On k-Coverage in a Mostly Sleeping Sensor Network Santosh Kumar Ten H. Lai Department of Computer Science and Engineering The Ohio State University Department of Computer Science and Engineering The Ohio State University [email protected] [email protected] ABSTRACT !"#%$"'&(!$)*+" ," ,-./01$)"2!.!32 465,7!7332"0& -,./8:9'&<;3=!+>" ?;@1 A$)"#B"%,0 01C $)*4EDFG,(", H,22JIJ3= ./K-E=G;;-8L!" M30 +*3 N"'>O'CP-.* N"+"H"Q,; -8 +**?R" $)4HSTHA 38IU= $)+),E;; "; !(3$E9V F " )<;L&W,-,./)9V,X+"/"$E3N"30& 9V-*$)FY-N,2!+*+;= *9$?4FZ?,!T;2;V 8I[ -*K= ,= )2HR;L&=$)'F" (>" (P \"W>30 M'3 %C7(] ^>_)] ^?+ K[I N"2=*$`3" $ba11 %"V^?;V"!'NcI 2(dQ**ea1%0,A=0P&B^fc4QZA=,= ;8&$)I8-, " BH-./(%0,\; "92"9!0P&)gfh[2=;V2='0&> "$i, , 4)5,JIUjA-!"$k,2L,E- 0!-"6.*!3E, 13-C ">^g2lJm/noQ*+Vap^g[c6!qT1 ,F./'"QOCr-L./ N"+*F"s.* & ;V"84)t:"e; L.!):3;;V B9V"32\*\,A%2=L` ,;,2T N0*U4uSR,*3+*,),T-*0*" ,T, =;8&=$)'L A$)!!L 8IUjE*9=N":, ;V 9V*3F" H, N2=*$v;L&=$)'H,2:,+ KW=;8&=$)'8IQ%,-N,W'-C -3 =3HA,B9V"32 &>-*0*U4RZP?,T;";V 8IVj " ;= L.=!=- -*A<;= .*30&W;39!,8@ 30E" A, + !e;!8&=$)'%$)w4xQ2"0&'I2j 3B!$(3!">),8 ,F313!Q*3= 2"0&=Y 8J;!8&=$)'6- *4 Categories and Subject Descriptors y(4 z|{ }~L2Qs11K QJ811/w:'-N,2"#-eds '- T4 '4!q{ [<Qf~8 [@GQQ*8KVfN(~"JwTj" S% -,0-3 "2+*V p¡j¢£¤ r¢N¥=¢*¦¢§"¨ General Terms 5," & 5,e"3=, 8© ?j" ª"e;2L !"0&«3;;V 8«9&ªFSR¬Fd[S -*' N"-FDHTCr¬Fx@®Exsz*z"¯=qL°8CP±=qCuCq8²"±=q"4 Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. To copy otherwise, to republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee. MobiCom’04, Sept. 26-Oct. 1, 2004, Philadelphia, Pennsylvania, USA. Copyright 2004 ACM 1-58113-868-7/04/0009 ... ³ 5.00. József Balogh Department of Mathematics The Ohio State University [email protected] Keywords =" P" VIJ= $)#-E=;8&=$)'8I2 N""$´=;8&C $)'8I=-L./ N+**I-*-!.0&'I=;V8 F$2"+"$)'84 1. INTRODUCTION 5, Y"R$)"."8Y9&Y 13$)'N"[; "9$µ-3= C '0&?p-+,(;L&=$)'R"u" j" T" ,(=C 3"(=-"(;;!