HAL Id: inria-00073000
https://hal.inria.fr/inria-00073000
Submitted on 24 May 2006
HAL is a multi-disciplinary open access archive for the deposit and dissemination of sci- entific research documents, whether they are pub-
L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non,
MTTF Estimation Using Importance Sampling on Markov Models
Héctor Cancela, Gerardo Rubino, Bruno Tuffin
To cite this version:
Héctor Cancela, Gerardo Rubino, Bruno Tuffin. MTTF Estimation Using Importance Sampling on Markov Models. [Research Report] RR-3672, INRIA. 1999. �inria-00073000�
a p p o r t
d e r e c h e r c h e
INSTITUT NATIONAL DE RECHERCHE EN INFORMATIQUE ET EN AUTOMATIQUE
MTTF estimation using importance sampling on Markov models
H´ector CANCELA, Gerardo RUBINO and Bruno TUFFIN
N ˚ 3672
April 29, 1999
TH `EME 1
#%$ '&("
)+*-,/.10325476984+:!;6<-=>?@21AB2DCE0GFIHJLK8MONPABQRCSJ2DTEQE0VU5HWXWYK8'Z
[\^]_'`IaXb cdef`Dgih-jO`k%efl@efkm]_'`e
nporqist`k%uYom_'qvo
c%giwxw3qvomk(yz`om`{r\^`or{r\^`}|~///b u%wxom vza1//b vLwBgi/`e
5RDxv `ofl{qv_+wx`jefl@efkm`_'eLq@{{hzo!|^q1gvy^gDl@eLhxkm`+om`h^`|@kfl|_5g|@l
kr`{r\z|^qvqvv{gipgor`Dgie gi|^ykf\^`lgior`7qvkm`|om`hxom`ykmq{qv_+wxl( km\\xv\yx`w`|^y-¡
gi¢x k£l¤efk¥g|^y^giomyxe¦§[q¤efkfh^yzl¨km\x`ogD©igigi¢x k£lgi|^yVor`g¢x k£l{r\Bgio¥gi{kr`omefkm{e
ª«giom¬vq1©@gi|_q@yx`e(giom` {qv_+_'qv|xlIh^ef`y¦¯®Yhx`kmq7km\^` et°`gi|xyO{qv_+wx`j- k£lqikf\^`
efl@efkm`_'e-gi|^y+yzh^`(krq km\x`orgiom k£lLqvefl@efkm`_}±gi hxom`e/¢qvkf\'gi|Bgi l@kf{gixefqvhxkfqv|xe²gi|^y
³ {ofh^yx`´et_Lhzgikfqv|9{Dgi|¢3` |^`µ+{`|¶kXqio `©v`||^qv|z¡·or``©igi|¶k¦Xu|¶hx_L¢3`oXqv²©vgom ¡
gi|^{`!or`y-h^{kfqv|ª9qi|@km`+¸¹giomq'km`{r\x|xh^`e \BgD©/`L¢3``|wxorqvw3q/ef`ykrqq1©v`or{qv_'`km\xe
yzµ+{hzk£l»º¼ _'w3qvomkrgi|^{`erg_'wx |^9_`km\^q@yxe7giom`gi_'qv|^km\x`5_q/etk7`µ+{`|¶k¦½[\^`
qv¢zst`{km ©/`qipkm\xeYwBgw`oYe%kmqefhzom©/`l`j-efkf|^+_+wqvofk¥gi|x{`ergi_+wx |^Iem{r\x`_'`e3krq
wxorqvw3q/ef`efqv_'`X_+wxorq1©/`_`|¶kre%gi|^ykrqIyzem{h^eme%qv|Okm\^`oyz ¾`or`|¶k(wxomqvw3`omkf`e¦
¿«ÀÁÂfÃLÄ ÅEv cYgor``©v`|¶k ef _Lhxgikmqv| -©igiofgi|x{`Xor`yzh^{kmqi| 3ª«giom¬vq1©@gi|ªOq-yx`e
yx`w3`|^yxgi¢x k£l+_`gvefhzor`e¦
ÆÈÇEɣʣË1Ì(ɲÍEÎÏÊ£ÐÒÑÓ
ÔRÕÖR×BØ-ÙÚ²ÛÜÝÞß±àfáâäã1àfÝÛ1ÕEå·æ1çæéèrêÛ¶ëíì¥î/ë·ïtðñìéò/ó»îôõöïí÷
Ë õ Ë1ø
ù
×Búpû1üýRÛ/þ3àfÿ·ÿ±ÝÞ ûDàtáDâñçÞÛvåÒèÞ à Ú ÒûDÙ²ÛéÚàfÞéÞ1àmÿtÛiå·èÞtàÛ Ë
î1ò
äÊ
ìõ
Ú ÒûDÙ¯ÛéÚàfÞÞ1àfÿfÛiå·èÞ àÛ tÎ Ë
î@ò
äÊ
ìõ
Unit´e de recherche INRIA Rennes
"!7""! (E7("P & +&("
#Y$'O
®Y`eefl@efkm]_'`e kmom]e {qv_'wz`j-`e |¶kr`om©¶`|x|^`|@kyz`L|^q/estqvhxome ordh^`_I¡
_`|@ky^gi|^e¹¢3`Dgh^{qvhzwyx`Xyxqv_gi |^`e¯kr`{r\z|^qvqvvh^`e`k¹yxqv©v`|¶k(efqvhx©/`|¶kYqi¾3ofo¯hx|^`
_+wqvofk¥gi|¶km`+exor`krd'yx`'qv|x{kmqv|x|^`_`|¶kD¦OnEqvhzoLdkfh^yz`o`hzore{gio¥gi{krdomefkmh^`eyx`
yzetwqv|z¢x kmdE`kRyz`^gi¢x kmdvD`e _q@yx]`eª«gom¬/q1©¶`|^eRefqv|¶k{qv_+_Lhx|^d_`|¶khzkm efde¦
|'o¥giemqv|+yx`g k¥g`¹`k¯yx`g{qi_'wx`j@kmd(yx`%{`e¼efl@efkm]_'`ez`k¼`|'o¥giemqv|+yx`g o¥gé¡
or`kmdXyx`e¹yxd±ggi|^{`e¶`eefqvhxkfqv|xe¹gi|Bgl¶km¶hx`e`k¹gef _Lhxgkmqi| ³ efk¥g|^y^giomy-´Yemqv|¶k
krqvhxkm`e yx`hzj |^`µ+{Dgi{`eE`k __`!wBgiom qie |^qi|¤gwxwx{Dgi¢z`e¦|{`ofk¥gi ||^qv_L¢xom`
yx`kr`{r\x|x¶hx`e²yx`ª9qi|@km`%¸¹giomq ordyzhxergi|¶kpgY©igiomgi|^{`¼qv|¶k²dkmdwxorqiwq/efd`epwqvhzo efhxot¡
_qv|¶km`o({`kmkm` yzµ+{hxkmdvºz`e_'dkf\^q-yz`e(yd{r\Bg|@kf qv|x|^gv/`Ywzorddor`|@kf` emqv|¶kwBgiof_'
`e%wxhxeX`µ+{Dgi{`e¦qv¢zst`{kf²yx`L{`kgomkm{`!`efkyx`w^gvemef`o `|or`©¶h^`L`e wzorq@{dyzde
yñd{r\Bgi|¶km qv|x|Bgvv`wxord dom`|¶km`%`j@etk¥gi|¶kmeyz`wzorqvw3q/em`o+yx`e+gi_'d qvorgikmqv|^e`k+yx`
yzef{hxkm`o%yx`eyz ¾dor`|@km`ewxorqiwxomdkrde(yx`{`e%_'dkf\^q-yz`e¦
Ä
é
Â
!
