{\rtf1\ansi \deff0\deflang1024{\fonttbl{\f0\froman Times New Roman;}{\f1\froman Symbol;}{\f2\fswiss Arial;}{\f3\froman Times;}{\f4\fswiss Helvetica;}{\f5\fswiss Chicago;}{\f6\fmodern Courier;}{\f7\fscript Script;} {\f8\fdecor London;}{\f9\fdecor Athens;}{\f10\fdecor San Francisco;}{\f11\fnil Cairo;}{\f12\fnil LosAngeles;}{\f13\froman Palatino;}{\f14\fnil Mathématiques;}{\f15\fnil Gregorian;}{\f16\fnil MT Extra;}}{\colortbl;\red0\green0\blue0;\red0\green0\blue255; \red0\green255\blue255;\red0\green255\blue0;\red255\green0\blue255;\red255\green0\blue0;\red255\green255\blue0;\red255\green255\blue255;\red0\green0\blue127;\red0\green127\blue127;\red0\green127\blue0;\red127\green0\blue127;\red127\green0\blue0; \red127\green127\blue0;\red127\green127\blue127;\red192\green192\blue192;}{\stylesheet{\s244 \f3\fs16\cf1\up6\lang1033 \sbasedon0\snext0 footnote reference;}{\s245 \f3\fs20\cf1\lang1033 \sbasedon0\snext245 footnote text;}{\f3\cf1\lang1033 \snext0 Normal;} {\s2 \f3\cf1\lang1033 \sbasedon0\snext0 page number;}}{\info{\creatim\yr1994\mo6\dy29\hr10\min44}{\printim\yr1994\mo6\dy22\hr17\min6}{\version1}{\edmins0}{\nofpages2}{\nofwords65922}{\nofchars67345}{\vern16433}} \paperw11880\paperh16800\margl1134\margr851\margt1417\margb1417\gutter0 \deftab709\widowctrl\ftnbj\ftnrestart\hyphhotz0 \sectd \linex0\headery1077\footery1077\colsx709\endnhere \trowd \trgaph80\trleft-80 \cellx5012\cellx9253\pard\plain \intbl \f3\cf1\lang1033 \par \pard \qc\intbl {\b D}{\b \'e9}{\b partement de Math}{\b \'e9}{\b matiques - Informatique}{\b \par }{\b Facult}{\b \'e9}{\b des Sciences de Luminy, Case 901}{\b \par }{\b 163, Avenue de Luminy}{\b \par }{\b 13288 MARSEILLE}{\b CEDEX 9. FRANCE}{\b \par }{\b __________}{\b \par }{\b \par }Jean MARION \par " : (0) - 91 26 90 80 \par FAX : (0) - 91 26 93 56\cell {\b\fs28 D.E.S.S. des Math}{\b\fs28 \'e9}{\b\fs28 matiques de l'Ing}{\b\fs28 \'e9}{\b\fs28 nieur}{\b\fs28 \par }{\b\fs28 -------------------------}{\b\fs28 \par }{\b\fs28 Corrig}{\b\fs28 \'e9}{\b\fs28 (succint) de l'}{\b\fs28 \'e9}{\b\fs28 preuve"FIABILIT}{\b\fs28 \'c9}{\b\fs28 "}{\b\fs28 \par }{\b\fs28 \par }{\b\fs28 Session normale 93-94}\cell \pard \intbl \row \pard {\b\fs28 \par }{\b\fs28\ul EXERCICE 1}{\b\fs28 . }{\b\fs28 \par }\pard \sb240 Il r\'e9sulte des d\'e9finitions que la fonction de d\'e9faillance F et la fiabilit\'e9{\fs16\up6 \chftn {\footnote \pard\plain \s245 \f3\fs20\cf1\lang1033 {\fs16\up6 \chftn }Ceci est une note en bas de page.}} R sont donn\'e9es pour tout r \'e9el t _ 0 par : \par \pard \qc F(t{\fs18 }) ={\field{\*\fldinst eq \\i( {\fs20 0,}{\fs20 }t, \\f(2,_(1{\fs20 }+ {\fs18\up6 s2))} ds)}{\fldrslt }} = {\field{\*\fldinst eq \\f(2,_)}{\fldrslt }} Arctg t , et R(t{\fs18 }) = 1 - F(t{\fs18 }) = 1 - {\field{\*\fldinst eq \\f(2,_)}{\fldrslt }} Arctg t \par \pard de sorte que le taux instantan\'e9 de d\'e9faillance{\f1 l }est donn\'e9 par {\f1 l}(t{\fs18 }) ={\fs28 - }{\field{\*\fldinst eq \\f((1 - \\f(2,_) Arctg t {\fs28 )',1} - \\f(2,_) Arctg t )}{\fldrslt }} soit : \par \pard \qc {\f1 l}(t{\fs18 }) = {\field{\*\fldinst eq \\f(2,_)}{\fldrslt }} .{\field{\*\fldinst eq \\f(1,(1 - \\f(2,_) Arctg t )(1 + {\fs18\up6 t2))}}{\fldrslt }} \par \pard \qj Par ailleurs un calcul \'e9l\'e9mentaire donne pour le MTTF {\f1 q} : \par \pard \qc {\f1 q} = {\field{\*\fldinst eq \\i( {\fs20 }0, {\fs20 +_,} t.f(t) dt)}{\fldrslt }} = {\field{\*\fldinst eq \\f(1,_)}{\fldrslt }}{\field{\*\fldinst eq \\i( {\fs20 }{\fs20 0,}{\fs20 }{\fs20 +_,}{\fs20 }{\fs20 \\f(2t,1}{\fs20 }{\fs20 +}{ \fs20 }{\fs20\up6 t2)}{\fs20 })}{\fldrslt }} = + _. \par __________________________________________________________________________________ \par \pard \sb240 {\b\fs28\ul EXERCICE 2}{\b\fs28 .