ITEM ToolKit Technical Support Notes

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1 ITEM ToolKt Notes Fault Tree Mathematcs Revew, Ic Mchelle Drve Sute 300 Irve, CA Phoe: Fax: Page 1 of 15 6/1/2016

2 Copyrght, Ic., All Rghts Reserved The Software Product, ay meda, prted materals, ole or electroc documetato, structoal materal, or smlar materals relatg the software are owed by ITEM SOFTWARE ad are protected by copyrght laws ad teratoal copyrght treates as well as other tellectual property laws ad treates. All other matters cludg use ad dstrbuto of the Software Product shall be accordace wth Item Software s SOFTWARE LICENSE AGREEMENT ad/or wth the pror wrtte permsso of (USA) Ic. The copyrght ad the foregog restrctos o the copyrght use exted to all meda whch ths formato may be preserved. Ths gude may ot, whole or part, be coped, photocoped, traslated, or reduced to ay electroc medum or mache-readable form wthout pror coset, wrtg, from USA. The formato ths gude s subject to chage wthout otce ad USA assumes o resposblty for ay errors that may appear ths documet. Item ToolKt s a trademark of USA Ic. All compay ad product ames are the trademarks or regstered trademarks of ther respectve compaes. Item ToolKt Notes Fault Tree Mathematcs Revew Revso JT2 Jue, 2016 Prted the Uted States of Amerca (USA) Ic Page 2 of 15 6/1/2016

3 Fault Tree Mathematcs Revew Ths documet seeks to clarfy the ofte msuderstood elemets of Fault Tree mathematcs. Whle ot a exhaustve revew, t covers the typcal trouble areas. Example Oe Two evets uder a OR gate, both usg the same falure model. Two other evets uder a AND gate, both usg a dfferet falure model. The OR ad AND gates are uder a OR Top Gate. (We are focusg oly o Uavalablty () results these examples.) TOP GATE Gate 1 =0.75 w=0.0 Gate 2 Gate 3 =0.5 w=0.0 =0.25 w=0.0 Evet 1 Evet 2 Evet 3 Evet 4 =0.5 w=0.0 =0.5 =0.5 w=0.0 =0.5 Methods/Models used uatfcato method: Rare (Esary-Proscha s the other opto dscussed later) Fxed models: Model 1 for Evet 1 ad 2, Uavalablty =.25 Model 2 for Evet 3 ad 4, Uavalablty =.5 Aalyss: Reta results at all Gates Cut Sets A cut set s a collecto of basc evets; f all basc evets occur, the top evet wll occur. (Kumamoto/Heley, p. 227) Cut set vew from Gate 2: E1, E2 (two cut sets, each wth oe Evet) Cut set vew from Gate 3: E3:E4 (oe cut set wth two Evets) Page 3 of 15

4 Aalyss Beg by calculatg the Uavalablty of each cut set, usg the umber of Evets each. The, at each Gate level, calculate the System Uavalablty at that pot, workg your way up to the top. Remember though you eed to cosder the vew of the cut sets the Gate has lookg dow the tree. Not just the result at that level. Cut Set Uavalablty: cutset = = (Kumamoto/Heley, p. 397) 1 = Evet Uavalablty = umber of Evets the cut set. Cut set #1 (E1): =.25 (due to oly oe evet the cut set) Cut set #2 (E2): =.25 (same) Cut set #3 (E3:E4): =.5 *.5 =.25 (two evets the cut set) Next, calculate the Uavalablty of the System at each Gate level. System (Gate level) Uavalablty: sys = = 1 cutset (Rare method, K/H, p. 412) = umber of cut sets uder the Gate Gate 2: = =.5 (the Gate sees two cut sets) Gate 3: =.25 (due to oly oe cut set) Now, use the vew of the cut sets from the Top Gate to calculate the overall System Uavalablty. I ths tree, Gate 1 sees all three cut sets. Cut set #1 (E1): =.25 Cut set #2 (E2): =.25 Cut set #3 (E3:E4): =.25 System Uavalablty: sys = = 1 cutset Top Gate 1: = =.75 Iterpretato Note that the mathematcs s completely focused o the cut sets. Addtoally, the values at each Gate level should ot smply be OR d or AND d together eve though, whe usg the Rare method, t looks as f you could. Page 4 of 15

