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11 . - multistate .

" :" multistate .

.

1 . : up ) ( - down ) .( , : up ) ( - down ) .(2 . ) ( , up ) ( , down) ( .3 . n n 1 nx x . , . up ) ( - down ) ( , ) / ( , .4 . , k-out-of-n k n ) n k .(5 . , consecutive k-out-of-n:F k .6 . , n 1 nx x . n . , .) n-out-of-n , consecutive 1-out-of-n:F ( .7 . , n 1 nx x . - n . n , .) 1-out-of-n , consecutive n-out-of-n:F ( .8 . n 1 nx x . ix ip , 1 i n . . ( )1, ,nR p p . - up ) ( . -1 np p . .9 . 1i iq p ) ix ( , 1 i n . . ( ) ( )1 1, , 1 , ,1n nR p p R q q . . . ( ) ( )1 1, , : 1 1 , ,1n nQq q R q q . . , . Q 1 np p . , ( )1, ,nR p p . - ( )1, ,nQq q . . , ( ) Qq - ( ) R p .10 . , 3-out-of-4 , , ( ) ( )3 4 3 441 4 33R p p p p p p| ` + . , ,( ) ( ) ( ) ( )3 42 3 41 1 1 4 1 3 1 6 8 3 Qq R q q q q q q + + .211 . , consecutive 3-out-of-4:F Q : :A - " 1 2 3, , x x x " , B - " 2 3 4, , x x x " , ( )( ) ( ) ( ) ( )1 4 1 2 3 2 3 4 1 2 3 4, , Qq q PA B PA PB PA B q q q q q q q q q q + + .( )2 3 1 4 1 4. q q q q q q + ( ) ( ) ( )( ) ( )( ) ( )1 4 1 4 2 3 1 4 1 4, , 1 1 , ,1 1 1 1 2 1 1 R p p Q p p p p p p p p . .( )( )( )2 3 1 41 1 1 1 . p p p p , ( ) ( )3 4 3 42 2 , 2 R p p p p Qq q q + .12 . , n 1 nx x . :1niiR p) ( ,( )11 1 1niiQ R q . , nR p , ( ) 1 1nQ q .13 . , n 1 nx x . Q :1niiQ q) ( , ( )11 1 1niiR Q p . , nQ q , ( ) 1 1nR p .14 . , - m , iin . 1miiR P, -iP i ) 12 ( , -11 1ini i ijjP Q q ) 13 ( , ( ) ( )1 11 1im niji jR p ,( )1 11 1 1im niji jQ R q , - ,ij ijq p j i .15 . , - m , i in . 1miiQ Q, -iQ i ) 13 ( , -11 1ini i ijjQ P p ) 12 ( , ( ) ( )1 11 1im niji jQ q ,( )1 11 1 1im niji jR Q p , - ,ij ijq p j i .16 . consecutive k-out-of-n:F , .1 111 11, 0( , , ) ( , , ) (1 ) , ( , , )0, 0y k n kn k y m yy m yj kRp kn p Rp kn mp p Rp k jj+ + + + < + '< : - " + " - " ." n " + " - " ." " " , . , 1 n k + " " , 1 k . 1 n kp + , . - 0 n k " " . - y " " , - m " + " " " . " + " , " " - y , . y - 1 1 n k + ; m - 1 y + 1 y k + ) k " " - y .( n m m , .311***** *****elements elements elements n m y myy m n + ++ + ++ +