}{\b\fs28 \par }\pard \qj {\b\fs20 {\pict\wmetafile8\picw10795\pich4445\picwgoal6120\pichgoal2520 \piccropl-2220\piccropr-2220 0100090000031e0500000c001c0000000000050000000b0200000000050000000c027e0032010500000031020100000008000000fa020500000000000000000007000000fc02010000000000000007000000fc02000000000000000007000000fc020000ffffff00000008000000fa020500000000000000000007000000fc 02010000000000000007000000fc0201000000000000001c000000fb0210000000000000009001000000000000000048656c7600008c900000000100000000ffffffff0000408108e640811eaa000008000000fa020500000000000000000007000000fc0201000000000000001c000000fb02100000000000000090010000 00000000000048656c760037d080002f48e4a03727ec002f48e40000408108e640811eaa000007000000fc020100000000000000040000002d010800040000002d010900040000002d010a00040000002d010b00030000001e00040000002d010400040000002d010500040000002d010600040000002d010700040000002d 010000040000002d0101000400000002010200040000002e011800040000002d010800040000002d010900040000002d010a00040000002d010b00030000001e00040000002d010400040000002d010500040000002d010600040000002d010700040000002701ffff040000002d010400040000002d010500040000002d01 0600040000002d01070005000000140200000000040000002d010800040000002d010900040000002d010a00040000002d010b00030000001e00040000002d010400040000002d010500040000002d010600040000002d0107000700000016047e00320100000000040000002d010800040000002d010900040000002d010a 00040000002d010b00030000001e00040000002d010400040000002d010500040000002d010600040000002d010700040000002d01080004000000f001040008000000fa0206000200020000000000040000002d010400040000002d0101000400000004010d00050000000102ffffff00070000001804220057000e004000 040000002d010a0004000000f00107001c000000fb0211000000000000009001000000000000000053796d626f6c006efffe2f0c4ebaf9dc1c00302efffe48c07203b280508f6f08040000002d010700040000000201010005000000090200000000070000002105020057201c00470004000000020102000500000014021c 00450005000000130238001d00040000002d0101000400000004010d00070000001b044c001c0039000500040000002d010a0004000000f00107001c000000fb020c0000000000000090010000000000000000546d7320526d6e00800000000000000000000000000000000000004800000048040000002d01070004000000 0201010005000000090200000000070000002105020072202b0027000400000002010200040000002d010a0004000000f00107001c000000fb020b0000000000000090010000000000000000546d7320526d6e0000393c24408000390000408108e640811eaa000000000000040000002d0107000400000002010100050000 000902000000000700000021050200532045000c00040000000201020004000000020101000500000009020000000007000000210502003120490010000400000002010200040000002d0101000400000004010d000700000018041d00e1000900ca00040000002d010a0004000000f00107001c000000fb02110000000000 00009001000000000000000053796d626f6c000000393c24408000390000408108e640811eaa000000000000040000002d010700040000000201010005000000090200000000070000002105020057201600d10004000000020102000500000014021b00c9000500000013023700a100040000002d0101000400000004010d 00070000001b044a00a00037008900040000002d010a0004000000f00107001c000000fb020c0000000000000090010000000000000000546d7320526d6e00800000000000000000000000000000000000004800000048040000002d010700040000000201010005000000090200000000070000002105020072202300ae00 0400000002010200040000002d010a0004000000f00107001c000000fb020b0000000000000090010000000000000000546d7320526d6e0000393c2418fa000008c3408108e640811eaa000000000000040000002d0107000400000002010100050000000902000000000700000021050200532042008d0004000000020102 