to describe below Simple Erlang Law To enable that a transition in a Markovian model follows a simple Erlang law the user enters directly in matrix mean value m in hours of distribution and parameter value k as follows The program will automatically generate the k 1 relevant fictitious states before processing the matrix see example 3 Generating Markovian Models SUPERCAB helps generating Markovian models typical of any transition laws So it is possible to process correctly fault rates relating to mechanical elements bath shaped curve or overcome the combinatory explosion encountered in the study of certain systems by replacing subassembly models by simplified models limited to a few conditions Such functionality is described in chapter 4 9 34 Modelling Entirely A System In order to facilitate the implementation of the fictitious state method SUPERCAB helps user achieve the Markov matrix of a complete system from various matrixes which model its transitions It generates automatically this matrix from a user defined simplified matrix in which the complex transition rates are replaced by references to defined Markov matrixe
17. lower right hand side cells of relevant matrixes separated by If argument T is not defined MARKOV function calculates in steady state at T infinite It calculates MTTF MUT MDT or MTBF if user enters MTTF MUT MDT or MTBF respectively as an argument T instead of a numerical value function may be used in matrix form To do so select a range of cells and press block capital and CTRL keys by entering the function Used in matrix form MARKOV function helps displaying time T probability Pr to be in the combination of states defined by vector STATE or probabilities to Pn if vector STATE is not defined It gives such results for intermediate time values between 0 and T if range of selected cells includes various lines Following examples specify the formalism adopted for such function when used in matrix form Vector STATE 15 defined 2 columns or more 1 column Time Pr Pr og 0 o T Nblines 1 100 0 82142808 200 0 78539121 300 0 76749154 0 76749154 b Vector STATE is not defined 1 column 2 columns P1 P2 0 1 P1 0 75790422 0 13709395 0 75790422 0 74332265 T 0 73803747 0 T Nblines 1 0 74332265 0 15914719 0 73803747 0 17235284 29 n columns P1 P2 Pn 0 o8 Jot Oo _ T Nblines 1 T n columns 1 or more Time P1 P2 Pn 0 o os _ ji on T Nblines 1 100 7 1 0 75790422 0 13709395 0 06352386 0 04147797 2 0
18. 2 M on 1 N M 2 2 1 N M 2 3 1 5 M Aoy N M Aore N M 2 4 1 TAT 5 N M 2 S 1 2 M Aon Aore N M 2 S 1 1 M on N M 2 S 1 1 TAT 3 Passive M N M 2 1 Reliability Availability 1 Final Loss of the system so less M units active N 4 2 Reliability Availability 2 or 3 Temporary unavailability so less M units active 5 2 3 Reliability Availability 2 Unavailability of TAT S 1 duration if S repairers 0 N M 2 S 1 12 Reliability Availability 1 ou Absorbing state 4 i MDT operators with lower or equal to the number of spares available 5 j TAT if j repairers S J MDT Redondancy M N Aoy Aore T Treconf MDT Nb operators S TAT Nb repairers Active passive Reliability Availability 63 Passive redundancy Treconf lt gt 0 OK Reconfiguration Loss 1 unit Reconfiguration Loss 2 units Reconfiguration Loss N M 1 units Reconfiguration Loss N M units Indisponible Reconfiguration Loss 1 unit Reconfiguration Loss 2 units Reconfiguration Loss N M 1 units Reconfiguration Loss N M units Indisponible Reconfiguration Loss 1 unit Reconfiguration Loss N M 1 units Reconfiguration Loss N M units Loss Syst me Redondancy M N Aoy Aore T Treconf MDT Nb operators S TAT Nb repairers Active passive Reliability Availability 1 spare 1 spare 1 spare 1 spare 1 spare 1 spare 2 spares 2 spares
19. 30049 2730497 0 787209028 0 1495686 0 108958 0 2404607 0 781536847 Ines ef 9 H 0 1353353 0 09107 0 2111002 0778511442 y i 01224564 0 075958 0 184787 0772061869 Loss of N Melements N M 1 01108032 063233 01813166 0 768124006 System Loss N M 2 u 0 1002588 0 052552 0 1404722 0 76464024 N 0 090718 0 043609 0 1220328 0 761559026 Reparable passive M N markovian model 0 082085 0 036139 0 1057797 0758834392 Return to the preceding menu hours 2 7 Optimization of system architectures Optimization of system architectures Complex systems can be the subject of Markovian treatments when they can be devided in independent subsets The fast speed of calculations facilitates their optimization possibility of coupling with optimization tools Passive Passive 7 8 2 Bloc 2 Unit 5 off if Unit 2 and Unit 3 ok Example Return to the menu 15 Architecture Optimization of a reception station of satellites Configuration of 24 real or integers parameters in blue Coupling with GENCAB tool Availability gt Objective 9 995 i Return to the preceding menu 16 3 Architecture Assessment SUPERCAB calculates the reliability and availability of one architecture from the features of its components preliminarily entered in one table and draws the block diagram automatically
20. 47 4 9 Approximation of One Law Menu command Approximation of one law helps making a model from a reliability curve probability density fault rates or a time sample The user chooses one type of model among the exponential normal lognormal Weibull generalized Erlang or any Markovian laws He specifies in the last two cases the dimension of relevant Markov matrix lt 10 and in the last case the number of parameters to be considered lt 10 He also specifies the nature of data considered the number of curve points or the size of relevant sample then presses button OK Then the program offers a format in which the user enters his data defines if necessary a Markov matrix by using 10 maximum parameters a to j and possibly initializes model s parameters Then he initializes menu command Processing and defines the number of calculation sequences to be performed Then parameter optimization of model is performed using the method of least error squares in the case of a reliability curve probability density or fault rate or is based on the method of maximum likelihood in the case of a time sample The user may reinitiate the calculation using command Processing as many times as desired after being informed of the evolution of the residual error or likelihood see example 8 Notes Normal and lognormal laws can only be used for modelling probability density curves or time samples due to the absence for su