-8L*U47yF!.*B" %9V7; -8[I ¶ ¢"¡¸·A¹"¨" E,*3!@9V>=;8&'8:G,2LE./ &@;V" @,A +"*º(-L./ 8@9'&@LE8#)O'CP" Ya1?p-NL -!"0»-8*<" N"-N+)"7' 32= NcI"2+*./?,%, A$E3#j!#1 %H;V-0»28)+",Y"J!$)8¼5,;= *9=C $´-N,2!+*+>9V-83A.K=32Q$)"(awY;V";3KL BC .-,26,*#6 T{0qL8c6-8)K#j"0&Yq8±*±LCqL"±B,"3 %ap½"C° &=Ncs*H;2 j"USFSX92 !jA,R"-!.*T$)=H{ °8wI=%,! , >H=0 8?Y!#1 ./ N"s$)*',4TZrFR 8%>,R%,?,H$)"R ?,H;+$)*I=,& -"3$)>*0&X±=4q8¾¿",? +"&À-*3$)8ºX,e"-./ $)"Á46SÂ3= N"[";; *"-N, Ã2=!+ ,Fj" V© !0C $)>I, 1 *I<;3A!.!32$)")"|\Li=3& -&=-*I%, YÄ"Å p¨Æ¨/Ʀ!F=»28Y)9V, N-*e"s$) ,2LRE$)"HR"-!.**4 5, :3e8Ç|9V\ *./8«9V" :,@"9VL./:";=C ; *"-N,>9V-*$)8"9*4jxQ0 #8I2($),)%8=8)A-*$AC ;3=R,T 8M'3 =3&E-&=-F1 %2.!3"V$)"I+*./, *+*.0& 8M'30 $)j"U,j" V4u=-*2[I+*./>E=3& -&=-*IV"-N,?$)"H8% $)F%,?)9VB"-./ 2 %,YB' ,T;+($)*475,0 NJI=",T$)"#=ÈEC -38ILju8H$),RT= $)Q,6";;= *;= KLj3$(9V "2$)"fT9V;8&'8H,2Q!$(30N"*30&E-,./Q,C 0 8A+**"jJ!"+".P&A2A-"'!3*36O'Cr-L./ N"+"*465,6j "' !.!"J; "9$9V-8"3H*,*,22[I=,-""$)!- -NL>$)!$)É+@,G3$E9V E" 8ªÇ, ;8&$)8f":*\,A, ,"2[If0 B'!"7?,28./ *3+",?"f,$µ=;8&'8,2LT-">+"32L N"' (,!+",eM'32!0P& "F-*JIu-K0»2-8L!"UI"2@ N-+G"T' 32 " *+(*3+*,);V )"J$)*475,%!"-NAQH*32A$),E1 -"$);3+A, ";; *;= ! =3$E9V %j" T8? 9V-"$)B-83%1" -*- -32[;L&=$)'4fZ2=8[I 0R!%3-,>E-*- >,%,"%$)"."8Y*3 %j" V4 Ê T3R* %,8 ;= *9$)6 $)!+)e";=C ; "; !3P&H-&=-2H,;+F-N,3*4sj*K= 7, 1"!8%+Y;!+-,$)*I=%,-,?jB-8"f¬T""$)!É8>ZC ÁË Ã=;V $)'N"NT,8%Q,2Q$)"Q-"E!#s1" 7$) u," (&' R*>qL¾=3&Y-&=-A{ °8r4 ;V2B=;!+<ap¬RZcJ5$)EB.!8GY;V n 4ES ,9V+"!+("7 ;V JI8"-N,?$)F2=;V2='0&-K= %,, ? $"@/)1 (,!B;V ªa1%0,@;= *9290& gVcT" B+">;|a1%0,:;= *9290&Àa#qTÌGgVcc4 tª,\,H;C ;= /-,JI,7Ã=;V-801$)6",6 F-8B9V6- 8" 9'&?)1"- H-* ?qLogfÍ[F8HIV+"!.*\Y=0 8eP >01C $))2G,)-./)$))"%2=.=!=32"7$)IQ0BB8"#& -"$);3%,R.""3T"gfh'%,-N,>!(,2;;Vu 9VT,TÃC ;V-8G3=&-&=- j"-N,e$)"*4RZ?*=0!"UIV,!F";; /-, f301 $=3&E-&=-! 92!"-,R +"&)-"3$);* >, V4 t:G"G, \,Y,G!8WE=.