#"
_Lhxgkmqi|Lyd©vd|^`_`|@kme¼orgior`e/ordyzh^{kmqi|'yz`¹g ©igiomgi|^{`v/_'q-yx]`e
ª«giom¬vq1©@`|^e^_`efhxom`eyx`e$xom`kmd!yx` qv|x{kmqv|x|^`_`|¶k
( ) I+&O¯!
`kh^e{qv|^etyx`o7g9_hxkf ¡£{qv_+wqv|x`|¶k or`wBgiorgi¢x`+etl@efkr`_«¦[\^`h^ef`oL\Bgieyx` x|^`y
(\Bgikgi|qiw`o¥gikfqv|Bg3etk¥gikm`Xexgi|^y5kf\^` ±gv{k¹km\^gik¹km\x`Xefl@efkm`_ e¼om`wBgi o¥gi¢z`Y_'`Dg|^e
km\Bgik!k{Dgi|V{qi_`+¢Bgv{r¬½gk7efh^{r\ getk¥gikm`gikr`oLkm\x`5q-{{hxofor`|^{`qv%g«±gi hxom`vEyzh^`
krq«efqv_'`'or`wBgi om|xO±gv{ k£l |^{ h^yx`y| ¦+*`5giom`'|¶km`or`efkm`y |¤`©igi hBgikm |^Oemqv_'`
efw3`{^{Lyx`w`|^y^gi¢z k£l_`kmom{e%orqv_ g'_'q-yx`Rqi¯kf\^`Letl-etkr`_«¦-,í/.10²yx`|^qikr`eXkf\^`
efk¥gkr`qizkm\x`¼_'q-yx`-gikEkf_'`32i(\^`or`4.106587¹ikm\x`efw3`{^{gikmqv|^e|^y-h^{`g%wBgiofkm kmqi|
qv(km\^`efk¥gkr`efw^gv{`+7 |¶kmq«k£q:9±yze stqv|¶k ;em`kre=<?>! km\^`5ef`k7qiYetk¥gikm`e(\x`or`5kf\^`
efl@efkm`_ e hxw@9Òyx` ©/`of|^ef`om©¶{`Xgie¯etw`{^`y ;zg|^yBA«{qv_+wq/ef`y+¢@l7kf\^q/ef`Yetk¥gikm`e
(\^`om`km\x`etl-etkr`_ eOyzqé(|C9±kf\^`yz` ©/`or`y em`om©¶{`¤yxq@`e|^qvk xk«gi|¶l@_'qvom`kf\^`
efw3`{^{Dgkmqi|^e ;¦
[\x`O_'q/efkI _'w3qvomkrgi|¶k7yx`w`|^y^gi¢x k£l¤_'`kmof{e'gor`<D9±;!km\^`DEF"@GH
&IJ%&'E yx` x|^`y§¢¶lLK3MON.QP 5R>-S@9·gvefefhx_+|x qvo|^etk¥gi|x{`km\^gikkm\^`_'q-yz` e
omom`yzhx{¢z`Lg|^y`om/q-y-{O;ºT9± U;¹km\^`VR9·ª9`Dgi|[ _`7[q+WBgi hxor`X;3yx` x|x`ygve
YNUZX[TS²(\^`om`-Z%[ ekf\^`]\Qqvkf\^` em`k$A^9Ògvemethx_+|^Lkf\Bgik.`_Y5a>b;zkm\^gik
ecZ%[dfehgjikO2lm.105Aon¶º49 U;%kf\^`o!p&'J%&'Egikkm _`q2`hBg¯kmqK3MXNUZX[srt2pSr
giefqgvefefhx_+|^Okm\Bgkq.`_u5s>!º9± ©v;Xkm\x` wbG&IJ%&'Egik!kf_'`2Ryx` x|x`y½¢¶l
KTM%N.1065D>-S¥ºx9±©j;¼km\x`y-efkfom ¢xhxkmqv|y9Òqio(km\^`X_'qv_'`|¶krep;qvEkf\^` o¥gi|xyxqv_ ©igiofgi¢x`]wH
F!zG&G&IJ%&'E!|8{}|j~2·@yx` z|^`y¢¶l
0
0
_ {. 45D>YhY(\^`om`{»e km\x`Y|xyz{gikrqvo
hx|^{kfqv|Oqvkm\x`wzor`y-{Dgkr`'¦
uor`¶hx`|¶kef kmhBgikfqv|Oe%km\Bgk km\x`_q@yx`9km\^`efkrq@{r\Bgvetkm{!wxomq-{`eme$.8;e hxkm`
{qv_+wx`j7g|^y7giom/`b9±kf\Bgik el7$l ;pgi|^y7kf\Bgik km\x`¹±gi`yLetk¥gikm`e¯gior`b!!p/km\^gik e
ZX[ Z%_%(km\O\xv\9wzorqv¢Bg¢x k£l/-(\^`om`Z=_Ye(kf\^`om`kfhxom|9km_'`Xkrq+km\^`|zkmgi3efk¥gkr`
3wxom`ethx_`ykrq5¢3`!kf\^`u9Òqi|xlj;%etk¥gikm`(\^`or`Igi Ekf\^`{qv_'w3qv|^`|¶kre giom`hxw9km\BgikYe
Z%_Ydehgi=kO2Yr|@l.106ds|j~p. 0d|n;¦[\^`Let°`qvpkf\^`!_'q@yx` _gDl_5gi¬v`yz µ+{hx k
qvoX_+w3q/emet¢x`!kmeX`jzgv{k|@hz_`om{gi `©vghBgikfqv|gi|^ykm\^`Lorgiom k£l«qi¼kf\^`L|¶kr`or`efkm |^
`©/`|¶kre{Dgi|yxq5km\^`ergi_'`!( km\g'|Bg©/`7ª9qv|¶kr`I¸¹gomq`efkf_gikmqv|añÒ¦,í|kf\^`xomefk
{Dgvef`vgi|Ggikm`om|^gikm ©/`Ogiwxwxomqgv{r\ yz`Dgie+( km\Vkm\^`9{qv_+wxhxkrgikmqv|VqvX¢3qvhx|^yxe'qv kf\^`
_`gvefhxom`eqv¹ |@km`om`efk¦u,í|¤kf\xe |^`vem``Èa/a=²qvo!gi|¤`µ+{`|¶kLef{r\^`_'`yz`©/qikr`ykrq
km\^`gi|Bgl@efe(qvEkf\^`!gvefl¶_+wxkrqvkf{gD©igigi¢x k£lv`j@kr`|^yx`y«|DÈa
krqIyx`gi (kf\9_'qvom`
/`|^`o¥giz_q@yx`e9·g|^y+giefqXkrqXkm\x`Yg|Bgil@etepqv3gvetl¶_'wxkmqvkm{¹w`omqvof_5g|^{`_'`Dgiefhxom`e ;¦
,í| kf\^`9ª9qv|¶kr`«¸¹giomqgior`gz¼y-¾3`or`|¶k+ _'w3qvofk¥gi|^{`ergi_+wx |^ef{r\^`_'`e+\BgD©/`¢``|
wxorq1©/`|¨krq«¢3`giwxwzorqvwxofgikm`vR |½qvomyx`oLkmqyx`etv| `µ+{`|¶kI`etkm_gikfqv| gi/qiomkf\x_'e¦
[\xew^giw`o q@{h^ef`eqv|§g¢Bgvef{gi|^y (yx`lVh^em`y yx`w3`|^yxgi¢x k£l¨_'`Dgvethxor`ikf\^`