21. 7 0 74332265 0 15914719 0 04206856 0 0554616 300 7 1 0 73803747 0 17235284 0 02945406 0 06015563 The treatment by Monte Carlo simulation is carried out in a similar way by means of function MARK SIM defined below ZMARK SIM MAT INIT STATES T N With the exception of additional argument N corresponding to the number of steps of simulation the formalism of this macro function is identical to that of the Markov function Its employment with SIMCAB tool makes it possible to introduce various laws of probability into a Markovian model Notes Algorithm of MARKOV function is based on calculation of matrix exponential It allows to process Markov matrixes of dimension lower than 254 80 for thr Excel versions before 2007 with a 10 15 resolution Accuracy of results is guaranteed at 10 13 for any value of T lower than the lowest mean transition duration of system opposite to maximum transition rate Beyond such value the guaranteed accuracy deteriorates proportionally 10 9 for a value of T equal to 10000 times the minimum transition duration and processing duration increases linearly Error value N A is returned if sum of probabilities of vector INIT is not equal to 1 or in case of typing error confusion between dot and decimal point especially in numbers In steady state the error value DIV 0 is returned if Markov matrix consists of various absorbing conditions zero rate line or absorbing loops
22. Cab Innovation 3 rue de la Coquille 31500 Toulouse Tel 33 0 5 61 54 68 08 Fax 33 0 5 61 54 33 32 Mail Contact cabinnovation com Web www cabinnovation com SUPEH CAB Version 14 using Microsoft EXCEL E Reliability Block Diagram 1 Passive 7 8 2 Passive Bloc 2 Reliability 7 availability Markov s Matrix MAT 2 3 4 5 6 7 8 1 i d cnb a 2 dnc b a 3 nd c ba 4 nd cnb a 5 ndnc b a 6 Lbda Lbdb dncnbna 7 ndnenbna 8 Unit 5 off if Unit 2 and Unit 3 ok ho Curves Jol x Reliability Availability Reliability availability amp markovian processing User s Manual FOREWORD The software SUPERCAB BASIC version 4 includes some of the SUPERCAB version 14 features It is not the subject of a specific user manual The copyright law and international conventions protect the SUPERCAB software and its User s Manual Their reproduction or distribution either wholly or partly through any means whatsoever is strictly prohibited Any person who does not comply with such provisions is committing an offence of forgery and is liable to prosecution and can be sentenced under the provisions prescribed by the law The Programming Protection Agency A P P references SUPERCAB at the LD D N Inter Deposit Digital Number index with the following reference IDDN FR 001 070017 00 R P 2000 000 20600 CONTENTS 1 The SUPERCAB Sof
23. S spares S spares 4 i MDT operators with lower or equal to the number of spares available 5 j TAT if j repairers 64 2 3 4 5 2 N M 1 2 N M 2 N M j 1 2 N M e2 2 N M e3 2 N M e4 2 N M e 5 2 N M 6 M Aon N M Aorr S Aorr 1 Treconf M Agy N M 1 Aoer S Aorr N M 1 Aoer 1 MDT S Aorr 1 Treconf S oFF 1 MDT 4 1 Treconf M Agoy AorrE x M on ore 1 Treconf M os 2 M Aon 2 M Aoy 1 TAT s 1 1 1 TAT M oN 1 TAT 1 TAT 1 TAT 1 TAT 1 TAT 1 TAT 1 TAT 5 1 TAT 5 1 TAT 5 2 1 Reliability Availability 1 Final Loss of the system so less M units active 4 2 Reliability Availability 2 or 3 Temporary unavailability so less M units active 2 3 Reliability Availability 2 Unavailability of TAT S 1 duration if S repairers J Treconf 1 Reliability Availability 2 1 ou Absorbing state 2 N M 7 2 N M 2 2 3 2 N M 2 2 2 2 N M 2 2 1 2 N M 2 2 2 2 2 1 2 N M 2 2 2 2 N M 2 2 3 2 N M 2 S 1 3 2 N M 2 S 1 2 2 N M 2 S 1 1 2 N M 2 S 1 S Aorr S Aorr
24. a same block 3 2 Calculating Reliability Availability Menu command Processing allows to calculate the reliability and the availability of the architecture preliminarily entered The user chooses in a dialog box the time unit to be considered hour day or year and quotes the value of time t for which it desires a processing He may possibly request the calculation of the equivalent lambda or the equivalent MTTF for this time value failure rate or MTTF of an element with same reliability at t than considered architecture If he wishes to get a reliability availability graph he should enter various time values in discrete form or define a calculation increment a starting value tmin and a final value tmax The program also helps calculate the asymptotic availability of repairable systems with infinite t together with the MTTF MTBF MUT and MDT parameters of the different subassemblies Acting on the OK button commands the processing Then the program adds at table of calculation columns in which redundancy formulas showed in paragraph 3 4 or probability formulas below defined are entered with their various arguments If architecture consists of logic sets relevant Markovian models are automatically generated in specific sheets see logic analysis program Notes The number of characters per cell being limited an error will be indicated if the size of a block s redundancy formula exceeds spreadsheet capacity
25. ary LAMBDA MU STATES MAT 2 3 4 7 8 1 5 6 CBA 1 bajo a T T OC On A 3 M a CnBnA a wema no BA no Bn e me ma on A w me nb Mw Mu Mua swe 1 11111 42 4 7 Logic Analyzer SUPERCAB generates automatically Markov matrix of a system from logic expressions featuring its behaviour and possibly optimizes the dimension of the latter a With No Optimization MAT 1 2 3 4 5 6 7 8 Ese zr c bna Cold c nb na nc b a nc bna Available states axb c ncnb a nc nb na Dependence relation a x b gt Ac CRT C TENE IEE WERT TERT States in which system is available are defined by a logic expression using operators OR AND x NOT n Any possible dependence relations linking certain transition rate values to system conditions conditional maintenance cold redundancy may be similarly expressed Dimension of matrix generated being 2 the number of elements is limited to 6 2 lt 80 b With Optimization If system elements are not all considered as individually repairable equivalent states may be regrouped by program for optimizing matrix dimension XT nc b a c nb na Cold nc nb na MAT N STATUSES 1 1 1 0 So as to identify the regrouped states to each of them is given the name of group stat
26. bda OSR r Cold No fault 1 eff Lbd r Lbdf 1 eff Lbd Loss of a channel 3 Lbd System loss 3 0 2 5E 06 hr 1 1 3 3 Lbdf 2 5E 07 hr 1 1 0 0000029 0 00000025 r 4E 07 hr 1 2 0 0 0000025 eff2 9096 3 0 0 Reliab t 8000hr 0 997796 INT 1 0 0 STATE 2 1140 53 Example 14 Redunded Communication System Example 11 X Cold redundancy equipment Lambda E Transmitter E Lambda R Receiver R Lambda L Linking L Lambda OFF Lambda Lambda 10 Reconfiguration on E L R in case of loss of L MAT 1 2 3 4 5 6 No fault 1 L 3L 0 E E R R 0 Loss 1 linking 2 L 2L E E L R R L 0 Loss 2 linkings 3 0 0 E E R R L L Loss 1 transmitter or linkings 4 0 R R L L Loss 1 receiver or linkings 5 E E L L All redundancy loss 6 System loss 7 MAT 1 2 3 4 5 6 1 04E 7 0 00E 0 2 64E 7 1 21E 7 0 00E 0 E 2 40E 7 9 60E 8 2 72E 7 1 29E 7 0 00E 0 R 1 10E 7 0 00E 0 0 00E 0 4 73E 7 C 8 00E 8 0 00E 0 2 09E 7 Reliability t 10 years 0 9997083 3 52E 7 54 5 Redundancy Formulas SUPERCAB offers a number of macro functions to calculate the reliability or the availability of a set of elements in redundancy m among n Such functions may be directly selected in the list of functions proposed by spreadsheet by following menu Selection then Insert a function 5 1 Active Redundancy n elements are simultaneously active but only m eleme