=!=!+@$)\'W;V 0C =Y"2À"8%+W$)"E:;ª3 +<"$)?;V )," 9VXÃ;" 8W; .="30&'4ÀSb=' &Cr92"@;VLj )$2"+"C $)R-N,$)T"T; ";V*8?W{0q½rI%, ("e-./B$)"H -"8?)Î p£¨4uÏF8.* 8I,F3"f,8ÇE-,FC =&=2$)!-"0&)*= ), *46Sª -u ?{0q*q ;= *;V"\-N,$)Y" ' &W-!"U4XZW,A-N,$)*Iu $)u9"-NQ"ÐA" 6(' .""/./ 0&B;= *;V" "2"'R0f C $+T +&A2B, 36F-*$)$E3-8L!" 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MODEL, CONTRIBUTIONS, APPLICATIONS, AND RELATED WORK 2.1 The Model and the Problem Definition t:-"! uFM'3 j +"*)"2306L 8T%, uL E9VB;!8&'>"--" N=!+EE,1*8%+$) á ] ^G_?] ^<§"£âwÄÄ'P¥2¦¢"¨"·) ¢#ã^XÎ΢"£NÎI%, H8"-N,> ,F^<+ K;V*'R,*#RE" 8 á £ ¹VÄ/¢"·bÅâ ã¢"£·iÄ'P¥2¦¢"¨"·) %¢ãR^ªÎ΢"£NÎIV%, B8-, "U,T^e E9V=;8&'8),"8M'32V!*!,' p!+:L)"'&<'-8*À@,Y;8&$) L 8=I62=C ;V'0&"s,", R 4 áºä ¢"â1Î΢"GÄ'P¥V¦¢¨"·) 6¡6â ¶ £¹" P^sI=%, F" =C ;8&'8:"-- N+?G?LCP$)!"2dQ*"W; '-4 F-<,?,<3$(9V "A" G=;8&'8Ç«, $)J8>"%9VHÃ-0&^sÍ t:-*K= ?; -3!L );V*'(åBW$)Y"2W3$)),2L LH$)EåF"-N,:"j,)=;8&'8G B!¹*Æ pâ1æ"F%,<; "9C 90&gfhV;V'0&Y *$b, ", IJ"WâV¹/Æ pâ1æ"Baw3 \0, )p"3 >" );+'c(%0,À; "92"9!0P&«a#q Ì:g[cÍQt: 3$)TB-Cr928Y+B$)%, T8"-N,Y"-./R" ,"RE+E N"37m="'&Y*9Þ#-%%,eA-Q N*=3 m- 8GB:-./E" H !"90&\-8\9'&>84 S ;V"R, +*"?TKYA9VHO[çÆ¢"æ"£Ä 0s0T%0,>, +W=#N"-?"OÀ $)" ?-./?" 4«Sµ +"!"| !e)9V O[çÆ¢"æ"£ÄE0u./ &;V*'Fe0FFOCr-L./ JV2? !:e9V?V¢" jOCr-L./ 8:0%0E-"'N" *$))*CPO'Cr-L./ 8 ;V"LapNc4 5,H; "9$)B" %!,%;2";V R ,*8%!+ q"4tÇ,7 !*$)*+^sI/mIgfI/"2EO "3!E9V3=È)-' Y+*32L N"'E,,E;= *9290&G"u,(' +*" 9V+FOCr-L./ 8E;; *"-N,q"7^Y";; /-,7»20&¼ '4tÇ,7 !*$)*+^sI/mIgfI/"2EO "3!E9V3=È)-' ?+"32 N'E,2LH,A;= *9"90&\"j,E +"*@9V+ TOCr-L./ >";; *"-N,HqBR^<";; *"-N,R»20&'4 5,e-*0*jeÀ"#8ª9V.*G ? 8X:" - -8-"20*(" E"#&=$);!-)-.* N"+"*4\5,)-"2=" "3+",fÂq7Q3=È)-'U-*0*H1 s"#&=$);="-7OCr-L./ N"+*"4 5,u-*-*0*H%!*9Vu3=È)-'f-*2=0!"1 s"#&=$);=C -*=CPOCr-L./ N+*a1%,!j7+/L!"E%;= .!T-#C L &E-*2=0!"E1 u"#&=$);!-OCr-L./ N+*Lc4s5,+/;A9VP ,FjA-"20*I 1 8YA"R¡6âVÄ/¢"¡ª¢#㥠¶ ¹"ÎH p£¹"ÎNâ1ç pâp¢"If! >+ N&@L 8%, A,A9V,8.