¼¦3*`«g|Bgil@°`«emqi_`¬¶|^q1(| _+wqvofk¥gi|x{`ergi_+wx |^em{r\x`_'`e5yx`efv|^`y krq
`efkf_gikr`k3¼`7`j@\x¢zkXefqv_'`_+wxorq1©¶|x'kr`{r\x|x¶hx`egi|^y¼`7yzef{h^efe qv|/`|^`orgi
wxorqvw3`ofkm`e(qvkm\xe¹±gi_+l+qv_'`km\xq-yxe¦
[\zeIwBgiw3`o+e+qvorg|x°`y gve+ qiq1e¦ *`9v ©/`km\^`O_q@yx`efw3`{^{Dgkmqi|^e+ |
" `{kmqi|5gi|^y`Iyx`em{om¢3`7/`|^`orgi ef _Lhxgikmqv|«km`{r\x|xh^`e | " `{kfqv|~-¦u eX¼`
efkmhxyzlL\xv\xl!yx`w`|^y^gi¢x`efl@efkm`_'e¶`(|¶kforq@yzh^{`gXorgiom k£lwBgiorgi_'`kr`o | " `{kfqv|
gi|^y `wxor`em`|¶k9km\^`_+w3qvomkrgi|^{`emgi_+wx |^½em{r\^`_`e9 | " `{kfqv|! -¦ " `{kfqv|
yx`Dgie(kf\_+wqvofk¥gi|¶k¯wxorqiw`omkm`e(qvkf\^` `etkm_gikmqvore=<R¢qihx|^yx`yom`gikm ©/`X`omorqio%gi|^y
¢qvhz|^yx`y|^qvof_5ggiwxwzorqDj@_gikmqv| ¦¸qv_+wBgiofefqv|^e qv@km\^`²g/qvofkf\x_egior`pkf\^`|!v ©/`|<
| " `{kfqv|!gvetl¶_'wxkmqvkm{Dgi lIgie¯kf\^`Yorgiom k£l7wBgio¥g_`kr`o¼/q@`e¯krq zgi|xy|¶hx_`om{Dgi l
| " `{kmqv|#"z¦Lª9qvor`q1©/`o¼`Ief\^q1 | " `{kmqv|5km\Bgik |@hz_`om{giRom`ethxkme {Dgi|`gvy
krq7(orqv|x'`efkm _5gkmqi|^e¼ |9efqv_'`{gvem`e%gi|^y`{qv|x{hxyx` | " `{kmqv|a
¦
$ +&("
[\^` etl-etkr`_ e¯om`wxom`ef`|¶kr`y89±_'q-yx``y ;R¢@l'g x|x kr`%{qv|¶km |¶h^qvh^e²km _`(\^qv_'q//`|^`qih^e
gi|^y½ofor`y-h^{ ¢x`5ª«giom¬vq1©½{r\Bgi |:. d k.10F~ 2% |n¶¦y*`yx`|^qvkr`5¢¶l7 km\x`efk¥gkr`
efwBgv{`qv .¤Bgi|^y`efhzwxwqvem`km\Bgik-'& l7$lN(*);¦
`kh^eLyz`|^qvkm`¢¶l+N-, ~. SYkm\^`'kmo¥g|^ef kmqi|orgikr`+orqv_ efkrgikr`/,½krqetk¥gikm`0.gi|^y
¢@l1 km\x`%y-em{or`kr`km_'`¹\^qv_'q//`|^`qih^eg|^y+ omom`yzh^{¢x`%ª«giof¬/q1©+{r\Bgi|'{Dg|^qv|x{Dgi l
`_L¢3`yxyz`y|@. gikYkre²s hx_+wkm _`e¦Y[\^`kmorgi|^etkmqv|«wzorqv¢Bg¢x k£l32N-, ~4.wS¼km\Bgik1
©@ef kre(efkrgikr`5.gikr`oYetk¥gikm`6,«©/`om^`e
2QN7, ~4. Sxd
+QN7, ~4. S
8394:9<;
=?>
+N-, ~A@SCB
`k(h^e¹wxor`{em`X\^`or` km\^`X_gi |{r\Bgiorgv{km`omefkf{eqvEkf\^`X_'q-yx`í¦x*¤`gvefefhx_'`Xkm\^gik
km\^`O{qv_+w3qv|^`|¶kmegiom`a9U;L` km\^`o+qvw`o¥gikfqv|^gi-9Òqio'hxw ;qio9 U;!hz|^qvw3`o¥gkmqi|Bgi-9±qvo
yxq1(| km\^gikIe¯±gi`y ;¦V[\^`«emgi_`\Bgwxw`|^eI(km\Vkm\x`O(\^qi`efl@efkm`_9¦Gu e+ergy
¢` qvor`ix7d >*D A (\^`om`B>}e kf\^`7em`kqv²hxwefk¥gkr`egi|xy8A e km\x`7em`kqv¯yxq1(|
efk¥gkr`e>EBA dGFx¦²[\^` {qi_'w3qv|^`|@kme\BgD©/`gemqLgo"&I qvobEp ¢`qv|xv|^krqkf\^`
em`kIH d k ~KJ~MLNLNL=~4Oan qv{gvemef`e¦ uY|qvw3`orgikmqv|Bgi»{gvefeQP'{qv_'w3qv|^`|¶k¼\Bgie±gi hxor`
o¥gikm`SRUTN-,S¯(\^`|Okm\x`_'q@yx`e¹|Oefkrgikr`6,¦
,í|+km\x`Yef`h^`·`( x¢Bgvet{Dglqv q1¨km\^`(|^qvkrgikmqv|Ih^em`y' |@Èa »gi|^yoÏaé»gi|^y
km\^`7giemefhz_'wxkfqv|xe _5giyx`km\x`or`i¦[\x`L(\^qv`em`kqi¯kfo¥gi|xefkfqv|xe%e wBgiofkmkfqv|x`y« |@kmq
k£qu9±yze stqv|¶k ;Rem`kme9¶kf\^`%em`kqv=&'!p gi|xy¶km\^`Yem`kqv ! v!p¦ [q!±gi{ krgikr`
km\^`²or`gvyz|xx¼`yz`|^qvkm`I+N-, ~. S d!R N-, ~. S»(\^`|uN-, ~. S/5gi|xy+N-, ~4.wS d N7, ~4.wS
(\^`|N7, ~4. S45 ¦Y*¤`giefqyx`|^qvkm`!¢¶l
>
kf\^`Lem`kXqvpefk¥gkr`eYkm\Bgik {Dgi|«¢`or`gv{r\^`y
orqv_ ,gikm`oYgL±gi hxor`ixgi|^y¢¶l
>
km\^`ef`k%qvEefk¥gkr`ekm\^gik{Dgi|O¢3`om`Dgv{r\x`y9omqv_ ,
gikr`o%g7om`wBgoD-km\Bgik(e
> d k .@lN-, ~. S/5?n ~ > d k .olN7, ~4. S/5n
B
c%`{Dgi Ekf\BgikX¼`7gvefefhx_'`Lkm\Bgk km\x`L|x kmgefk¥gkr`e zj-`ygi|xyyx`|^qikr`y¢¶l ¦ " |^{`
gi(kf\^`«{qv_'w3qv|^`|¶kre'gior`hxwG | kf\Bgik'efk¥gkr`v¹¼`giemefhz_`@|L5 >¦L*`«giemq\BgD©/`
$_d F 9±km\^gikXe3|^q'or`w^giomeYorqi_ et|^{`L`©v`ofl@kf\x|xeXgvefefhx_'`ykmq¼qvom¬9(\^`|