27. between conditions During a MTTF calculation the error value DIV 0 is returned if Markov matrix consists of an absorbing condition or loop among the combination of conditions retained 30 Example 1 MAT 1 8 00E 06 8 50E 04 2 a 5 00E 03 3 00E 07 4 INIT 0 8 0 1 0 1 0 STATE 1 0 1 0 Probability 0 75250553 t hr 500 0 95 0 9 T hr PR 0 0 9 A 100 0 82142808 200 0 78539121 0 45 PR P1 300 0 76749154 0 7 400 0 75793647 0 65 500 0 75250553 0 100 200 300 400 500 infinite 0 05858807 MTTF hr 952 399732 MUT hr 206 160984 hr 3312 6611 MTBF hr 3518 82208 MAT 1 8 00E 06 8 50E 04 2 3 5 00E 03 3 00E 07 4 INIT 0 8 0 1 0 1 0 C Cl a Probability 0 73481079 t hr T hr 0 100 200 300 400 500 infinite 500 P1 0 8 0 75790422 0 74332265 0 73803747 0 73589341 0 73481079 0 00211764 P1 P2 0 1 0 13709395 0 15914719 0 17235284 0 18035253 0 18528851 0 94123024 31 0 1 0 06352386 0 04206856 0 02945406 0 02204306 0 01769473 0 05647043 P4 0 0 04147797 0 0554616 0 06015563 0 061711 0 06220596 0 00018169 Example 2 Redondancy 1 among 2 No failure Loss of unit 1 Lbd1 Mu1 Loss of unit 2 System loss Unit2 MAT Lbd2 Mu2 1E 05 hr 1 Lbd2 0 0005 hr 1 Mu1 0 0208 hr 1 MTTR 48 hours 0 00001 0 0005 02083 0 0005 0 0 01389
28. ch laws of an analytical expression of reliability and fault rate In the case of the approximation of a curve by a Markovian model the processing is slightly quicker if time values start from 0 and follow an arithmetic progression In the case of any Markovian model seeking an optimum may prove to be very hard as error shows generally many local minima 48 Example 8 Markovian model generalized Erlang 0 1 1 2 3 4 5 1 a 2 b 3 C 4 d 5 a0 0 009447231 a 0 009187634 0 001432604 0 001253136 0 0 000354237 0 000309561 dO 0 019258912 d 0 019306467 Error 0 031726406 Error 0 027472586 1 05 0 95 Reliability 09 Initial 0 85 0 8 Final 0 75 0 7 0 10 20 30 40 50 Matrix 1 2 3 4 5 0 000309561 0 000309561 0 009187634 0 000309561 0 001253136 0 000309561 Approximation of a reliability curve using a generalized Erlang law 49 4 10 A Few Examples of Markovian Modelling Example 10 Cold redundancy initiated by a switch OKon1 1 exAl A2 1 e xA1 OK on2 2 A1 ex A2 1 2 Loss 1 3 A2 Loss 2 4 Al Unavailable 5 u2 ul e efficient when initiated A1 A2 fault rate in off condition of elements 1 2 50 Example 11 Cross strapping with common resources Example 8 b b ressources Lambda Lambda Lambda Lambda Lambda Lambda a in hot condition a a in cold condition a a 10
29. dundancy Utilisation ratio Characterize ON OFF operation of the active elements Spare 5 additional elements of replacement MDT Mean Down Time Average duration of unavailability detection repair or standard exchange The redundancies can be the subject of analytical simple cases or Markovian treatments Formulas Redundancy M N Xon Xoff T Reliability Availability 4m E Example Return to the menu Markovian models 14 EJ D_Redondance Redundancies Passive Reparable Reliability Availability of redundancies M N M N T 1 1 1 0 3048374 0 996799 0 9986833 0 9988595 0 8187308 0 979414 0 3909783 0 99314729 0 7408182 0 943895 0 3738757 0 982393009 dicen ana a 0 0001 PANEM 5488118 0 756928 0 8865047 0333008083 M 2 Mesa 0 4965853 0 682361 18167165 0 915040381 N 4 2 5 Reparable passive MAN 0 449329 0 607922 7629676 0 897662757 AQFF AQN 10 0 4065697 0 53612 0 7069871 0 881270988 0 0002 E 0 3578794 0 468662 06502853 0 866078485 0 3328711 0 406586 05941186 0 852173543 10000 15000 0 3011942 0 350407 0 5394868 0 839562504 0 2725318 0 300256 0 4871488 0 828200981 0 246597 0 255989 0 4376477 0 818015322 0 2231302 0 217287 0 3913421 0 808916967 02018965 0 18372 0 3484366 0 800811774 0 1826835 0154807 0 3090114 0 793605846 S ii 01652989 01
30. e comprising most of failing elements cnbna cbna cnba cnbna nc nb na nc b na nc nb a nc nb na Repair rates relating to blocks may be subsequently introduced in reduced matrix example repair rates of set ab The number of elements is limited to 26 a to z and that of regrouped states to 80 The operator NOT n cannot be used in logic expressions when optimization is requested 43 Menu command Logic analyzer helps obtaining entry formats see example 7 in which system elements are designated by characters a to f User defines the combination of states being sought i e the in which system is available by entering a logic expression between the states of different elements as follows n a b n c e nf Then he enters the time transition rate values in hr 1 and any possible dependence relations from logic expressions identical to previous one Many dependence relations may affect a same rate and user may insert if necessary additional lines in the table In case of superimposing states between different logic expressions the lowest suffix rate is considered Markov matrix is generated then processed by the program as soon as user initiates menu command Processing Notes Example 7 processed with optimization leads to a matrix of dimension 12 instead of 64 Such dimension may still be reduced to 8 as shown below if logic expressions defining dependence relations are modified for considering the latte
31. e condition is entered in second column 3 The number of series elements or sets 1s entered in the third column 18 4 The type of redundancy to be entered in the fourth column may be Active m n Passive m n Series Logic defined by an expression featuring the system availability or Complex This last type does not lead to any processing but allows the user to complete manually the generated calculation tables and Block Diagrams An active or passive redundancy 1 2 is assumed by default if m n is omitted A probability value can be taken by default by entering it in this column 5 A MTTF OFF equal to MTTF On 10 or a fault rate OFF equal to the fault rate On 10 is assumed by default if the latter is omitted in the fifth column 6 A 100 operation rate is assumed by default if the latter is omitted in the sixth column 7 A reconfiguration time in hour may be entered in the seventh column Parameter k of an Erlang law used for featuring this time may be related with it in parenthesis see MARKOV program 8 MTTR may be entered in the same way in the eighth column The following utilities bar Blocks facilitates such entry It is displayed or disappears using menu command Utilities bar bloks vx b M Meaning of buttons is as follows Adding a series block Initiating this button allows to display the following dialog box enabling to insert a series block in the table or many ident
32. efore it is limited at values of M and N defined by relation N M lt 39 if k 1 1 2 1 M 2 No fault 1 ES 1 Reconfiguration 2 M 1 4 A element loss 3 Reconfiguration 4 Sysemloss 208My2 IT TIT IT lambda A lambda Off tr 1 Treconf if k 1 5 4 Redundancy of Repairable Elements Availability is obtained by following formula REPAIRABLE REDUNDANCY M N lambda_on lambda_off T MDT k MDT mean repair time in hour Notes Such function considers only one single repair officer and is processed by performing following Markov matrix with dimension N M 2 It is therefore limited to values of M and N defined by relation N M lt 78 if k 1 56 u if k 1 5 5 Repairable Redundancy with Reconfiguration Duration In case of failure of an active element the mission is stopped for all reconfiguration duration Elements are repairable Availability is obtained by the following formula ZM AMONG REDUNDANCY M N Lambda on Lambda off T Treconf kr MDT km Treconf reconfiguration meantime in hour kr parameter of the relevant Erlang law MDT mean repair time in hour km parameter of relevant Erlang law Notes Such function is processed by performing the following Markov matrix It is limited to values of M and N defined by relation N M lt 39 if kr and km 1 2 5 pa M 1 ELE M 2 wm No fault