0004000000020101000500000009020000000007000000210502003120460091000400000002010200040000002d0101000400000004010d00070000001b044f0028013c0011010500000014021800e0000500000013023c001101040000002d010a0004000000f00107001c000000fb020c00000000000000900100000000 00000000546d7320526d6e004bda0039308800393056f098000000000000001c00390002040000002d010700040000000201010005000000090200000000070000002105020072202500f7000400000002010200040000002d010a0004000000f00107001c000000fb020c0000000000000090010000000000000000546d73 20526d6e0000393c24408000390000408108e640811eaa000000000000040000002d01070004000000020101000500000009020000000007000000210502005320480014010400000002010200040000000201010005000000090200000000070000002105020032208600ce00040000000201020004000000020101000500 0000090200000000070000002105020032204c001b0104000000020102000500000014024300a00005000000130243001301040000002d010a0004000000f00107001c000000fb0211000000000000009001000000000000000053796d626f6c00807b1e0039306a00008d0c50f1000000004acc0000906c4080040000002d 010700040000000201010005000000090200000000070000002105020072205200c900040000000201020004000000020101000500000009020000000007000000210502004c20630014000400000002010200040000002d010a0004000000f00107001c000000fb020b000000000000009001000000000000000053796d62 6f6c000000393c24000000000000408108e640811eaa000000000000040000002d0107000400000002010100050000000902000000000700000021050200312066001c00040000000201020004000000020101000500000009020000000007000000210502004c206100bd0004000000020102000400000002010100050000 00090200000000070000002105020032206800c4000400000002010200040000002701ffff040000002d010400040000002d010500040000002d010600040000002d01070005000000140200000000040000002701ffff040000002d010400040000002d010500040000002d010600040000002d0107000500000014020000 0000040000002701ffff040000002d010400040000002d010500040000002d010600040000002d01070005000000140200000000030000000000}}{\b\fs28 }{\b\fs28 \par }{\b 1) }Le calcul de R{\fs20\dn4 1} et R{\fs20\dn4 2 }est imm\'e9diat{\fs16\up6 \chftn {\footnote \pard\plain \s245 \f3\fs20\cf1\lang1033 {\fs16\up6 \chftn }Et en voici une deuxieme.}} ; on trouve : \par \pard \qc\sb240 {\field{\*\fldinst eq \\x(R1(t{\fs18 }) = r(t{\fs18 }))}{\fldrslt }} et {\field{\*\fldinst eq \\x(R2(t{\fs18 }) = {\fs18\up6 r2(t}{\fs18 }) + 2r(t{\fs18 }).r(t{\fs18 }).(1 - r(t{\fs18 }{\fs28 )))}}{\fldrslt }} \par \pard \qj\sb240 \par {\fs20 {\pict\wmetafile8\picw11253\pich5750\picwgoal6380\pichgoal3260 010009000003140600000c001c0000000000050000000b0200000000050000000c02a3003f010500000031020100000008000000fa020500000000000000000007000000fc02010000000000000007000000fc02000000000000000007000000fc020000ffffff00000008000000fa020500000000000000000007000000fc 02010000000000000007000000fc0201000000000000001c000000fb0210000000000000009001000000000000000048656c7600b44080a4a040814ae20000403cfcf55cd1408108e640811eaa000008000000fa020500000000000000000007000000fc0201000000000000001c000000fb02100000000000000090010000 00000000000048656c760037d044002f48e4a037296c002f48e40000408108e640811eaa000007000000fc020100000000000000040000002d010800040000002d010900040000002d010a00040000002d010b00030000001e00040000002d010400040000002d010500040000002d010600040000002d010700040000002d 010000040000002d0101000400000002010200040000002e011800040000002d010800040000002d010900040000002d010a00040000002d010b00030000001e00040000002d010400040000002d010500040000002d010600040000002d010700040000002701ffff040000002d010400040000002d010500040000002d01 