. . .. ( , , ) (1 )y m yRp kn mp p , y " " , m y " " , " ."17 . , consecutive 4-out-of-6:F , .3 3 43 3 11 1 25 62 2 3 33 4( , 4, 6) ( , 4, 6 ) (1 ) ( , 4, 6 ) (1 )( , 4, 6 ) (1 ) ( , 4, 6 ) (1 )yy m y my m y mm mm mRp p Rp mp p p Rp mp pRp mp p Rp mp p+ + + + ++ + 44122 31 1152 232 2 2 2 31 1 1( , 4, 6 ) (1 )( , 4, 4) (1 ) ( , 4,3) (1 ) ( , 4, 2) (1 ) ;( , 4, 6 ) (1 )( , 4, 3) (1 ) ( , 4, 2) (1 ) ( , 4,1) (1 ) ;( , 4, 6 )mmqmmRp mp pRp p p Rp p p Rp p pRp mp pRp p p Rp p p Rp p pRp mp + + + + . . .. . .63 343 3 2 3 31 1 1(1 )( , 4, 2) (1 ) ( , 4,1) (1 ) ( , 4, 0) (1 ) .mmpRp p p Rp p p Rp p p ' ' + + . . .2 3 4 52 2 8 7 2 . p p p p p + + 18 . consecutive 2-out-of-n:F , .1201( , 2, ) (1 )nj n jjn jRp n p pj+] ] ] + | ` . ,. : - " + " - " ." j " ." - n " + " - " " " . " - 1 j " " " " , 1 j " + " , - " " ) ( ; " 1 n j + , . 1 n jj + | ` . , , " ." j 2 n n , -( )1 2 n + n - .19 . , consecutive 2-out-of-6:F , .36 6 5 2 4 3 307( , 2, 6) (1 ) 6(1 ) 10(1 ) 4(1 )j jjjRp p p p pp p p p pj | ` + + + . ,.420 . k-out-of-n , . ( ) 1n n iii knR p pi| ` . ,. : - X . ( ) ~ , X Bnp , -( ) ( )( ) 1n n n iii k i knR PX k PX i p pi | ` . , .21 . , 4-out-of-6 , :( ) ( ) ( )6 6 24 5 6 4 5 641 15 1 6 1 15 24 10iiinR p p p p p p p p p pi| ` + + + . , multistate .22 . multistate :1 ( " " ) short ( - - ,2 " ( ) " open ( - - ,3 ( , . , :1 " ( " - - ,2 " ( " - - ,3 ( , .23 . n 1 nx x . , " " ) ( ; " " ) ( ; " " . " " , " " , ) .( " ) , " ( . - " " ." - " " ."24 . , multistate k-out-of-n " " k n " " ; " " 1 n k + " ." .) 4 (25 . , multistate consecutive k-out-of-n:F " " k " " ; " " " " , ) ( k . ) 5 (26 . , multistate n 1 nx x . " " n " "; " " " ." ) . 6 (27 . , multistate n 1 nx x . " " - n " " ; " " n " " . ) 7 (28 . multistate n 1 nx x . -1, ,o onq q . " ) " open ( , -1, ,s snq q . " ) " short ( . 5 -oQ " " -sQ " ." , 1 o on oq q q . ,1 s sn sq q q . . 1m o sR Q Q . mR multistate , - R . 1 Q R " - " ) 9 ( .29 . , ,o sQQ Q - R : ( )1, ,o o onQ Qq q . ; ( )1, ,s s snQ Rq q . . , 9 : ( )11 1 , ,1o o onQ R q q . ; ( )11 1 , ,1s s snQ Q q q . .30 . , multistate 3-out-of-4 , , ( )2 3 46 8 3o o o o oQ Qq q q q + ;( )3 44 3s s s sQ Rq q q ;2 3 4 3 41 1 6 8 3 4 3m o s o o o s sR Q Q q q q q q + + .) 10 (31 . , multistate consecutive 3-out-of-4:F( ) ( )1 4 2 3 1 4 1 4, ,o o o o o o o o oQ Qq q q q q q q q + . ; ( ) ( )( )( )1 4 2 3 1 4, , 1 1 1 1s s s s s s sQ Rq q q q qq . . , 3 4 3 42 2 , 2s s s s o o oQ q q q Q q q + -3 4 3 41 1 2 2 2m o s o o s s sR Q Q q q q q q + + . ) 11 (32 . , multistate n 1 nx x . :1ns siiQ q, ( )11 1no oiiQ q . , ns sQ q , ( ) 1 1no oQ q . ) 12 (33 . , multistate n 1 nx x . :1no oiiQ q, ( )11 1ns siiQ q . , no oQ q , ( ) 1 1ns sQ q . ) 13 ( 62 . ) tie sets and cut sets .(1 . n 1 nx x . . ) path set, tie set ( , . , ) " .( ) cut set ( , . , ) " .(2 . , , 3 " , { { { , , , , , , , A D E B D E CE ; 3 " , { { { , , , , , A B C CD E .3 . , k-out-of-n k " , 1 n k + " . nk| ` . , " -1nn k| ` +. , " .4 . , consecutive k-out-of-n:F k " , ) ( k , " .5 . , n 1 nx x . " , { 1, ,nx x . . 1 ) , ( " .6 . , n 1 nx x . 1 " , " , { 1, ,nx x . .7 . -1 mt t . " ) tie sets .( -iT " it ,1 i m . . ( )1miiR P T..8 . -1 lc c . " ) cut sets .( -jC " jc ,1 j l . . " - " ( )1ljjQ P C..9 . : ) ( , " " , .10 . , 2 ,{ { { ( ){ ( ){ ( ), , , , , , , , ,up up up up upR P A D E B D E CE P A D E P B D E + +{ ( ){ { ( ){ { ( ){ { ( ){ { { ( ), , , , , , , ,, , , , , , , ,up up up up upup up up up upP CE P A DE BDE P A DE CEP BDE CE P A DE BDE CE+ + B A C DE 7{ ( ){ ( ){ ( ){ ( ){ ( ){ ( ){ ( )3 3 2 44 4 52 3 4 5, , , , , , , ,, , , , , , , , , , 2 3 ;up up up upp p p pup up upp p pP A DE P BDE P CE P A BDEP A CDE P B CDE P A B CDE p p p p + + + + +. . . .. . . { { { ( ) { ( ) { ( ){ ( ) { { ( ) { { ( ){ { ( ) { { { ( ){ ( ) { ( ){ ( ) { ( )3 2 4, , , , , ,, , , , ,, , , ,, , , , , ,downdown down down downdown downdown down downdown downdown down downdowndown down downqq q qQ P A B C CD E P A B C P CDP E P A B C CD P A B C EP CD E P A B C CD EP A B C P CD P E P A B CD + ++ + + + .. . . { ( ) { ( ) { ( )4 3 52 4 5, , , , , , , , , 2 .down down downq q qP A B CE P CD E P A B CD E q q q q + + +