33. em having to respect a certain objective of availability 59 5 6 Active and passive redundancy with stock of spares One second generic formula of redundancy is proposed to treat the active or passive redundancies of type M among N with spares stock of S dimension This one allows evaluating the reliability or the availability of such a redundancy by considering the duration of reconfiguration Treconf the duration of repair per standard exchange MDT and the Turn around Time TAT corresponding to the duration of spare replacement Active M N Passive Ee Redondancy M N AON AOFF T TreconfoMDT Nb operators S TAT Nb repairers Acti ve passive Reliability A vailability T Treconf MDT 1 A 1 AOFF e Same untt of time OFF defect Nb operators e 1 1 operator carries out the standard exchanges in series e 2 N operators carry out the standard exchanges in parallel Nb repairers e 1 1 repairer carries out repairs in series e 2 Nrepairers carry out repairs in parallel Active passive e 1 Active e 2 Passive Reliability A vailability e 1 Final loss of the system so less M active elements e 20u3 Temporary unavailability so less M active elements e 3 Final loss of the system so less M active elements and absence of spare The Markovian models used by this formula are provided in the following pages 60 Example 15 UNITS MTTF Kind of Stock Unit
34. guments and may use the as variable of 1 to n n being the matrix dimension If different relations define rates in a same position of matrix only the last relation is considered The user may command additional dialog boxes if necessary to enter all transition rates When matrix is completely defined initiating the button OK displays a new dialog box used for defining the combination of states being sought states at 1 Then initiating the button OK commands the creation of the personalized function then its storage in the predefined file Notes To validate a personalized function the user may use it on a numerical example and display the relevant Markov matrix using menu command Matrix used In EXCELS personalized function files are displayed in reduced size windows 46 Example 9 Crating a personalized function Redundancy m among n repairable with final absorbing state N M N M 1 N M 2 Markov Matrix 1 2 3 a s Lelementloss 2 p MA N M aya f Log 0 7 j 2 element loss 3 LI _ 31 o Formula REDUNDANCY WITH ABSORBING STATE M N IbdON IbdOFF Mu T Procedure PERSONALIZED FUNCTIONS REPAIRABLE REDUNDANCY WITH ABSORBING STATE N M 2 M bdON N M 1 lbdOFF 1 N M 2
35. he blocks which compose them can be defined by logical expressions 17 3 1 Entering an Architecture Menu command Entering architecture allows getting the following dialog box which helps obtain a blank table as that shown below the last two columns of which are optional used if system is unavailable during certain reconfigurations or if it contains repairable elements ARCHITECTURE ENTRY 2 x Name MTTF hours Failure rate System unavailable during reconfigurations Repairable system V Example of architecture V English language Explanatory comments On a new sheet C In a new workbook From the selected cell OK Cancel 5 MTTF Nb Kind of MTTF Operation Switching MTTR ON redundancy OFF rate delay hour hour hour r 96 hour Elements may be featured by their MTTF in hour or their fault rate in hour or fit hour 15107 9 Table and documents made may be requested in the English or French language and one example of architecture preliminarily entered may be displayed for illustration purposes Then the user enters features of system elements either directly in table or by using the utilities bar Blocks the use of which is specified below The architecture entry table is filled in as follows The name of the element is entered in the first column 2 The MTTF or the fault rate of element in activ
36. iagrams and calculation tables by using specific modelling This dialog box is used as previous ones Result of the table The activation of this button allows to display the following dialog box making it possible to insert in the table a logical expression defining the operation of the system from that of the blocks which make it up by defect the blocks are in series Each block of the table is then identified by a small letter automatically inserted and the expression is built by simple selection in a drop down list of the logical operators OR AND NOT the symbols of bracket and the various blocks The logical expression is recopied in the table with the activation of button OK RESULT OF THE TABLE Enter the logical condition A B C D A F actea b c d e UNITS MTTF Nb Kind of MTTF ON redundancy OFF hour hour a Unit 1 833333 33 5 Unit 2 a 1250000 Unit 3 b 2000000 Unit 4 c 1000000 Unit 5 d 133333 33 b k AND OR c Unit 6 1055966 2 Unit 7 455996 35 Correction Unit 8 27777778 Passive 7 8 e Unio 11363636 Serie 5 a a b c d e 21 PR Structure on the N41 The activation of this button allows to display the following dialog box making it possible to insert on the sheet a logical expression defining the operation of the system in nominal or degraded modes from that the different ones subsystems which make it up each one being defined by
37. ic generation of models Markovian models generated starting from parametric formulas or logical expressions Redundancy M among N formula 1 2 3 2 p e O Redundancy M Aon T gt Loss of 1 element 2 M N M D Loss of 2 element 3 ey dem Loss of N M elements N M 1 Loss of the system N M 2 Structure Available States a oub etc E Relation of dependence c off sia et b nc nb na Available States Return to the preceding menu 10 LOGIC EXPRESSIONS Available states c d e a esc b Lbda 00000007 000000007 Lbdb 00000008 0 00000008 4 Lbdc 0 00000045 e dncnbna Lbdc nend c bna endncnb a 10 0 00000075 ne d cnbna 11 i Lbdd nendncnbna 12 C M CO Cn KWH Lbde 00000009 INIT Lbde STATES Lbdi Lbdi Morr Loss of C or D use of A Lass of E or D and A use of B Return to the preceding menu 2 5 2 Regrouping of equivalent states Cil Regroup Regrouping of equivalent states Ok 1 Loss of c 2 Loss of aor b 3 total loss 4 Available States Calculation of availability Correct regrouping only if the repair duration of the chain ab 1luab is independent of the concerned unit Example Return to the preceding menu 11 DAR a b at Ab Matrix of dimension 4 instead of 23 8 ACog AAtAD Nat