5 Example Oe.b Aother Fault Tree quatfcato method s the Esary-Procha (E-P) equato for evaluatg cut-sets ad values. Ths wdely accepted approach provdes more precse results tha does Rare. It s also the default settg for ITEM Toolkt Fault Tree ad RBD modules. Usg the same tree Example Oe, but rug the Aalyss usg E-P, results are: Top Gate =.5781 Gate 2 =.4375 Gate 3 =.25 The questo rased s: Why does the value for at Gate 2 ad the Top Gate chage, but the value for at Gate 3 does ot? Beg by uderstadg the E-P equato. [ cutset ] = 1 j = m System Uavalablty: = ( 1 ) = of commo evet cutsetj = of cutset j, excludg commo evets m = umber of commo evets all cut sets = umber of cut sets Evaluate at Gate 2: (1-.25) * (1-.25) =.5625 (=2, 2 cut sets), the =.4375 m No eed to do the frot product ( = Evaluate at Gate 3: sys 1 1 (Kumamoto/Heley, p. 390) 1 ) due to o commo evets =.75 (=1, 1 cut set) =.25 No eed to do the frot product due to o commo evets Evaluate at the Top Gate: (1-.25) * (1-.25) * (1-.25) =.4218 (=3, 3 cut sets vewed from the Top Gate) =.5781 No eed to do the frot product due to o commo evets j Page 5 of 15

6 Example Oe Cocluso: You ca ow see the dfferece results betwee the Rare ad Esary-Proscha methods of quatfyg Evets. The table below shows the dffereces: at Gate 2 at Gate 3 at Top Gate Rare Esary- Proscha Whle t may be tempg to just AND or OR the results at each gate level together, t s far more cosstet to perform the proper summato ad product equatos that are actually beg used Toolkt to arrve at results. Page 6 of 15

7 Example Two The subtle chage ths example s that Evet 1 has bee repeated uder Gate 3. Ths chages the cut set vew from the Top Gate. At frst glace, the =.5 at the Top Gate s uexpected. TOP GATE Gate 1 =0.5 w=0.0 Gate 2 Gate 3 =0.5 w=0.0 =0.125 w=0.0 P1 R P1 Evet 1 Evet 2 Evet 3 Evet 1 =0.5 w=0.0 =0.5 Methods/Models used uatfcato method: Rare Fxed models: Model 1 for Evet 1 ad 2, Uavalablty =.25 Model 2 for Evet 3, Uavalablty =.5 Aalyss: Reta results at all Gates Cut Sets The cut set vew from Gate 2: E1, E2 (two cut sets, each wth oe Evet) The cut set vew from Gate 3: E3:E1 (oe cut set wth two Evets) Page 7 of 15

8 Aalyss Cut Set Uavalablty: cutset = = 1 = Evet Uavalablty = umber of Evets the cut set. Cut set #1 (E1): =.25 (due to oly oe evet the cut set) Cut set #2 (E2): =.25 (dtto) Cut set #3 (E3:E1): =.5 *.25 =.125 Next, calculate the Uavalablty of the System at each Gate. System (Gate) Uavalablty: sys = = 1 cutset (Rare method) = umber of cut sets uder the Gate Gate 2: = =.5 (the Gate sees two cut sets) Gate 3: =.125 (due to oly oe cut set) Now, use the vew of the cut sets from the Top Gate to calculate the overall System. I ths tree, Gate 1 oly sees the E1 ad E2 cut sets due to the Repeat of Evet 1. Ths s due to the absorpto rule of mmal cut set aalyss. (Kumamoto/Heley, p. 248) A mmal cut set s such that f ay basc evet s removed, t s o loger a set. A cut set that cotas other sets s ot a mmal cut set. (Kumamoto/Heley, p. 229) The cut set E3:E1 s ot a mmal cut set sce t cludes the cut set E1, due to the Repeat of Evet 1. Therefore t s removed from the cut set aalyss above the level of Gate 3. Cut set #1 (E1): =.25 Cut set #2 (E2): =.25 System Uavalablty: sys = = 1 cutset Top Gate 1: = =.5 Page 8 of 15