. . . , R Q , 1 R Q + , ) ( . . ) ( 1 R Q + .11 . , consecutive 3-out-of-4:F 2 " { { 2 2 3 4 1 1 2 3, , , , , c x x x c x x x - 3 " { { { 3 1 4 2 3 1 2, , , t x x t x t x . R - Q ) .( ( ) ( ) ( ) ( ) ( ) ( ) ( )( ) { ( ){ ( ){ ( ){ ( ){ ( ){ ( ){ ( )2 23 3 41 2 3 1 2 3 1 2 1 3 2 31 2 3 2 3 1 4 2 33 41 2 4 1 3 4 1 2 3 4, ,, , , , , , , 2 2 .up up up upp pp pup up upp p pR PT T T PT PT PT PT T PT T PT TPT T T P x P x P x x P x xP x x x P x x x P x x x x p p p + + ++ + + + +. . . .. . . ( ) ( ) ( ) ( ){ ( ) { ( ) { ( )3 3 41 2 1 2 1 23 41 2 3 2 3 4 1 2 3 4, , , , , , , 2 .down down downq q qQ PC C PC PC PC CP x x x P x x x P x x x x q q + + . . . ) ( 1 R Q + .83 . . 1 . n 1 nx x . . up - 1 ,down - 0 . n 1 nx x . ) , ( , ( )1, ,nx x . , .2 . ( )1, ,nX x x . , 2n ) ( , 0 1 ( )X .3 . , , ) , .( : ) ) ( ( .4 . , k-out-of-n k " " 1 nx x . , ( )1, , 1nx x . ) - up ( ; ( )1, , 0nx x . ) - down ( .5 . , consecutive k-out-of-n:F k " " 1 nx x . , ( )1, , 0nx x . ; ( )1, , 1nx x . .6 . , n 1 nx x . : ( ) ( )1 11, , min , ,nn n iix x x x x . . .) - ( )1, , 1nx x . 11nx x . ; ( )1, , 0nx x . .(7 . , n 1 nx x . :( ) ( ) ( )1 11 1, , max , , 1 1n nn n i ii ix x x x x x . . .) - ( )1, , 1nx x . 1 nx x . 1 ; 10nx x . ( )1, , 0nx x . .(8 . ) ( 0 - 1 . :{0,~1,qp .2 1 3 1 2 30 0 0 00 0 1 00 1 0 00 1 1 11 0 0 01 0 1 11 1 0 01 1 1 1x x x ( )( )1 2 31 2 1 2 31 3 2 3 1 2 3.x x xx x x x xx x x x x x x + + 9 ( ) 0 1 E q p p + . ,{0,1 ~1,pq , , ( ) ( ) ( ) 1 1 E E E q .9 . ix ip , 1 i n . , - R ) ( , 1 nx x . - ,0,~1,iiiqxp', -{0,~1,QR . ( ) ( ) ( ) ( ) , , ,i i i iE Q E R Ex q Ex p .10 . , 3 , 1 3 2 3 1 2 3x x x x x x x + :( ) ( ) ( ) ( ) ( )1 3 2 3 1 2 31 3 2 3 1 2 3 1 3 2 3 1 2 3 1 3 2 3 1 2 3pp p p pp pR E Ex x x x x x x Ex x Ex x Ex x x p p pp p pp + + + . . . ) 3 - 1 :(( ) ( ) ( )( )1 2 3 1 2 31 2 31 2 3 1 2 31 2 3 1 2 3 1 2 3 1 2 3 1 3 2 3 1 2 31 0 1 1 1 0 11 1 1 .qp p p q ppp pR P Px x x Px x xPx x x q pp p qp p pp p p pp p