km\^`efl@etkr`_ eYetk¥gikm`e ;¦
`k!h^e!yx`|xqvkr`+¢@l TN-,S km\^`I|¶hx_L¢3`oqv¹qvw3`orgikmqv|Bgi²{qv_+wqv|x`|¶kre!qv¹{gvefe P
(\^`|km\^`7_'q-yx` efkrgikr`Ie,¦'[\^`I|¶kmhzkm ©/`Lyx`Dgqv¯±gi hxor`Lgi|^yor`w^gio kfo¥gi|^etgikm`
|¶krq7km\x`Xqvqé( |^ qvof_5gor`gikfqv|xef\x w^e%<Rqio%gi ,@587¹
N-, ~4.wS/5 P~ TN-,S % TN .wS¼gi|^y Pe¦kD¦ TN-,S r TN.wS ~
N-, ~4.wST5 P~ TN7,S TN-. S²gi|^y Pe¦k¦ TN7,S ( TN .wS
B
[q z|xet\ km\^`'yx`ef{ofwxkfqv|qv¹kf\^`'_'q-yz`í `kh^eetw`{l\xqé}kf\^`'kfo¥gi|xefkfqv|xe
q-{{hxo¦Eu%kr`okf\^` ±gi hxor`pqizefqv_`²qvw3`orgikmqv|Bgiv{gveme PX{qv_+w3qv|^`|¶k(\^`|km\x`¯etl@efkr`_
efk¥gkr`Xe ,zkm\^` etl@efkr`_ s hx_+w^ekmq+etk¥gikm`.'( km\wxomqv¢Bgi¢z k£l! cN .", ~KPwS¥¦ [\xegiq1e
krq¨k¥gi¬v`|¶kmq gi{{qihx|¶k«km\x`{gvem`qv#=&'!p!ppvO^ kf\Bgike km\x`etkmh^gikmqv|
(\^`om`kf\^`±ghzor`qv!emqv_'`{qv_+w3qv|^`|¶kO|^y-h^{`eX(kf\ efqv_'`wzorqv¢Bg¢x k£l/km\^gik
gefhx¢^ef`k+qv{qv_'w3qv|^`|¶kreLeef\¶hxkIyxq1(| 9±qvo|^etk¥gi|^{`vpkf\^`5±gi hxor`+qvkm\x`5wq1¼`o
efhxwxwzl½{gi| _gi¬v`Oefqv_'`Oqikm\^`o'{qv_+wqv|x`|¶kre7hx|^qiw`o¥gikfqv|BgU;¦ [\^`Owzorqv¢Bg¢x km`e
cN .", ~KP SYgor`'gvemethx_`y¤krq¢3`'yx` x|^`y¤ qvo!gi.m~, ~KP»ºE|/`|^`orgií|_qvefkqvkf\^`
{Dgvef`e$ cN .", ~KP Sxd |z¦
% ¢^ef`of©/`km\Bgik
N-, ~4.wS/5o~ RxN7, ~4.wS d &
'
T = TN-,S RUTN-,S( N .", ~KP S
B
¸qv|^{`om|x |^km\^`Lom`wBgorekm\^`Iqv|zlO|^``yx`y¤gvemethx_'wzkmqi|eXkm\^gik orqv_ `©v`omlefk¥gkr`
yz¾3`om`|¶k(orqv_ km\^` |zkmgi»qv|^`ixkm\^`or`egik(`Dgvetkqv|^` or`wBgiokmo¥g|^ef kmqi| -kf\Bgike
,
d |~
>
dGF
B
[\xe(`jz{h^yz`ekm\^`{Dgiem`qvQ&IEQ!p v!pz{qvomom`efw3qv|^y-|^IkrqIefl@etkr`_'e(\^`om`kf\^`
or`w^gio(±gv{ km`e¼gior`gv{km ©vgkr`yqi|xl5(\x`|Okm\^`or`gior`
³
`|^qvhxv\z´X±g`y5hx|x kre¦
((R $ 7 E"
[\^`or`/`|x`o¥gkm©v`Lgiwzwxorqgi{r\krq`©igihBgkr`!kf\^`1V}{qv|^etetkre qi h^ef |^'kf\^`qv qi¡
(|^I`j@wxom`efefqi|c<
d
YN 1ehgcNUZX[4~Z%_=SS
9ta;
(\^`om`
d 2NZ%[ (fZ=_%S1 Ò¦«[q«`efkm _5gkr` N `egcNUZX[~Z%_%SS gi|xy
R`'/`|x`o¥gkr`
|^yx`w`|^yx`|@k{l-{`e
D²¦¦¦ kf\Bgike em`h^`|^{`eLqv¼gvyisfgi{`|¶k!efkrgikr`eetk¥giofkm |^
gi|^y`|xyz|^'(km\etk¥gikm` gi|^y|^qvkX{qv|¶k¥g|x |^' k |gi|¶l«qvkf\^`o wq/etkfqv| 3gi|^y¼`
`efkf_gikr`Ykm\^`{qiomor`efw3qv|^yz |^+`j@w`{krgikmqv|^e¦
% ¢^ef`of©/` xomefk kf\BgikYkm\^`|@hz_`o¥gikmqvo gi|xy«km\^`!yx`|xqv_' |Bgikmqvo|«kf\^`o¦\ ¦e¦Yqv 9ta;
{Dgi|¢3`7{qv_+wxhxkm`yyzom`{kmlOorqi_ km\x`7`_L¢3`yxyz`yyzef{or`kr`Lkf_'`L{r\Bgi |#1km\BgkXe
qvof¬¶|^|yzem{or`km`7km_'`v¦L[q qvof_5gi °`!km\ze `kXh^eyx`|^qvkr`7¢¶lVkm\x`+ef`kqvgi
km\^`{l@{`e gi|^y¢¶l km\x`!ef`kYqvRkf\^`!{l@{`e%w^gvemet|^Ikm\xomqvh^v\@A«¦[\^`wzorqv¢Bg¢x k£l
qvpg7{l-{`$5e
N%S d
> "!$#%
2QN7, ~4.wS
B
uY|9`efkf_gikrqio(qv& e¹km\^`|
'
( d
8*)
+= -, N.
+S
8 )+= 0/
N.
+S
(\^`om` qvogi|¶l¤{l-{`11E¼`5yx` x|^`
,
N%S gie!km\x`5efhx_Sqv¹kf\^``j@w`{k¥gkmqi|^eLqv¹kf\^`
emqistqvhzom|km _`e |gi R kre efkrgikr`eXhx|¶km por`gv{r\x|xoA qio ¢`|^5¢Bgi{r¬krq g|^y
/
N%S
e+`hBgi%krq a«2a53(gi|^yVkrq
qvkm\^`om(em`v¦ % ¢^em`om©v`«km\BgkD¹krq`etkm_gikm`kf\^`
yx`|^qi_' |Bgikrqio+ | kf\^`«`j@wxor`emefqv| qvXkf\^`4((\^`|§g{l@{`9om`Dgv{r\^`euA«kf\^`
wBgikm\O{qv|^efkfomh^{kmqv|9e(efkmqvwxw3`yOef|x{` `!gi or`Dgiyzl¬¶|^q1 kf\BgikZ%[ (Z%_1¦
%ef |^7km\^`!¸`|¶kfo¥gi_+k[\^`qior`_9x¼`\BgD©v`
5 6 N '
(87((S
9;:
/ ) < =
N|j~XSp~
(km\
/ ) d 6 )
'+= /
N.