9 Iterpretato Be careful of Repeat Evets ad Gates your Fault Trees. Remember that they wll cause cut sets to be removed from the aalyss (mmal cut-set rule), perhaps causg you to doubt the valdty of the results. The AND/OR logc of the tree also has dramatc mpact whe Repeat Evets are preset. Go back ad look at the tree used for ths example: At the Top Gate, perhaps you expected sys = at Gate 2 + at Gate 3. But, as you dscovered va the cut-set mathematcs (Rare method), the at the Top Gate s actually.5. Ths makes you suspcous sce t s also the value for Gate 2. Gate 3 Evets seem to be gored. Look aga at the cut-set vew from the Top Gate. If Evet 1 (E1) happes, the Top Falure happes. Or, f Evet 2 (E2) happes, the Top Falure happes. The OR Gate 2, s allowg ths. If ether E1 or E2 happes, the Top Falure happes. (Not a good thg f oly oe evet ca cause your etre system to fal, ad you have two of those, E1 ad E2!) If Evet 3 (E3) occurs, t has o mpact uless E1 (repeated) also occurs due to the AND Gate 3. The source E1 mmedately causes the Top Falure to occur, so the cut set E3:E1 becomes rrelevat at the Top Gate vew. Cut-set E1 (ad E2) are the sets that mpact the system modeled wth ths tree, ot E3:E1. Returg ow to the results, you ca see how =.5 at the Top Gate makes sese, whch happes to be the same value for Gate 2. I effect, oly the Evets uder Gate 2 have ay mpact o the Top Gate. Those uder Gate 3, the AND Gate, are made rrelevat due to the logc of ths partcular fault tree ad the cut-set mathematcs. Page 9 of 15

10 Example Two.b Now take the dagram ad use the Esary-Proscha method of calculatg the values. Usg the same tree Example Two, but rug the Aalyss usg EP, results are: Top Gate =.4375 Gate 2 =.4375 Gate 3 =.125 [ cutset ] = 1 j = m System Uavalablty: = ( 1 ) = of commo evet cutsetj = of cutset j, excludg commo evets m = umber of commo evets all cut sets = umber of cut sets sys 1 (Esary-Proscha) Evaluate Gate 2: (1-.25) * (1-.25) =.5625 (=2, 2 cut sets) =.4375 No eed to do the frot product due to o commo evets Evaluate Gate 3: 1-.5 =.5 (=1, 1 cut set, 1 commo evet - E1, =.25, removed) 1-.5 =.5.25 *.5 =.125 (due to a sgle commo evet, E1) Evaluate the Top Gate: (1-.25) * (1-.25) =.5625 (=2, oly 2 cut sets vewed from the Top Gate due to the Repeat) =.4375 No eed to do the frot product due to o commo evets Example Two Cocluso Oce aga, we see how the results chage slghtly betwee Rare ad Esary-Proscha. Addtoally, ths example shows the mpact repeated Evets, ad the logc surroudg them has o the results. at Gate 2 at Gate 3 at Top Gate Rare Esary- Proscha j Page 10 of 15

11 Example Three I ths example Evet 1 has bee repeated, but Gate 3 s ow a OR gate. Ths also chages the cut set vew from the Top Gate. TOP GATE Gate 1 =1 w=0.0 Gate 2 Gate 3 =0.5 w=0.0 =0.75 w=0.0 P1 R P1 Evet 1 Evet 2 Evet 3 Evet 1 =0.5 w=0.0 =0.5 Methods/Models used uatfcato method: Rare Fxed models: Model 1 for Evet 1 ad 2, Uavalablty =.25 Model 2 for Evet 3, Uavalablty =.5 Aalyss: Reta results at all Gates Cut Sets The cut set vew from Gate 2: E1, E2 (two cut sets, each wth oe Evet) The cut set vew from Gate 3: E3, E1 (two cut sets, each wth oe Evet) Page 11 of 15