= B"R"#&=$);="-E-L.'C N"+*B%3- N"J47Z>!+E- 0-8f-"20*1" T#&$);=C ->-L./ N+**Ij0))0 8W,2LE,%2=L"R,>;,2 N0*À-*./ +*G±\Ep#)E;V*9!",)p# A, NL%"[-"=.* +*-*I",,2L ;V 6, N"0!"U465,%!8 Q;,2"R N"">%"(K 8M'3'0&3Y%"Y2-8L" % ,HÃ=;V-8Y9V,28." %>»0-84 2.2 Notations and Assumptions t:H*=*;=R,F1"!8%+"N"%, *3+*,*3,;2;V 84 áWèEé apêVcs,B-HQ N*=!3%mE-' >%,;V*'Rêf4 á ds L{ ë%w7,;= *9"90&Y,2L%./'ëÂ'--3= 4 á:ì { í?w%,HÃ=;V-.""37í?Í á t@\8&Ç,?"«./'ë(apîVc\¹*¦ ·A¢"Î Y¹*¦ ¡u¹"¨Î:¢8ÆÆÅ£NÎ0 !$ïLðBñºd7 L{ ë apî[cPFòóq*Í6ZPT0))-8 )%,-,À;2L N"$) +*'?º»0&'IR,ªj:$);0&Ç&ª,?d7 L{ ë% ";=C ; /-,Hq*4 á t@ 0?ô[apî[c)òkõ=aröfapî[ccE0ÐÇ$ ïLð ñ ô[apî[co"öfapî[cò÷±h ô[apî[csòXøFaröfapîVccJ0Ð!$ïLðBñ@ô[apî[co*öfapîVcsòÂùªh""Bô[apî[csú öfapî[cj0Ð<$ ïLðBñ ô[apîVco*öfapî[csòq*Í á t@Y3>ûVapêUhü=c <", Ë 3-!!=8"X#N"->9VC ?;V*'êG"ü2Í á 5,E; N$) m=ap^UcIUmýap^UcI=gQap^UcIJþ*ap^UcIU"\þýap^UcF "R3-*)H^sÍx" 8e" 8"9P&'I>3emI m"ýwIgfI[þ"IJ2eþý7e,0 H;!"-*IV%,!E#!Q N+, $)8"!+(,,B 13-"%"7^s4 x " T8"Bu;= 'N"UIjB3E] ^<" ÿr] ^ "2 ] á 1" \ÿ ] "Ís!-A"3 B-8!-3!* A *93#8IJ,B' "RLÐ[-R*3= % 304 á t@B"3$)H!$À3; ðBñ gfap^fcÂq*h23%#N?", C %!"4ut:H""3$)F,2LR^=góù T^¿ùª4 á t@H3$)Hm=ap^fc v±A"%^óùÇ4 2.3 Contributions S¸13-* sap^=gVcTAΦ¢¡%¦ ¨§"£¢¡6â§E00HH$)*"*-8"0& - "+=I2+*'TE=»20P&>%^¿ùªI"2YTõ=ap*+j"+Vap^=gVcc4 5, "3+",*3=7,s;2;V 8I sap^g[cQ"08&=6"uRL%0&E+" 8C +E3-*U4 Ê þ"ap^Ucuò ^g2lJm n Í *+Vap^g[c a#qLc Z),!j;2";V 8I'$/,T*8%!+A-" 93**, 3H7-L./ N"+"H"6EM'3 +"!"?"s,30R 8 q*4x" %,H+ !?;L&=$)'8IB;= .*F,2LT0s, HÃ=# \8%0&X+ 8%!+:13-" sap^g[c 3-,º,2L)þ*ap^U c Üq c *q g%*+[ap^g[c O"+j*+Vap^g[c F!"+[ap^=gVcI sap^g[ ,G!U,B;V*'Fe,H +"!"\ ("$)*#"0&=HO'C -L./ 8"R^:;; *"-N,%»0&'4 4jt@A>; L./B1 H,(+" !<=;8&=$)'T,H0jþ*ap^Uc qAÌ sap^g[c q gT"+Vap^=gVc *+j"+Vap^=gVc E*+Vap^g[c 1" A*$)\ap!8%0&À+" 8%+G13-*2! c sap^g[cI7,ºA^ ";;= /-,T=»20P&Y, B"$)"#&RÃ=#FA;V" !B,6 +*"(,2LsQqCr-L./ 8J4sj N"0&'I"0VT;V" !