+
S ~Bg|^y
9 d 9
> N ,
S7 J?(¸qé©
> N , ~ /
SA@B(
9 > N /
S ~
(\^`om`
9
> N qS yx`|^qikr`e!km\^`©igiofgi|x{`qi hz|^yx`okm\^`+wxorqv¢^gi¢x k£l_'`Dgiefhxom`
¦9u
{qv| xyx`|^{` |¶kr`of©igi {Dg|9kf\@hxe%¢3` qv¢xkrgi|^`y¦
[\x``efkf_gikmqv|qv(km\^`'|@hz_`o¥gikmqvo |9faO;wxom`em`|¶kre7|^q«wxomqv¢x`_S`©/`|¨|¤kf\^`
o¥giom``©v`|¶k{qv|¶km`j@kef |^{` |km\^gik {Dgiem` N 1egcNZ%[4~ Z=_=S S NZ=_%S¥¦[\^`7`efkm _5gkmqi|
qv
\^q1`©/`oD eYyzµ+{hzkXqio `©v`|_+wqvemef ¢x` h^et|^5kf\^`Letk¥gi|^yxgioryª9qi|@km`+¸¹giomq
em{r\^`_`7|kf\^`Lo¥giom`7`©v`|¶k{gvem`i¦`,í|^yx``y km\^`I`j-w3`{k¥gikfqv|qv¯km\x` zorefk kf_'`km\^gik
`©/`|¶k ³ Z%[ ( Z%_f´¹q-{{hxome e gi¢3qvhxk4
:
/km\^`|Igiom/`qvop\zv\x l!om` gi¢x`etl-etkr`_e¦TWBqio
kre²`efkm _5gkmqi| v¼`Y{gi|qv q1 gi| _'w3qvofk¥gi|^{`(ergi_+wx |^giwzwxorqgi{r\ ¦ [\^`%yx`g e krq
{r\Bgi|^/`(kf\^`(hx|^yx`oml¶ |^X_`gvefhzor`efh^{r\'km\Bgk&&»kf\^`{l@{`e |Ikm\x`¹|¶kr`or`etkm |^em`k&
or`{` ©/`Yg\xv\^`o `v\¶kD¦²[\ze e |^qvk²w3q/emet¢x`¹|+/`|^`orgií@gi|^yI(\Bgik²¼`Y |+±gi{k¯yxq
ekrq{r\^gi|^/`kf\^`!kfo¥gi|xefkfqv|Owxomqv¢Bgi¢x km`e 2QNS e% |¶krq02NS eY( km\g|giwxwxomqvwxofgikm`
{r\^qv{`(efh^{r\Ikf\Bgik ¼`(`j@w`{k²km\^gikpkm\x`¹`v\¶k
NFS qvxO3 4U\wYw!p q"E "&h
(3|^{or`gvem`i¦
[\x`qvq1(|^_`km\^q@yE{Dgi `y ª " ¡£®$, " qvo`a ! "4"Em1" 1H
!#" &'Xgi|^y |@kforq@yzh^{`y5| Òzh^ef`e¹ |^yx`w3`|xyx`|¶kef _Lhxgikmqv|^e qvokf\^`
|@hz_`o¥gikmqvo!gi|^y¤yz`|^qv_+|^gikrqvoqi-9fa;¦ % |¤kf\^`Ikrqvkrgi
6
ergi_+wx`Iw^gikm\^e
6
gior`Ior`¡
em`of©/`y« qvo(km\x`!`efkm _5gkmqi|qvQN 1e g NZ%[~ Z=_%S Sgi|^yDN 7S
6 qvokf\^``efkf_gikmqv|qv
¦ uYeXkm\^``efkf_gikmqv|qiN 1e gcNZ%[4~ Z=_=S S¹e et_+wx`i3¼`h^ef`Lkm\^`7{omh^yx``efkf_gikrqio
, ) d N
6
S
8
)
+= , N.
+>
S(\^`or`?
+>
ekm\^`±km\Lw^gikm\ergi_+wx`y!hx|xyx`opwzorqv¢Bg¢x k£l
_`gvefhxom`
¦ [\^`%_+wqiomk¥g|^{`erg_'wx |^ km`{r\x|xh^`Ye¼gwxwx`yIkrqkm\^`%`etkm_gikfqv|+qv
¦ u |^` `efkm _5gkrqvo(qiEkf\^` e¹km\^`|
'
( d
,
)
/ !)
(km\
/ !)
dtNN 7S
6 S !) '+= /
N.
+S N.
+>
S
N +> S
h^ef |^½|^yx`w`|^yx`|@kOwBgikf\^e
+>
q N 7S
6
Yergi_+wx`y hx|^yx`o9kf\^`|^`
wxorqv¢^gi¢x k£l!_`gvefhxom`
-gi|^y+|^yx`w`|^yx`|@k¹qvkf\^` +>
!
6 ¦ *`Y\^g1©v`Ykf\^`|
5 6 N '
87( S
9;:
/ !) < = N|j~XS
(km\
9 d 9
> N , S
@ N ( S 9
>
N / : S
7 B
u yzl¶|Bgi_+{ {r\^qv{`qv I{Dgi|«giemqL¢3`_gvyx`Xkmq7or`y-h^{`
9 ¦
,í| " `{kmqv| -¼`Ior`©¶` kf\^`I_5gi |em{r\^`_`e!wxomqvw3q/em`yqvokf\^`I`efkf_gikmqv|qv
gi|xy¼`+wzorqvw3q/em`Iefqv_`L|x` gDl-eqv¼w`om qvof_' |^km\x`+`efkf_gikmqv|^e(\z{r\(
¢`%ef\^q1(|5kmq!¢3`\BgD©v`Y¢3`kmkm`o¯|5giwxwxomqvwxomgikr`(etkfhBgikmqv|^e¦Y`j-k¼em`{kfqv| xoretk¹yzem{hxeme
km\^`qvom_gi °Dgkmqi|Lqv»km\^`o¥gor``©v`|¶k(ef kmhBgkmqi| v|'qvomyx`o²krq¢3` gi¢z`¹krqyx`©/`qvwkf\^`
gi|Bgi l-ete(qvkm\xq/em` kr`{r\x|x¶hx`e¦
E OR ¹(
*¤`X_Lhxefk(qvof_5gi °`%kf\^` ±gv{k(kf\Bgik¹±gi hxom`e(gor`Xo¥gor` qvo(etq1^gi|xykf\Bgik(om`wBgi oregiom`
±gvefk¦qW^qvq1(|^@Ïa ·¼`7|¶kmomq-y-h^{`Ig!!zEqv!F!@¦ *`+gvemethx_`Lkf\BgikXkf\^`
±gi hxor`eo¥gikm`eqvE{gveme P5{qv_+w3qv|^`|¶kme\^g1©v`kf\^` qvq1(|^qiom_o<
R TN7,S d CTN7,S +
> !