0600040000002d01070005000000140200000000040000002d010800040000002d010900040000002d010a00040000002d010b00030000001e00040000002d010400040000002d010500040000002d010600040000002d010700070000001604a3003f0100000000040000002d010800040000002d010900040000002d010a 00040000002d010b00030000001e00040000002d010400040000002d010500040000002d010600040000002d010700040000002d01080004000000f001040008000000fa0206000100010000000000040000002d010400040000002d0101000400000004010d00050000000102ffffff000700000018042900b1000c009100 040000002d010a0004000000f00107001c000000fb0211000000000000009001000000000000000053796d626f6c006efffe2f0c4ebaf9dc1c00302efffe48c07203b280508f6f08040000002d010700040000000201010005000000090200000000070000002105020057201e009b000400000002010200040000002d0101 000400000004010d00070000001b047100180060000700040000002d01080004000000f001040008000000fa0206000200020000000000040000002d0104000500000014022200940005000000130261001900040000002d01080004000000f001040008000000fa0206000100010000000000040000002d01040004000000 2d0101000400000004010d00070000001b0471004a005f003900040000002d01080004000000f001040008000000fa0206000200020000000000040000002d0104000500000014026800190005000000130268003a0005000000140268004b0005000000130268005d0005000000140268005e0005000000130268008f0005 000000140268008f0005000000130268009f00040000002d01080004000000f001040008000000fa0206000100010000000000040000002d010400040000002d0101000400000004010d00070000001b047100b20060009f00040000002d01080004000000f001040008000000fa0206000200020000000000040000002d01 04000500000014026800b2000500000013026800c300040000002d01080004000000f001040008000000fa0206000100010000000000040000002d010400040000002d0101000400000004010d00070000001b047000d6005e00c300040000002d01080004000000f001040008000000fa0206000200020000000000040000 002d01040005000000140226009a00050000001302600044000500000014022800a3000500000013026100a3000500000014022500a5000500000013025f00c50005000000140228009e000500000013024a007a000500000014022900a20005000000130252008e00040000002d01080004000000f001040008000000fa02 06000100010000000000040000002d010400040000002d0101000400000004010d00070000001b0472003a0160002601040000002d01080004000000f001040008000000fa0206000200020000000000040000002d0104000500000014026800260105000000130268000e0105000000140268000e010500000013026800f2 000500000014026900d5000500000013026900e6000500000014026a00e80005000000130268000d01050000001402a0009d00050000001302a1009d000500000014021e00b20005000000130260002f01040000002d010a0004000000f00107001c000000fb020c0000000000000090010000000000000000546d7320526d 6e004c920039308800393056f098000000000000001c002f4b2c040000002d010700040000000201010005000000090200000000070000002105020053206b0008000400000002010200040000000201010005000000090200000000070000002105020053206b003b00040000000201020004000000020101000500000009 0200000000070000002105020053206d00a0000400000002010200040000000201010005000000090200000000070000002105020053206700c4000400000002010200040000000201010005000000090200000000070000002105020053206c0029010400000002010200040000002d010a0004000000f00107001c000000 fb020b0000000000000090010000000000000000546d7320526d6e0000393c24408000390000408108e640811eaa000000000000040000002d010700040000000201010005000000090200000000070000002105020031206e000e000400000002010200040000000201010005000000090200000000070000002105020032 