%"BqCr-L./ JI,0T%TOCr-L./ JI"%p4 z=4x" , N"2=*$ 301 $Û;L&=$)'8Iue; L./,2 7þ"ap^U" c Âq { 7ap^g[c O*+j"+Vap^=gVcPoQ!"+[ap^=gVcu1" F*$) 7ap^g[cI=,e"f,H;V*'T>,B +**GL ($)*#F"0C &=TOY-L./ 8"%^<";;= /-,T=»20P&'4 ½4jt@u"T;= L./6" Q,6 N2=*$Â30" $Ú=;8&=$)'[,2 Tþ*ap^U c ÜqHÌÚ{ sap^g[c *+u*+Vap^g[cPoQ!"+[ap^=gVcB1 *$) 7ap^g[cIJ,: ^X";; *"-N,B»0&e, )"$)"# "0C &=jÃ=#B;V*'),% +**,2Lj"TqCr-L./ 8[4 °4x" 6dQ"!"# 93=!"UI"; L./u,$)-*0* "1 T,3" $)0&# 93=84 ¯=4xQ!"0&'I*jR;= L.=!=-" -*7;= .!"30&(;39!,8 304et@,L ,2LA?-L &W-*0*º= ./8 1 0.9 0.7 0.6 0.4 0.3 c = 1+ψ(np) 0.5 c = 1−ψ(np) Pr[All Points 1−Covered] 0.8 0.2 0.1 0 1 c−−> # %$2f~L&('")Ye11Q~L8KV<(*U,+f -+f ~Q88KV<)U U.u / 1F 0 ap^g[c2 ±e2^g3Û ùÇÍ546+UÇþ7¸qTÌ80Fap^g[ch 2f~~L=8Q8ÔQ~L=Q1*`,f + Õ}~:9;Y ) 1À7VJ8</& *:V= ~L?> ,U + VQ1ªQQ~L22@+B: A QXCf + =XþD« q T 0 ap^g[chJ,f + Vf1 QQ~L2E+B&(.DFGf + 12$ W7=N ==XqfÌT 0 ap^g[cj2Q|q T 0 ap^g[c (,f + )1QU( * 5f + 2~L2Q8V7 H IU + =~L,f + >7:f + J[= K2~T * ,U + eQ~L2QQ11N Ç1)Qf*~21. |{0q8àLFawds *;V"0!"À4qLcH='HB,*![4)5, "eLC 1-H,)-!"$Ü$"A|{0qàj, #&$);="-A-"-C .P&'j$);0&)#&=$);"-%-L./ N+**4u5,!j-!"$ 8« $!%*;VJ4 ¤ L:MONPM7x ,R+" !>=;8&=$)'8I*j,28./T"--*3'8 K ·)¹£! " ,(9V"3 &>-*0*J4Ze,(-8E6 N2=*$k=;8&C $)I,j=3$E9V Q"V" 6(,- è é apê[cf" u"'& ;V*' ê(s!;V2f"f!'-*J47ÏF8.* 8I*B,6+" !E=;8&C $)%$)=pI23$(9V 7 TGA=!- è é apê[c;V%* ,'-8*UêUÍ/5, 1 *I1 u,+" !);L&=$)'8IL%"9C NGA-/L #$LB" T,H3$(9V %"7" T! è é apê[ch %,-,Y 30Y,Fà NH $?I sap^=gVc g%*+Vap^g[cÍ=5,!0C -7,2Ls, 7*$)%=0Ð[ -9VjE, $)#2 N2=*$$)=%%0,> ;V-%A-L./ N+**4 T"-\,29V",«- 0-8B3-*+*./«9'&ª3" ?, -"R"JO'CP-.* N"+"T2E*O'Cr-L./ N"+"%-*./ +"Eq*465, C " *I7" )O'Cr-L./ N+**I7,)- -8.*3"%^g2lJm n os"+Jap^g[c q"46ZPfj (#NH"3 %-"20*R"%!$Xþ*ap^U2 c QÒq"2 $|3;þ*ap^UR c Qq"h2,G"0,"3+*,e,&>'"?$); 8I2,&?L "F./ &33Q1 H;;-8L!"HT,&e=Y"T; L.! "'& =1" $L*>"Y,T NL"f-*./ +"-*4uDFY,F, ,"JI "3 Y-"20*; L.K=e<%2=8´H;,"? N"0!"|,2L ;= L.=!=Y@+*3!"