12 Aalyss Cut Set Uavalablty: cutset = = 1 = Evet Uavalablty = umber of Evets the cut set. Cut set #1 (E1): =.25 (due to oly oe evet the cut set) Cut set #2 (E2): =.25 Cut set #3 (E3): =.5 Cut set #4 (E1): =.25 Next, calculate the Uavalablty of the System at each Gate. System (Gate) Uavalablty: sys = = 1 cutset (Rare method) = umber of cut sets uder the Gate Gate 2: = =.5 (the Gate sees two cut sets) Gate 3: = =.75 (the Gate sees two cut sets) Now, use the vew of the cut sets from the Top Gate to calculate the overall System. I ths tree, Gate 1 oly sees the E1, E2, ad E3 cut sets due to the Repeat of Evet 1. Ths s due to the absorpto rule of mmal cut set aalyss. (Kumamoto/Heley, p. 248) The cut set E1 s ot a mmal cut set sce t cludes the cut set E1, due to the Repeat of Evet 1. Therefore t s removed from the cut set aalyss above the level of Gate 3. Cut set #1 (E1): =.25 Cut set #2 (E2): =.25 Cut set #3 (E3): =.5 System Uavalablty: sys = = 1 cutset Top Gate 1: = = 1 Iterpretato Aga, the Repeat of a Evet had a mpact o the cut sets vsble at the Top Gate, but so dd the OR Gate 3. Evet 3 (cut set) ow appears at the Top Gate. Page 12 of 15

13 Example Four Why s the Top Gate = 0? TOP GATE Gate 3 =0.0 w=0.0 TOP GATE Gate 1 =0.0 w=0.0 W orkg House Evet 1 Gate 2 Workg =9.724e-5 w=9.724e-5 =0 Gate 3.1 Gate 5 =9.724e-5 w=9.724e-5 Evet 2 Evet 3 Evet 4 Evet 5 =1.78e-8 w=1.78e-8 r=1.78e-8 =3.07e-8 w=3.07e-8 r=3.07e-8 =9.72e-5 w=9.719e-5 r=9.72e-5 =0.0 w=0.0 r=0.0 Aswer: The Workg House evet (=0, R=1), whe cosdered the cut-set aalyss for Gate 1, s the domat force at ths level the FT. The cut set vew from ths gate s zero cut sets, resultg =0. AND of a 0 results a 0. If however, you chage the Workg House to a Faled House (=1, R=0), the model chages, resultg Gate 1 havg a o-zero value for. Addtoally, the logc/falure models uder Gate 5 eed to be cofrmed as t s provdg =0 results as well. (I the real model, Gate 5 was a Trasfer Gate.) Page 13 of 15

14 Example Fve Workg wth Rate/MTTF models, ad MTBF results ca be cofusg. I partcular, the dfferece betwee the Mea Tme To Repar ad Repar Rate. Rate Model: Falure Rate = 1e-5 (oe falure 100,000 hours) Repar Rate = 0 If the Repar Rate s 0, ths assumes that o repars are beg made, ad oly oe falure wll occur durg the lfetme of the devce. MTBetweeF s the a very large umber, ad does t really exst because there s o tme betwee falures sce oly oe wll happe. MTTF Model: Mea Tme To Falure = 100,000 hours Mea Tme To Repar = 0 If MTTR = 0, ths meas that the repar s happeg stataeously. MTBF = MTTF + MTTR, so MTBF = MTTF ths case. The pot here s that you eed to be careful of the value you assg to the Repar parameter. 0 ca mea ether a very short tme, or a very log tme, depedg upo the model you are usg. Page 14 of 15

15 Example Sx Aother msuderstood area s that of System Urelablty ad Relablty. Commoly, people try to use the followg smple formula: R λt ( t) = e (Kumamoto/Heley, p. 286) Whle t s true for costat falure rates, t s ot applcable for systems, whch by ature, have a umber of falure rates due to the varous compoets that make up the system. It s ot always possble to plug the system lfetme, ad the calculated falure rate back to ths equato ad obta the same value for R(t) that a program lke ITEM Toolkt arrves at. (Kumamoto/Heley, p. 415) Rather, whe workg wth systems, the followg equatos should be used. System Relablty: R( t) = e ( 1 ( t) ) System Urelablty: F( t) = 1 e ( 1 ( t) ) Lookg at these equatos, you ca see how the relablty of a system s based upo the (Uavalablty) of the system, whch from the frst few examples ths documet, you ca see that s based upo cut sets. Page 15 of 15

1-D matrix method. U 4 transmitted. incident U 2. reflected U 1 U 5 U 3 L 2 L 3 L 4. EE 439 matrix method 1

1-D matrix method. U 4 transmitted. incident U 2. reflected U 1 U 5 U 3 L 2 L 3 L 4. EE 439 matrix method 1 -D matrx method We ca expad the smple plae-wave scatterg for -D examples that we ve see to a more versatle matrx approach that ca be used to hadle may terestg -D problems. The basc dea s that we ca break