206d004000040000000201020004000000020101000500000009020000000007000000210502006b206f00a900040000000201020004000000020101000500000009020000000008000000210504006b2b31206f00c600040000000201020004000000020101000500000009020000000007000000210502006e206f002f01 0400000002010200040000002d010a0004000000f00107001c000000fb020c0000000000000090010000000000000000546d7320526d6e0000393c243098000c9ca4408108e640811eaa000000000000040000002d0107000400000002010100050000000902000000000700000021050200722040004d0004000000020102 00040000000201010005000000090200000000070000002105020072203600e7000400000002010200040000000201010005000000090200000000070000002105020072203e00b50004000000020102000400000002010100050000000902000000000700000021050200722043006e000400000002010200040000000201 0100050000000902000000000700000021050200722049007d0004000000020102000400000002010100050000000902000000000700000021050200722048008f000400000002010200040000000201010005000000090200000000070000002105020072204c00a0000400000002010200040000002d010a0004000000f0 0107001c000000fb0211000000000000009001000000000000000053796d626f6c000000393c24408000390000408108e640811eaa000000000000040000002d010700040000000201010005000000090200000000070000002105020072207900210004000000020102000400000002010100050000000902000000000700 00002105020072207a0051000400000002010200040000000201010005000000090200000000070000002105020072207a00b500040000000201020004000000020101000500000009020000000007000000210502007220780016010400000002010200040000000201010005000000090200000000070000002105020072 207a00dd000400000002010200040000002701ffff040000002d010400040000002d010500040000002d010600040000002d01070005000000140200000000040000002701ffff040000002d010400040000002d010500040000002d010600040000002d01070005000000140200000000040000002701ffff040000002d01 0400040000002d010500040000002d010600040000002d01070005000000140200000000030000000000}} \par {\b 2)} L'analyse du syst\'e8me propos\'e9e par l'\'e9nonc\'e9 pour d\'e9crire la fiabilit\'e9 du syst\'e8me {\f1 L}{\fs20\dn4 n}, n > 2, et qu'il est facile de justifier, conduit \'e0 l'obtention de la formule r\'e9cursive : \par \pard \qc\sb240 R{\fs20\dn4 n}(t{\fs18 }){\fs20 = }{\fs28 \{}R{\fs18\dn4 n-1}{\fs20 (t ).}({\fs20 r(t}{\fs18 }{\fs20 ) + }{\f1\fs20 r}{\fs20 (t}{\fs18 }{\fs20 ) - r(t}{\fs18 }{\fs20 ).}{\f1\fs20 r}{\fs20 (t )}){\fs28 \}}{\fs28 }+ {\fs28 \{}{\fs20 r(t ).}{\fs28 (}{\fs20 1 - }{\f1\fs20 r}{\fs20 (t}{\fs18 }{\fs20 )}{\fs28 )}.{\field{\*\fldinst eq \\i\\su(j=1,j=n-2, {\fs18\dn4 Rj(t).r(t)n-j+1)}}{\fldrslt }}{\fs28 \} }+ {\fs28 \{}{\fs20 r(t ).}{\f1\fs20 r}{\fs20 (t )}{\fs18\up6 n-1}.{\fs28 (}{\fs20 1 - r(t )}{\fs28 )}{\fs18\up6 n-1}{\fs28 \}} \par \pard \qj\sb240 {\b 3)} Pour n = 3 avec r(t) = {\f1 r}(t) = e {\fs20\up6 -}{\f1\fs20\up6 l}{\fs20\up6 t,} la formule \'e9tablie en 2) conduit \'e0 : \par \pard \qc\sb240 {\field{\*\fldinst eq \\x(R3(t{\fs18 }) = {\fs20\up6 r3(t}{\fs18 }).[ 7 - 9r(t{\fs18 }) + 4 {\fs18\up6 r2(t}{\fs18 }) - {\fs18\up6 r3(t}{\fs18 })] = {\fs18\up6 e-3lt}{\fs18\up6 }[ 7 - 9e-lt{\fs20\up6 } + 4 {\fs20\up6 e-2lt} - { \fs20\up6 e-3lt}{\fs20\up6 }])}{\fldrslt }} \par \pard \qj\sb240 Dans les "networks" du type \'e9tudi\'e9 les MTTF des connexions sont tr\'e8s grands et donc {\f1 l} est petit ({\f1 l }< 10{\fs20\up6 -4}). A 10{\fs18\up6 -3} pr\'e8s, pour 0 _ t _ 100, 1 _ k _ 3 et on a donc : e{\fs20\up6 -k}{ \f1\fs20\up6 l}{\fs20\up6 t }{\f1 @} 1 - {\field{\*\fldinst eq \\f(kt,10 000)}{\fldrslt }}, et l'on trouve alors que : \par \pard \qc\sb240 {\field{\*\fldinst eq \\x(R3(t{\fs18 }) @ 1 - \\f(2t,1 000) - \\f(3t2,1 000 000) \'e0 10-3 pr\'e8s, 0 _ t _ 100)}{\fldrslt }} \par \par \pard \par \par }