38. ical ones just above the line selected SERIES BLOCK 2 x Unit 9 Number of identical series blocks 2 11000 hr Name MTTF Operation MTTR ON OFF rate hour r 10 hour hour Unit 9 11000 fio By default Fault rate OFF fault rate ON 10 Operation rate 100 OK Cancel S rie 2 19 Adding redundancy blocks Initiating this button allows to display the following dialog box enabling to insert a redundancy block in the table or many identical ones Just above the line selected REDUNDANCY BLOCKS Number of different blocks per chain C Active Passive M F among N e edundancy Unit 7 Unit8 Dur e de reconfiguration hrs MTTR 2000 hrs 5820 hr 10000 off 2856 hr 60 Number MTTF MTTF Operation ON OFF rate eon mn m Mttr 2000h tr 800h Unit 7 5820 10000 By default Fault rate OFF fault rate J 10 Operation rate 100 2 4 Number series redunded sets Passive 7 8 Cancel 9 The considered redundancy may concern a unique block or chain of various blocks as in example above 2 Unit 7 4 Unit 8 the whole in redundancy 7 8 The type of redundancy of the complete chain is then informed in the entry table at next line according to those of relevant Units Adding a logic set Initiating this button allows to display the following dialog box enabling to insert a logic
39. its own table UNITS MTTF Kind of ON redundancy hour 833333 33 Active 1055966 Passive A TOTAL UNITS MTTF Nb Kind of ON redundancy hour ARCHITECTURE ON THE N 1 LEVEL Enter the logical condition A B H C D A F Passive 7 8 B TOTAL UNITS MTTF Kind of ON redundancy hour 11 UM 833333 33 AND OR C TOTAL Correction ARCHITECTURE ON THE N 1 LEVEL Logical condition A B G OK Cancel Each subsystem is then identified by a capital letter automatically inserted and the expression is built again by simple selection in a drop down list The logical expression is recopied in an additional table corresponding to the system N 1 level with the activation of button OK Drawing of the logic structure The activation of this button makes it possible to draw a logic structure in the shape of a block diagram starting from its expression defined in the selected cell the symbol corresponds to the negation of the element concerned a a b c d e 22 Block Diagram The activation of this button has the same effect as the command Block Diagram of the menu Notes Lines may be inserted or deleted in the table provided the character is maintained at the bottom of the latter menu Edit insert or Edit cancel Blank lines may be especially used for separating elements or blocks in the table but they should not be inserted within
40. ium 7 Reconfiguration Loss of 2 unit Reconfiguration M among N redundancy system unavailable during reconfigurations 58 Type 12 1 1 repair officer ul ul ul 0 Type 1 2 n repair officers ul 1 wl uP ul 0 Type 1 23 duration of reparation independent of the number of units in failure ul 0 ul wl ul ul if R False Type s true repairing starts at the beginning of reconfiguration duration Re true Reliability gt and 0 The parameter Na allows to assess the probability of the state of row Na in case of the mission is stopped during reconfiguration or not The Nb parameter allows per example to calculate the reliability or the availability of a set of units in redundancy M among N with a stock S of spares as shown in the following figure M Number of units necessary for the mission 9 0 Global number of units N D MDT MDT 8 a S Stock of spares MDT 1 M among N with a stock S of spares The system is available during the Nb N M S first reconfiguration duration but unavailable during the following ones Example 15 of the following page uses this formula of redundancy within the framework of an optimization of batches of spare The coupling of software SUPERCAB and GENCAB indeed makes it possible to automate this type of treatment of minimizing for example the total cost of a syst
41. ive ES 5 2 3 Reliability Availability 2 Unavailability of duration if i repairers N M 2 S 1 12 Reliability Availability 1 ou 3 Absorbing state 4 i MDT operators with lower or equal to the number of spares available 5 if j repairers S 1 Factory Redondancy M N Agy Aore T Treconf MDT Nb operators S TAT Nb repairers Active passive Reliability Availability 3 MDT 3 TAT Passive Redundancy Treconf 0 OK Loss 1 unit Loss 2 units Loss N M 1 units Loss N M units ndisponible Loss 1 unit Loss 2 units Loss N M 1 units Loss N M units ndisponible Loss 1 unit Loss N M 1 units Loss N M units Loss Systeme 1 spare 1 spare 1 spare 1 spare 1 spare 1 spare 2 spares 2 spares S spares S spares 2 3 N M N M 1 N M 2 N M 3 N M 4 N M 5 2 2 2 N M amp 2 2 1 N M 2 2 N M 2 3 N M 2 4 N M 2 S 1 1 N M 2 S 1 1 M Aon N M Aorr S Aorr 2 1 1 MDT S Aore 3 1 MDT 4 S Aorr N M M Aoy Xorr S orF N M 1 M os 2 1 MDT 4 M Aon 1 N M 2 E 1 4 S Aoer N M 3 1 TAT M on N M AoFE S 1 AorF N M 4 1 TAT z M Aon N M 1 Aorr 1 MDT S 1 Xorr N M 5 1 TAT 1 MDT 4 N M 2 2 2 1 TAT M Aont Aorr N M 2 2 1 1 TAT M Aon
42. l to version 4 the tool allows obtaining the block diagram in the form of image Of variable size according to enlarging or reduction ratio s the Block Diagram is then drawn in line and appears on the right of the corresponding table 24 4 Markov Matrix 4 1 Reminder on Markov Matrixes Markovian processes are used in reliability availability analyses for modelling the behaviour of various systems The example below illustrates the way followed Alp System conditions 2 h 4 1 2 3 Xu N No fault d d fault rate 2 Element loss u 2p repair rate 3 System loss Markov graph The analyst identifies different system conditions then defines transition rates between conditions Certain equivalent conditions may be regrouped as the loss of one or another element in this example System may also be represented directly in the form of matrix 12 3 2 0 No fault 1 Lossof onedemett 2 Hu A System loss 3 0 2 u Markov matrix Coefficients Aij of matrix correspond to transition rates of conditions i in line towards conditions j in column Na gt L Such matrix representation is more synthetic than the graph and limits risks of errors forgotten transition Transition rate Aii located the main diagonal are not generally informed 4 27 together with the zero value rates
43. nd Processing Notes n addition to the graph of results and relevant numeric values entered in two columns of sheet detail of calculations is given in the form of one table in depth configuration for each calculation sequence and in width configuration if events occur between two successive sequences Sum of probabilities of vector INIT should be equal to 1 In the event where the processing is requested for a time value corresponding to one forcing obtaining the calculation result before or after the latter is possible 40 Example 6 EVOLUTIONARY SYSTEM Tmax 800 hours Stp 4 Tinit Period Duration O 24 8 Tinit Period Duration 3 24 16 Tinit Period Duration Tinit Period Forcing1 P1 P2 P3 1 2 0 Availability 1 05 1 0 95 0 9 0 85 0 8 0 75 0 7 SP AY LPP PY P en PP ap SL PO EPO hours System only maintained on working hours being subject to a specific periodic maintenance action every 2 weeks on the weekend 41 4 6 Principle of Insertion Menu command Matrix entry helps obtain the matrix of a system of n elements n x 6 each with two possible states proper operation or failure in applying the principle called Insertion which is described below System elements are designated by characters from a to f Table is updated automatically as soon as user enters elements fault or repair rates Such rates may generate fictitious states if necess