(\^`om`` km\^`o kf\^`!om`DgiCTN7,S¹e%efkfom{km lwqvefkf©v`!g|^y«km\x`!|¶km`/`o TN-,Se%vor`gikr`o
km\Bgi|¨qio7`hBgikrq½a/²qvoCTN-,Sqd TN-,Sqd |z¦¤[qef _'wzlkf\x|x/ep¼`5|Bgikfhxo¥gi l
em`kCTN7,Sd | TN7,Sd |z¦YqwBgiofkm{hxgio gvefefhx_+wxkfqv|eX|x`{`ememgiomlgi¢3qvhxkXkf\^`
cN .", ~KP S ñexemqzx` (om kr`
cN-. ", ~KP S dATN-, ~4.wS
> "!
(km\or`DgATN7, ~4. S %L|-z|¶kr`/`oMTN7, ~4. S %L|-xgi|^yMTN-, ~4.wS d |!(\x`|ATN-, ~4.wS6d |-¦
¸qv|^{`om|x |^7km\^` om`wBgi o(o¥gikm`ex`ef _'wx letk¥gikm`
N-, ~4.wS6dN Sm~
(\^`om` NS]d! #"_`gi|^eLkf\Bgikkf\^`om``j@etkreLk£¼q{qv|^etk¥gi|¶kmeP
~KP ?r |«efh^{r\
km\Bgik P
! l$6NSXl P %! +9±om`{Dg¯km\Bgik qvo`©/`oml¤efk¥gkr`/, d |zRkf\^`om``j@efkreLgik
`Dgiefk%qv|^`efkrgikr`S.e¦kD¦ N-, ~4.wS4rL|;¦*`!{gi|«kf\¶h^e qv¢xem`of©/`km\Bgik%km\x`orgiom k£lqvRkf\^`
|¶kr`or`etkm |^I`©/`|¶k ³ Z%[ (Z%_m´ |^{or`Dgiem`e(\^`|7yz`{om`Dgvef`e¦
[\x`'qvof_ qv¹km\x`'±gi hxom`Lo¥gikm`eqi¹km\^`'{qv_+w3qv|^`|¶kme\Bgie!km\x`' qv q1(|^{qv|^ef`¡
¶hx`|^{`!qv|Okm\x`±ghzor`Ykfo¥gi|xefkfqv|xe¹|+. < qvo%gi 3N7, ~4. S/5«
RxN7, ~4.wS6d
!&
> !'"
(\^`om`
N-, ~4.wS d `eg
T : > !
> "!
_ k TN7,S@ MTN7, ~4.wSpn
9Òqv¢^ef`of©/`km\Bgik( >
dGFxzkm\x`|« qio%gi .B5 >
¼`|x`{`ememgiom l5\BgD©v` N7, ~4. S %O;¦
`k h^e%q@qv¬5|^q1gik%km\^`kmorgi|^ef kmqv|9wzorqv¢Bg¢x km`e¹qi1¦YW^qvoYgi|¶l, ds|zBef|x{`
`gvefefhx_'` km\Bgik > dGFx-`\BgD©/`
N-, ~. S/5 d 2N-, ~. S d &
> "!" ~ N-, ~4.wS %~
gi|^y
N-, ~. S/5 d 2QN7, ~4.wS d N S
B
WBqio¹km\^` |xkfgi3efkrgikr`ix`\BgD©/`km\Bgk( qvo%gi .B5 x_1
2N|~4. Sxd !&
_ ! & _ 9 ! " B
% ¢xem`of©/`¯\^`om`¯kf\Bgikzgiori_' |
9
#
N|j~4@S d5 A«ékm\x`|¼`¼\^g1©v`I2N|~ S6d NFS
gi|^y km\^`or`pe|^q(o¥giom` `©v`|¶kwxorqv¢z`_9¦[\xe\^giwxw3`|^e | w^giomkf{hzgio» 6_ E > d Fx¦ " qz
km\^`p |@km`om`efkf|^{gvem` qvo h^e/9 km\^` o¥gor` `©v`|¶kEefkfhBgikfqv|m;^ekf\^`²{Dgvef`¯qi 2N|j~ S6dN S
qvogi 5 x_ E?A«¦3,í|Oqvkf\^`oqvomyxexkf\^`{Dgvef` qv|¶kr`or`efke(\x`|a9±U;$6_ E@>
d F
gi|^yy9± U; ?.B56_ E >e¦kD¦ 5 6_ EoA9 N|j~4.wS( N|j~Sr¦
uef _'wx`{qv|^ef`h^`|^{`Iqvpkf\^`!wzor`©¶qvh^eXgiemefhz_'wxkfqv|xeYekm\^gik qvo gi|¶lO{l@{` 1
kre¹wzorqv¢Bg¢x k£l
N.XS²e
N.XS d
"
(\^`om`kf\^` |¶kr`/`o! Oe" %/¦3,íE`yx` x|^`
d k 5l
N%S d
"vn ~
km\^`|9¼` \BgD©/`QÏa d '
%##
N%S3d N'$zS
(\^`om` 1d giori_' |k% l d Fjn%Cv¦Y*`ef`` qvof_5gl'|^q1 km\Bgk
yx`{or`gvem`egve
<
|z¦
& ) ! "
[q7et_+wx lLkm\^` yz`em{om wxkmqv|qv km\x` yz¾3`om`|¶k(em{r\x`_'`ex`k¹hxe|¶kmomq-y-h^{` kf\^`Xqv qi¡
(|^5|^qik¥gikfqv| ¦ W^qvogpetk¥gikm` ,`7yz`|^qvkm`7¢@l >
N-. S¹km\^`Lkfo¥gi|^etkfqv|wzorqv¢Bg¢x k£l
2QN7, ~4. Sr(qvo`Dgi{r\G. 5
> ¦ ,í| km\^`erg_`«gDl/ qiogi efkrgikr` ,%`khxeyx`|^qikr`
> N .wS6dG2N-, ~. S qio¼`gv{r\0.5 >
¦%ef |^gi|' _'w3qvofk¥gi|^{`(ergi_+wx |^ ef{r\^`_'`%_'`Dg|^e
km\Bgik |^efkm`Dgvy7qi 2 `(h^ef`Yg yz ¾`or`|@k _gikfom j2/`Dgvyz |^XkrqX|x`
>
NFS ñe²g|^y
>
NFS ñe¦
[\^`Xkmorgi|^ef kmqv|wxomqv¢Bgi¢z km`e¼gvemefq-{gikm`y5(kf\km\^` efk¥gkr`e(qi A gor` |^qik({qv|x{`of|^`y
|Pkf\^`«`efkf_gikmqv|Vqv
ef |^{`O(\^`|§g{l@{`9or`gv{r\^`euA«¹kIe ³ efkrqiwxw`y-´«gve'¼`
`j@wxg|^`y | " `{kmqi|«~-¦
"!#%$&!
[\xe7eLkf\^`_qvefkIefkfo¥giv\¶km qvof¹gory½_`km\^q@y<5kmq |^{om`Dgvef`km\^`wxomqv¢Bgi¢x k£l¤qv%or`¡
/`|^`o¥gikf©v`{l@{`eY|^{h^yz |^+efl@efkm`_ ±ghzor`e^` |^{om`Dgvef`!kf\^`!wxomqv¢Bgi¢z k£lqvpkf\^`
±gi hxor` kmorgi|^etkmqv|^e¦ *`!_h^efkY{r\^q@q/em`7g+wBgio¥g_`kr`o(' 5 N|j~XS¥^(\x{r\«e `hBgikrq
> N >
SEqvo¼gi , d |9k£l¶wx{gi l/ |
B*)
+'L|
B*,
;¦ [\^`%kmo¥g|^ef kmqi|'wxomqv¢Bgi¢z km`e giom`
km\^`|«{r\Bg|^/`y9gve( qv q1e¦
-
,5D> ~ ,
d |j~ .B5
>/.
