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\par \pard \qc\intbl {\b D\'e9partement de Math\'e9matiques - Informatique
\par }{\b Facult\'e9 des Sciences de Luminy, Case 901
\par }{\b 163, Avenue de Luminy
\par }{\b 13288  MARSEILLE}{\b   CEDEX  9. FRANCE
\par }{\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\'e9matiques de l'Ing\'e9nieur
\par }{\b\fs28 -------------------------
\par }{\b\fs28 Corrig\'e9 (succint) de l'\'e9preuve"FIABILIT\'c9"
\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 . 
\par }\pard \sb240 Il r\'e9sulte des d\'e9finitions que la fonction de d\'e9faillance F et la fiabilit\'e9 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 .
\par }\pard \qj {\b\fs28 {\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  
\par }{\b 1) }Le calcul de R{\fs20\dn4 1} et R{\fs20\dn4 2 }est imm\'e9diat ; 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 {{\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 \{}{\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 }