44. nd Used Matrix helps displaying following processing the matrix generated by program including all fictitious states 35 Example 3 Simple repairable redundancy Lbd fault Mttr Loss of one element 2 Sysem loss 3 0 00002 1 Lbd 0 00001 hr 1 2 48 4 0 00001 Mttr 48 hours 3 4 gt typical deviation 24 hours INIT 0 8 0 2 0 Availability at T 2000 hours 0 999885 STATE 1 1 0 Program Generated Matrix Mat 0 00002 0 00002 0 0 0 0 0 08334 0 083333 0 0 00001 0 0 08334 0 083333 0 00001 0 0 0 08334 0 083333 0 00001 0 0 0 0 08334 0 00001 0 Init 0 8 0 2 0 0 0 0 State 1 1 1 1 1 0 36 Example 4 Arborescence Availability at T 4000 hours 0 999984971 MAT No fault Loss of D Loss of Reconfigured on E Sytem loss INIT STATE M1 A No loss Repairable deterioration Definitive loss of A M2 B No fault Repairable deterioration Definitive loss of B M3 C No fault Loss of A Loss of C M4 E No fault Repairable deterioration Non repairble deterioration Definitive loss of E 37 Q G N 9E 06 4 06 48 1 2 8E 06 1 2 3 8E 06 3E 06 24 1 6E 06 1 2 3 4 9E 06 4E 06 1E 06 8E 06 5E 06 2 6E 06 Example 5 Elements undergoing wear passively redundant MAT No fault 1 Loss of active element 2 System loss 3 0 96338
45. nts are necessary to ensure mission Reliability is obtained by the following formula by replacing arguments in the indicated order ACTIVE REDONDANCY M N lambda T Ej lambda fault rate of element in hour T mission duration in hour ACTIVE REDONDANCY 5 7 0 00001 5000 0 9965 Note Reliability of such redundancy may be expressed as follows N M i R gt se 0 5 2 Passive Redundancy Only m elements required for ensuring the mission are simultaneously active The reliability is obtained by following formula PASSIVE REDUNDANCY M N lambda on lambda off T lambda on fault rate of element in active condition lambda off fault rate of element in passive condition Note Reliability of such redundancy may be expressed as follows N M tice i l 1 ree Pa V with lambda_off i i l A j 0 55 5 3 Redundancy with Reconfiguration Duration In case of failure of an active element the mission is stopped for all reconfiguration duration Availability is obtained by the following formula REDONDANCY WITH DURATION M N lambda on lambda off T Treconf k Treconf average reconfiguration duration in hour k parameter of an Erlang law random variable of mean value Treconf and with typical deviation Treconf k 1 by default exponential law Notes Such function is processed by performing the following Markov matrix with dimension 2 N M 2 Ther
46. of any disagreement over the interpretation and performance of any whatsoever provision of this Agreement and if parties fail to reach an agreement under an arbitration procedure only Toulouse s Courts will be competent to settle the dispute despite the plurality of defendants or the appeal for guarantee 68
47. ontinuing the customer will return the C D ROM to CAB INNOVATION at CAB INNOVATION s Head Office at his own expense and with registered mail with acknowledgement of receipt by specifying exactly the troubles encountered Within the three months of reception of consignment set forth in preceding paragraph CAB INNOVATION will deliver at its own expense a new product version to the customer This new version will be benefiting of the same guarantee as benefited the first version The customer looses the benefit of the guarantee if he does not comply with the instructions manual recommendations if he performs modifications of configuration set forth in Article 2 above without obtaining a prior written consent from CAB INNOVATION or if he performs modifications additions corrections etc on software package even with the support from a specialized service company without obtaining a prior written consent from CAB INNOVATION ARTICLE 5 PROPERTY RIGHT CAB INNOVATION declares to be holding all the rights provided for by the intellectual property code for SUPERCAB package software and its documentation As this operating right granting generates no property right transfer the customer abstains from any SUPERCAB software package reproduction whether it is wholly or partly carried out whatever the form assumed excepting the number of copies authorized in Article 2 any SUPERCAB software package transcription in any other language
48. p Time MDT Mean Down Time MTBF Mean Time Between Failure MTBF MUT MDT Calculating the probability to be in the various states or a combination of states at time T is performed by using menu command Processing Initiating the latter generates the display of a dialog box in which the user enters the value of T together with a number of intermediate values if he wishes to get the evolution curve between 0 and In addition to Markovian calculation resolution of the equation of Chapman Kolmogorov the treatment can be also carried out in transient state by Monte Carlo simulation The user then defined a number of steps corresponding to the number of simulations to carry out between 0 and T Pressing button OK commands the processing see example 1 If vector STATE is not defined the program calculates the probability to be in state 1 and displays all probabilities from P1 to Pn if a calculation for intermediate values was requested 28 If various matrixes were entered on the same sheet the user may choose one of them by selecting the relevant cell MAT prior to command the processing The probability calculation may also be directly carried out in any cell of calculation sheet by entering the MARKOV function as follows MARKOV MAT INIT STATE T This function meets the formalism of spreadsheet macro functions a separating each arguments and matrix arguments being defined by references of top left hand side and
49. r only when concerned elements are operational A Logic expressions Avail eera MAT Aa Ab Ad c 2 e d c bna Ad Xe a Ac Ad end c b a Ab e d cnbna Ab e d a c b endnc nb a nend c bna Ac ne nd nc nb Ab Ac o Ac Ad Ae 2 Ab AC ONOa b G N o gt m x Ad 44 Example 7 Logic expressions Available states c d e a e c b Mua Loss of C or D gt use of A Lbdb Loss of E or D and A use of B Mub Probability 0 9999768 t hr 5000 Muc Lbdd Mud Lbde Mue 45 4 8 Personalized Functions Menu command Personalized Functions helps creating and using personalized Markovian functions with the same formalism as redundancy functions m among n shown in previous chapters Such functions are stored in files of directory PERSO the opening and closing command of which is directly accessible by pressing button Loading Button Creation allows to create such function either in an already existing file or in a new file The user defines in a dialog box the names of function and each of its arguments maximum 10 the dimension of Markov matrix then the value and position in the latter of the different transition rates see example 9 To do so he enters relations made from different ar