>
N-. S d0'
> N .wS
> N > S º
-
,5D> ~ , d |j~ .B5
>/.
> N .wS dtN7' S > N .wS
> N > S ¦
[\x` _ONS epgiom`¼|^qikR_'q-yz^`yQ9Òet|^{`¼¼`(gi or`Dgiyzl \BgD©/` _Nx_%S d1NSF;¦ % ¢xem`of©/`
km\Bgikkf\^`kmqvk¥g wxorqv¢^gi¢x k£lqvR±gi hxom`Xorqv_ ,e|^q1 `hBgkmq1'9±kf\Bgike^ qvoYgi|¶l
,5D>87 k|n¶
> N >
S3d2' ;¦
43 56778:9<; >=? @A BC@D56EFGHI!
[\^`'yx`Dg\^`om`'ekmq«ef`wBgiorgikr`Ikm\x`'±gi hxom`Lkmo¥g|^ef kmqi|^e orqv_ ,N7,V5 >-SX |¶krqk£q
yze stqv|¶kEem`kme%<km\^q/ef`({qv|xefetkm |^ qvBkf\^` T!p±ghzor`¯qv»g {qi_'w3qv|^`|@kpqvBemqv_'`{gvefeP
9·gi|^y{gi`yqw& =&'!p ;gi|^y km\x`por`_gi |x|^qv|^`e/9±{Dgi `y]#wHw& =&'!p ;¦
WBqiq1( |^+kf\xekm\x`Lem`kqv¯efk¥gkr`e!
>
e w^giomkfkmqv|^`y9|¶kmq5k£q89±yzestqv|¶kz;em`kreKJ >
gi|^yMLNJ >
x(\^`om`
J >
dk .@lN-, ~4.wS²eg|Dw&E±ghzor`On ~
LNJ > d k .?lN-, ~4.wS²egu#wHw&±gi hxom`On
B
[\^`yx`Dg7e(kf\^`|OkrqI|^{or`gvem`km\^`wxomqv¢Bgi¢x k£lqvpgI|xqv|z¡í |xkfgi»±gi hxom`vzkf\Bgikexkrq
_5gi¬v`¼kf\^`¼±gi hxom`¼qi^emqv_'`{gvefe P!{qv_+wqv|x`|¶kreR_'qvom`¯wxomqv¢Bgi¢z`²km\Bgi||Lkf\^`qvofv |Bgi
_q@yx`·@ km\^`or` e(gk(`Dgiefk(qv|x` {qv_+wqi|^`|¶kqvkm\^gik{gveme¹kf\Bgik(\Bgvegor`gvyzl+±gi`y¦
[q _'wx`_'`|¶k²km\xe¶`Y_Lh^efk{r\^q@q/em`%k£qLwBgo¥gi_'`km`ore'/~85N|j~XS49±k£l¶wx{Dgi l/
| B)
+'T~ L|
B,
S¯g|^y{r\Bgi|^v` km\^`Xkfo¥gi|xefkfqv|'wxorqi¢Bgi¢x kf`e² |5km\x` qvq1(|^¹gDlm<
-
,5D> ~ ,
d |j~ .B5#J > ~
> N .wS d '/N 7 S >
N-. S
>
N4J > S
gi|^y.B5 L6J > ~
> N .wS d0' >
N-. S
>
N<LNJ > S º
qvo,Vd |zp`5hxem`kf\^`5erg_` qiom_Lhxgv`I(km\F'd /º | km\^`emgi_'`'g1lv
J > d Fxx¼`hxem`d gi|^y L6J >
dGFxz¼`!ef`kd |-¦
-
,5D> ~ , d |j~ .B5 > ~
> N .wS dtN 7 '3S >
N-. S
> N > S ¦
,í|kf\xeef{r\^`_'`vvgve| kf\^`TW«_'`kf\^q-yDkm\^`pkmqvk¥giv±gi hxor`wxorqi¢Bgi¢x k£le
> N > S d
'¯¹¢xhxk|^q1¼`«\BgD©v`ghxomkf\^`oor` z|^`_'`|¶kD(`Dgvyz |^kmq
>
NL6J >
S?d ' gi|^y
>
NHJ > S d '/N 7 S¥¦
56778:9<; >=? @A BC@ = B7A7 6 9
D56EF5
[\^`7_+wx{kYgvefefhx_+wxkfqv| | " W e kf\Bgik km\x`+{om kr`ofg'hxem`ykrqyx` x|^`'gi|qvw3`o¥gé¡
kmqi|Bgi²efkrgikr`o9±kf\Bgikekf\^`'k£l¶w3`qv({qv|^etyx`or`yefl@efkm`_B;e!{q/ef`+kmq9km\x`'etkmh^gikmqv|
(\^`om`kf\^` efl@etkr`_ ehxw gi|^yqv|x l' - qvo(`gv{r\O{qv_+w3qv|^`|¶k{gvefe P3-kf\^`X|¶hx_L¢3`o
qv qvw3`orgikmqv|Bgi^{qv_+wqv|x`|¶kre¯e¯vor`gikr`oqvo¼`hBgefqv_`km\zor`et\^qvyTi-g|^y3|^` km\^`o
km\^` |zkmgi|¶hx_¢`o qvE{qi_'w3qv|^`|@kme
=
T |^qvokm\^``©/`T!gor` ³ ©v`omlyz`w3`|^yx`|¶kf´!qv|
P3¦ Yq13gvemethx_`km\^gikYkf\xegiefkwBgiofk qvEkm\^`gvemethx_+wxkmqv|^eyxq@`eY|xqvkY\xqvyxkf\Bgik%e
gvemethx_`5kf\Bgik7omqv_Skf\^`yx`w`|^y^gi¢x k£lw3qv|¶k7qi%©¶`²km\^`etl-etkr`_ e7gem`om`e7qv
T¡íqvhxkt¡£qv¡
=
TL_'q-yzhz`e¢zhxk!kf\Bgikkm\^`
= T ñegi|^ykm\^`T e!gior`+efkforqv|^ilyx`w`|^yx`|¶k qv| P3¦pu§or`gvemqv|^gi¢x`%¹gDl+krq _'wzorq1©/` " W e¼kmq!_gi¬/`%_qvom`%wzorqv¢Bg¢x`%km\^`%±gi hxor`e
qvEkf\^`{gveme P{qv_+w3qv|^`|¶kmeqio((\x{r\ TN-,S¯e({q/ef`o(krq7kf\^` km\xom`ef\xqvyTv¦
¸qi|^efyx`oLg9efkrgikr`,L5> gi|xy¤yx` x|^`gO{gieme P"!" &T ,½ TN7,S 7TBd
`eg T = & N T
N-,S7T
S¥ºBqvkm\x`om(em`i»kf\^`!{gvefe%e#wH "!z" &ö¦ %qéBqvoYgIefk¥gkr`
.5 >
²km\^`5kfo¥gi|^etkfqv|N7, ~4. SXe@"!z" &kf\^`om`e7emqv_'`{ofkf{Dg¹{gveme/P¤| ,
efh^{r\kf\Bgik# TN .wS ( TN-,S¥¦ *¤`yx`|^qvkr`¢¶l
>%
km\^`efhx¢xem`k qv >