-\"|%,2LY9V,28." Y@Ã=;V-Y|»20 -"4?x" #N-*If1" YqCr-L./ N"+*:,E N"*$Ü301 $ =;8&=$)'-8*I[{ sap^g[c "+j*+Vap^g[cPos!"+Vap^g[cj%,%C =8Õ;,"A N"0!"@1 qCP-.* N"+"A,2 -"./ +*B?± uS ^ ¿ùªh'2ALu,"$)$); L.!7313+*3K"-%! »0R-8"46xQ+"3 BqT3# NLj,u3*)" ,R;= *9C 9!0P&B"JqCr-L./ N+**4u!$(3!"B 30QA-*EÙ!3# NL ,F;V*'F3 , F%0,<"-3"f 30F,2LÃ=,9F9V,28.'C !$)! )G,Y*?,8%À|xQ+*3= <q"4|S$)" 1 $" =-3*>*>;,2F N""?-8"9VF1"3Y>0 NL3 H* ;V -*!"?, &:{0q8¯Ls2Y N2=*$µ+" N;, { ½ t:",2LA0,*3+*,º Y$)"."8:$)"#0&@9'&:, 8À\»22À*3=),?3$(9V AH" ^Â<9Ve;8&'8 +*./FgfI"m=I/"(OVh"F"-N,./%O'Cr-L./ N+**I**3 7-*2=0!"6"8 ,)-*$);3=N*<"R&\");2L N"$) B*3 %^7I2gfIQmIs2 Oe+*./e,(, T, *IJ3$)+),(3$(9V T"u ! +"*4 xQ2!0&'I/jT",2Lu j,%.*'uL T8[I/,%0C »2-*?8=%E9VH'A,9"H#N"U4j*M3'0&'I ,6 H$E3#f9Vu-*-J4sx" Q,6 N2"$301 $ÚC ;8&=$)'8I8,u-*0*Q" 6O'Cr-*-.0&B,"Q9VE= ./8 X{ zIjqL°87%,Yg<òµq"Íf5,A-"2="H-8:!>9VE,8% ,*!)1 RdQ**># 93"U47ÏF8j./ 8I0#U";V 3%,, (!$)! 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FURTHER RESULTS AND RELATED RESEARCH PROBLEMS 8.1 Further Results t@6$)'*P% !8H 30Q"(-.* N"+"j ,f-"U4 q*4ZPH,? +**ÇW9Ve-L./ ª!: -N+"!G%0,Ç3 !+",\2 %!=,UI[#8*e"jM3 Yaw"T,2"F9V "3$)<, Þ "3+*,*3=(,);2";V cI6"2@0 m=IQ, Þpo 0.9 0.9 0.8 0.8 0.6 0.5 0.4 c 0 0 0.5 1 0.5 0.4 0.3 = 0.2 0.1 low low 0.1 0.6 c = 0.2 chigh = 1.44997 0.3 0.7 chigh = 1.44997 0.7 0.499832 Pr[All Points 1−Covered] 1 0.499832 Pr[All Points 1−Covered] 1 1.5 2 2.5 c−−> 3 3.5 4 4.5 5 ' gH~"K82Ú(* }~z9;)Y1}RV1[8G :=V~L?>F)K,+ # %2$ f~LrqiS ==~8V1ÂþI*r2~?~LQf[ QQ%*r2~" U=Q f<=[n. gGò ±Íq:h7mBòDZ=Í ±½h72f|^« = ~"KG*P~L[ LhifA(A:CjAkh A(A(A?. ,-*0*A1" -.* N"+"> $"W,"$)*IuÃ=-;= 1 8»0*"7þ*ap^Ucj þ _ ap^Uc7ò Þ ^=glUm n Í *+Vap^g[c '4ZP,A%!=,@,A +"!"W(-"Y?É I7p4 *4?0, ; "9$`"J-L./ N+*RuH-L./ ",R;V"6*Y30 +,)!*I',A,%"$)%; 'J+*./A,6;2";V *+* Hj" 4(5,E 8*<!T,2F,(%,*E +"*< 9VR-L./ 8)p!32 u,T9V"32 &)-*2=0!"UI$)! B,F!=F"f(M'3 *47ZP,-8""I=,T-*2=0!"" O'CP-.