50. r unspecialised designers efficient reliability and availability assessment tools An architecture assessment program allowing to compute the reliability and availability of systems from features of their components draw evolution curves and draw Block Diagram automatically A program for processing Markov matrixes capable of assuming uncontinuous transition rates using the fictitious state method In addition to the availability in transient and steady state this program computes MTTF MUT MDT and MTBF program for assessing evolutionary systems whose state or behaviour is modified at determined times Such program enables to model limited lifetime elements maintenance being periodical or limited to working hours and any other type of determinist evolution A logic analyzer enabling to build automatically the Markov matrix of a complex system from logic expressions featuring its operation n approximation function allowing to obtain a Markovian model or various laws of probabilities typical of whatsoever transition law Such a model may be used for representing the behaviour of an element undergoing a wear curve of bath type fault rate or to replace a subassembly in a complex model so as to avoid a combinatory explosion Reliability availability calculation functions for elements in redundancy M among N assuming reconfiguration or repair times and a program for creating personalized Markovian functions allowing
51. re authorized to perform a number of software package documentation copies equal to the maximum number of users allowed to use it CAB INNOVATION will be in a position to perform inspections either itself or through a specialized entity purposefully authorized by CAB INNOVATION at customer premises to verify if customer has met its requirements number of software package copies used location of such copies etc Parties will agree as regards the practical modalities of performance of such inspections so as to disturb minimally customer s activity ARTICLE 3 DELIVERY INSTALLATION AND RECEPTION The software package and attached supplies will be delivered to the customer on mail reception date The customer installs at its own costs the software package using relevant manual delivered by CAB INNOVATION The customer performs the inventory and shall inform CAB INNOVATION within three working days of the delivery of any apparent nonconformity with respect to the order The customer is liable for any loss or any damage caused to supplies as from the delivery ARTICLE 4 TESTING AND GUARANTEE Guarantee is effective as from the mail delivery date set forth in Article 3 and has a three month validity During the guarantee validity if the customer experiences a software package operation trouble he should inform CAB INNOVATION about it so as to receive any helpful explanations with the purpose of remedying such trouble If the trouble is c
52. refer to discrete values or to contents of sheet cells The box especially enables to give a name to contents of cell selected and possibly recopy such name on sheet at the left hand side of such cell Using names in a formula Initiating this button allows to display the following dialog box which helps use names of predefined variables in a formula to enter in selected sheet FORMULA 21 Names used Mu 0 01 Formula of active cell C2 ponen E f Select one name in list to recopy it in Formula Next Next gt OK Cancel Next lt Next v Selecting one name in the list generates its immediate copy in formula Selected cell may be incremented in any direction so as to successively enter the different rates of matrix Displaying formulas Initiating this button allows to display immediately all formulas entered in sheet instead of the value of their result replacement of by or inversely Displaying deleting square pattern Initiating this button allows to display or delete sheet square pattern 27 4 3 Processing Markov Matrixes SUPERCAB processes Markov matrixes in transient and steady state and calculates the MTTF MUT MDT and MTBF parameters PROBABILITY MAT 1 05 I 0 95 0 9 0 85 Proba co i O 1000 2000 3000 4000 5000 INIT 0 8 0 2 0 0 hours STATE 1 1 0 0 MTTF MDT MUT 0 gt MTTF Mean Time To Failure MUT Mean U
53. s 1 2 3 4 1 a b 2 C d Automatic 1 ibis 2 3 4 3 g generation of 112 113 a b 4 overall matrix 123 a b Mi 112 113 g 121 123 131 132 simplified matrix may contain up to 10 different references to 10 and relevant matrixes preliminarily entered on calculation sheet are entered as an additional argument of Markov function as follows ZMARKOV MAT INIT STATE T M1 M2 M10 This functionality enables to achieve automatically the Markov matrix of a complex system from the matrixes of its constituents Such matrices may possibly be linked each other and represent an arborescence see examples 4 and 5 Notes Relating To Using Fictitious States transition rate tij is replaced by a Markov matrix transition from state J to state 1 is also defined by the latter and relevant cell may remain empty Transition rates placed on the same line as a matrix with transition Mi or as a simple Erlang type transition remain operational for the whole fictitious states generated Using various transition matrixes or simple Erlang type transitions on a same line should be proscribed as it results in conflicts between transitions The Markov program 18 currently limited to the processing of 80 states matrix including the possibly generated fictitious states The error value DIV 0 is returned if mean transition duration m is zero in the expression E m k Menu comma
54. set in the table or many identical ones just above the line selected LOGIC SET HE E Unit 2 Unit 3 i 1500 hr 20000 off 1300 hr 2 Logic condition of operation a b c d 3 example a b c 5 zi Name Number MTTF MTTF Operation Logic MTTR ON OFF rate conditionon hour 15300 hr hour hour 35 OFF condition Unit 2 1500 20000 1 500h Nc ico FT T d units 2560 a b 2560 hr By default fault rate OFF Fault rate ON 10 Operation rate 100 I MTTR global Number of series logic sets Bloc 2 Unit 5 off if Unit 2 and Unit 3 ok 20 The condition of proper operation of the set is defined by a logical expression between its various constituents So in this example the expression a b c d means Unit 2 OK AND Unit 3 OK OR Unit 4 OK OR Unit d OK For each block of logic set a similar expression may be used to define the possible conditions of setting to OFF condition cold redundancy So in this example Unit 5 d is in OFF condition as long as condition a b is verified that is Unit 2 OK AND Unit 3 OK 5 Adding a complex set Initiating this button allows to display the following dialog box enabling to insert a complex set in the table or many identical ones just above the line selected This last type of set does not lead to any specific processing but allows the user to complete manually the generated Block D