{qv_+wq/ef`y9qvEkf\^`
{om km{Dgi3±gi hxor`e-kf\Bgik(e
>% d k .Q5 >
lN-, ~4.wS e({om km{gi n B
*`7giemq'yx` x|x`
>
3kf\^`Lef`k qv #wH "!z" &R±gi hxom`eB¢¶l
>
d > 7
>% ¦
[\^`| gefw3`{gi °`y " W _`km\^q@y»(\x{r\¼`7{Dg " W " {Dg|¢`yx` x|x`y¢¶l9kf\^`
qvqé( |^_'q-y-^{gikmqv|Xqv-km\^` >
NFS ñe/9±¼`²qv_+kkf\^` orqv|¶km`ome D{gvem`²(\x{r\e\^gi|^yz`y
gve( qvo
"
W;z<
-
,5D> ~ .u5
>
~
>
N-. Sxd0' N 7 S >
N-. S
> N
>
S
gi|^y.B5
>% ~
> N .wS6d0' >
N-. S
> N
>% S ¦
-
.B5 > ~
>
N-. S d N 7'3S >
N-. S
> N > S ¦
" `` " `{kmqv|' qvoYkm\^`!|@hz_`om{gi¢`\BgD©@qvo qv kf\xeY_`km\^q@ygi|^ykf\^`kf\^`Lgi |
km\Bgik{gi|O¢`qv¢xkrgi|x`y(\^`|Oh^et|^7k( |^efkm`Dgvyqv " WY¦
$ 56778:9<; #=? @A# @ = A 7 # 9 D56E
[\xee km\x`'yzh^gi²qv
"
W
"
¦[\x |x¬qv(gOefl@efkm`_ ¼qvom¬¶|xgveem`kre!qv T¡£qvhzkf¡£qi¡
= T _q@yzhx`eX|wBgiorgi`·3 P O ¦¸qv|^efyx`og9etk¥gikm`, 5 > gi|^yyx` x|x`5g9{gvefe
PL"!z" &/ ,V TN-,S% Tvº²qvkf\^`of(em`ipkm\^`'{gvemee#wH "!z" &ö¦ %qép qio7g
efk¥gkr`S.?5 >
^km\^`kmorgi|^ef kmqv|DN-, ~4.wS¯eq"!z" &R Ekf\^`om`!e%efqv_'`!{omkf{gi {gveme'P |
,efhx{r\«km\^gik# TN.wS ( TN-,S¥¦²uYe%¢3` qvom`vzkf\^`em`k%qvEefkrgikr`e .5 >
efh^{r\9km\Bgk]N7, ~4.wS
e({ofkf{Dgí¶e(yx`|^qvkr`y9¢¶l
>%
Bgi|^y
>
d
> 7
>% ¦
u xoretk yx`gYeRkmqX qvq1 km\^`¹gi|Bgiq//qvh^eEem{r\^`_`giepqvoRkf\^`
"
W
"
{gvem`<Eh^et|^X |
km\^`¹emgi_'`¹gDlLk£¼q wBgiorgi_`kr`ore '«g|^y vkm\^`¹wzom|x{wz`¼¼qvhxyL¢`¼krqgv{{``orgikr`kf\^`
{om km{Dgi»kmorgi|^ef kmqv|^e zorefkzkm\x`|km\^`X|xqv|z¡£{om km{gi»qv|^`ex¢¶l_'`Dgi|xe(qvkm\^` om`etw`{km ©/`
`v\¶kre'¤gi|^y '/N 7 S¥¦²[\xe¹`Dgiyxe(krq7km\^` qv q1(|xomhx`e=<
-
,5D> ~ .u5
>
~
>
N-. Sxd0' N 7 S >
N-. S
> N
>
S
gi|^y.B5
>% ~
> N .wS6d0' >
N-. S
> N
>% S ¦
-
.B5 > ~
>
N-. S d N 7'3S >
N-. S
> N > S ¦
uYeY¼`!( em``| " `{kmqv|@»kf\^`om`!e|^q+|^``y«qvo%km\^` ¤wBgio¥g_`kr`o»gi|^y9kf\^`
_`km\^q@y`{Dg " W¹n e¹kf\^`|9yx` x|^`yO¢¶l5km\^` qv q1(|xomhx`e=<
-
,5D> ~ .u5
> ~
>
N-. Sxd0'
> N .wS
> N
> S
gi|^y.B5
>
~
>
N-. SxdtN 7F' S >
N-. S
> N >
SA@ > N
>
S ¦
-
.B5 > ~
>
N-. S d N 7'3S
> N .wS
> N >
SA@ > N
>
S ¦
uYe%¼`ef``iB¼`qi|xlgi{{``o¥gikm`kf\^`{om km{Dgikfo¥gi|xefkfqv|xexkm\x`|xqv|z¡£{om km{gi qv|x`e
gior`X\Bg|^yz`yO |km\^` emgi_` gDlOgve¹km\^` om`wBgi ore¦
4
9 8 D B778 9 =? A B
56"!
*¤`giemefhz_`kf\Bgikkf\^`om`_5gDlV¢`«emqv_'`«wxomqvwBgvgkmqi|Gqv±gi hxom`eI|Gkf\^`efl@efkm`_9¦
WBqio+gi,s5f>!¼ kreIyzetk¥gi|x{`? N-,SkrqDA eIyx` z|^`yPgve7km\^`_+|x _5gi¯|¶hx_L¢3`o+qv
{qv_+wqi|^`|¶kre((\^qvem` ±gi hxor`Ywxhxkkm\^` _'q-yz` |«g7yxq1(|9efkrgikr`ixkm\Bgike
N-,S d 1e g
# [ ' T N TN7,S7 TN.wS S B
% ¢¶©¶qih^ef l/ qvo9gi|¶l . 5 >
`\BgD©/` N-. S N7,S¥¦ u ±gi hxom`:N-, ~4.wSIeergy
1mwYgi|^yqv|x l N-,S rC N-. SYgi|xy ke1#wH 1mwY ¾: N-,Sd N .wS¥¦
[\^`B"!" &'E5qvN-, ~4.wS/5 e
N7, ~4.wS6d N-,S7: N .wS%L|
B
[\^`yz`Dg'qipkm\xeXgi/qvofkm\z_ e%krq+k¥gi¬v`|¶krqgv{{qvhx|¶kXkf\^`yz ¾`or`|@k {ofkm{Dgi kf`e¼krq
{qv|¶kmomqvx_'qvor`yx``wxl!km\^`(±gi hxom`¼kmorgi|^etkmqv|^e|7km\x` _'w3qvofk¥gi|^{`emgi_'wz|xXem{r\^`_`i¦
,ík¹egvefefhx_'`yxqv{qvhzorem`ixkm\Bgk¹km\^`Xhxem`o%{gi|O{qv_+wxhxkm` km\^` y-efkrgi|^{`eY N-,S qvo(gi|¶l
qvw`o¥gikfqv|^gietk¥gikm`5,9(kf\Oq1 {q/efk¦
®Y` x|^`or`{hxomef ©/` lkm\^` qv q1(|x!wBgiofkm kmqi|qv > <
>
_ d k .5 > l N7, ~4.wS d |n ~