55. ss 1 3 22 z 0 00002 Loss 2 4 A1 s s 0 00001 Total loss 5 u2 Initial State Available States _ 1 Return to the preceding menu 2 4 Treatment of the Markov matrices ER DIDAC 4 Treatment of the Markov matrices Probability Vector P P1 t P2 t Pi t Pn t Piitedt i M EN Pi t dt Pi t X Pj t X Pi t P M Matrix of Markov t 4 dt n Equation of Chapmann Kolmogorov System of first order differential equations P M 0 in asymptotic mode Example Return to the menu s D Traitement Treatment of the Markov matrices Probabilities MAT t h 1000 Availability P1 P3 P4 5 _ Availability Transient Asymptotic infit j 0 54408821 0 0043109 2 546E 05 02 76 MTTF hr MUT hr 0 MDT hr Return to the preceding menu MTBF d 2 5 Advantages limitations and palliative solutions Ey DIDAC 5 Fast calculation and accuracy applicable to the very low probability values Delicate Modeling risk of error Combinative explosion 2 states for n elements with 2 states each one Exclusive use of constant Palliative techniques Automatic generation of madels Regrouping of equivalent states Coupling between Markovian treatment and Fault Tree Method of the fictitious states Treatment by phases Return to the preceding menu 2 5 1 Automatic generation of models bs G n Automat
56. tached to these provisions ARTICLE 14 ENTIRETY OF THE AGREEMENT The user s manual defining the SUPERCAB software package features is appended to these provisions The provisions of this Agreement and his Appendix express the entirety of the Agreement entered into between the parties They are prevailing among any proposition exchange of letters preceding its signing up together with any other provision stated in documents exchanged between the parties and relating to the Agreement s subject matter If any whatsoever clause of this Agreement is null and void with respect to a rule of Law or a Law in force it will considered as not being written though not involving the Agreement s nullity ARTICLE 15 ADVERTISING CAB INNOVATION could mention the customer in its business references as a SUPERCAB software package user ARTICLE 16 CONFIDENTIALITY Each party undertakes not to disclose any kind of documents or information about the other party that it would have been informed of on the Agreement s performance and undertakes to have such obligation fulfilled by the persons it is liable for ARTICLE 17 AGREEMENT S LANGUAGE This Agreement is entered into and drawn up in the French language In the event where it is translated into one or more foreign languages only the French text will be deemed authentic in case of any dispute between the parties ARTICLE 18 APPLICABLE LAW DISPUTES The French Law governs this Agreement In the event
57. than that provided for in this Agreement see Appendix any adaptation to use it in other equipment or with other basic software packages de base than those provided for in this Agreement To ensure this property protection the customer undertakes especially to maintain clearly visible any property and copyright specifications that CAB INNOVATION would have affixed on programs supporting material and documentation assume with respect to his staff and any external person any helpful information and prevention step ARTICLE 6 USING SOURCES Any SUPERCAB software package modification transcription and as a general rule any operation requiring the use of sources and their documentation are exclusively reserved for CAB INNOVATION The customer holds the right to get the information required for the software package interoperability with other softwares he is using under the conditions provided for in the intellectual property code In each case an amendment of these provisions will set out the price time limits and general terms of performance thereof ARTICLE 7 LIABILITY The customer is liable for choosing SUPERCAB software package its adequacy with his requirements precautions to be assumed and back up files to be made for his operation his staff qualification as he received from CAB INNOVATION recommendations and information required upon its operating conditions and limits of its performances set forth in user s manual
58. to achieve its own library of various redundancy calculation function 1 2 Installing SUPERCAB on Hard Disk Please comply with instructions shown in CD ROM 1 3 To Start SUPERCAB In EXCEL open SUPERCAB XLA file Software s functionalities are then accessible using menu Dependability spreadsheet functionalities remaining always available pO ee rec 1 I UE T UE m eon R vision Affiche Help Teachware Other menus Utilities bars Architecture entry Block diagram Matrix entry Matrix used Evolutionary system Logic analyzer Approximation of one law Personalized Functions PROCESSING Menu on Excel versions prior to 2007 A help and a teachware are proposed in the menu 2 Teachware The teachware presents Markovian process by means of various boards and demonstrations 2 1 Basic concepts EN DIDAC 1 Basic concepts MTTF Mean Time To Failure average duration before the first failure MUT Mean Up Time average duration of correct operation MDT Mean Down Time average duration of unavailability MTTR Mean Time To Repare average duration of repair MTTR MDT MTBF Mean Time Between Failure average duration between two failures Density of failure Function of distribution P KO with t Ff fJdt 1 R Reliability P OK with t o d MTTF tf t dt R dt ERHI MTTF 0 0 0 0 Return to the menu 2 2
59. tware 1 1 General Presentation 1 2 Installing SUPERCAB on Hard Disk 1 3 Starting SUPERCAB 2 Teachware 2 1 Basic concepts 2 2 Rate of transition 2 3 Markovian process 2 4 Treatment of the Markov matrices 2 5 Advantages limitations and palliative solutions 2 5 Automatic generation of models 2 5 2 Regrouping of equivalent states 2 5 3 Coupling between Markovian treatments and Faults Trees 2 5 4 Method of the fictitious states 2 5 5 Treatment by phases 2 6 Formulas of redundancy 2 7 Optimization of system architectures 3 Assessing System Architecture 3 Entering an Architecture 3 2 Calculating Reliability Availability 3 3 Generating Block Diagram Automatically 4 Markovian Processing 4 Reminder on Markov Matrixes 4 2 Entering an Markov Matrix 4 3 Processing Markov Matrixes 4 4 Fictitious States Method 4 5 Evolutionary System 4 6 Principle of Insertion 4 7 Logical Analyzer 4 8 Personalized Functions 4 9 Approximation of One Law 4 10 A Few Examples of Markovian Modelling 5 Redundancy Formulas 5 1 Active Redundancy 5 2 Passive Redundancy 5 3 Redundancy With Reconfiguration Duration 5 4 Redundancy of Repairable Elements 5 5 Redundancy Repairable With Reconfiguration Duration 5 6 Global Redundancy 5 7 Active and passive redundancy with stock of spares Licence agreement 1 The SUPERCAB Software 1 1 General Presentation SUPERCAB makes available to users whether they are reliability experts o
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