* N"+"RJF307!RE,-8RV N"*$¸301 $ ;8&$)R HdQ*"@;8&$)%N/ ,H1*8%+ 1 $? èT¦ ¡u¹"¨ÎTOJç ^ ¢"æ"£#¹8§')¢#ãE¹![7â1V,W%5,(30R! á èT¦ ·)¢"Î R"$)*#T"08&=TOCr-L./ 80 ^g2m «q *+Vap^g[c sap^g[c awO(ÌXq8c=*+u*+Vap^g[c Í *+Vap^g[c èT¦ ¡u¹"¨"t Î sH¢"ç ^ ¢"æ"£#¹8§')¢#ãBC ¹ [sâ1[, WjSR$)*#T"0C á èT¦ ·)¢ÎN &=, HÃ#RE*Cr-L./ 8;V"!'R*,30%! 0 ^g2m 7ap^g[c *+j"+Vap^=gVc «q%Ì Í "+Vap^=gVc *+Vap^g[c t:%!J ;V" %,HN!%e", R;2;V 84 8.2 Related Research Problems 5, HL F./ N"[ #+( 8 -N,>;= *9$)u,2 \ H F3+¬RZP&;V "u;!+4(DF(;= *$)0C '(M3#*@H,2L "RO'CP-*-.0&'4YZ@=-"º4 z=Isj "8:,E?3È)-!(-"20*W" EOCr-"-.=0P&@-8W9V = ./8:K *$÷*3= A3ÈA-'(-*2=0!"º*ºO'CP-.* N"+"*4GÏF8C ./ 8I, HFA-*;VH1 H !.+AE!+",' F3=È)-'R-*0C ">1 O'Cr-*-.0&1 R*3 R$)=p4 S E% #+? 8L -,W;= *9!$) $) +*A%,W* -*!= ?,\! +* e; "9$ A=+*!+X;= "'-"?1" e 0 0 0.5 1 1.5 2 2.5 c−−> 3 3.5 4 4.5 5 ' gH~8V( * }~:9;Y ) 1}RV1[8\ :V= =~L=\T > ),+ # %V$ f~L u?S ==~VÔ1Úþ*r2~<}TV8VÚQ~LJ*=8efs= Q<=[n.GgWò ±=Íq;h7mHòDZ=Í ±"½Dh62Q|^Y= = ~=Cr* ~L[ v hifA: A wjv A h A(A(A\. " TP" ,2LR3H¬RZP&=;V"7;+4DFH9# N-C *eu3-,?AP" YEjCr9V,8./8>$)"9 " TC V4?5,)9V,28." ",!Bj" \! $)KL Be?$)*9! *Cr,'- VI/%, %À+*,9V j;;V8L 2E"KA+*,=C 9V" Q";;V8 s N2"$)0&'4fÏTLj./ 8I sP =f3+ ¬RZ&=;V "u;+IJ,(";;V N"-A2G;;V8L N"-A" !+",9V" E*L%?j9V,28./8@; JI7%,-,º";V(3; ,";;V" 30&<1 .**;+e!$); A;= "'-"!(1" (, >,2«,:"?38ÇÒ;3 \$)*9@"=Cr,'-\C 47¬T"3 -%-"# N!j, '6H j! *=./'-8LG" Y,G=./*;$)')"B$);! >2|+*,'Cw!+",' ; '-*4SY,\"$)e$)*IT; "'-*>,2Le3$)\3C -N,2"+*+B+*,9V" uL ;L&=$)'6$&E"u9V%30N"9! 1 E,j" 4\5, 1 *Iu, A?;V"!j1 )C ./";+\; "'-*eap*4 +4ºCr#N?9"8º; '-*) 0Cr#N9!É!+@; '-*NcE )$);0&=+e,Y; '-*)-3= C '0&38Y>$)*9!*=Cr,'-P =46SRÃ$);Fs3-N, e$);0»2-8L!"W(,E"R+*,'Cw+*,'E;,2Cr928@; C '-"!B;= *;V*<X{0q"q61" B+*,9V" =-L./ &\"G "3+=I %, ;= "'-"!E#N %,º9= N"2<Õ!0K#NA(, 9V+"!+Bf.* &);V J4st:%9V!./T,6, RH;V"=C !"Q" H$) B 8 -N,<e;= "'-"u=./";$)'%,2LFÃ=;!"0 , N2"$¸&'70Cr9V,28./A23= "J" j" 7,2 3H¬RZA&=;V7;+4 9. CONCLUSION ZÇ,!>;2";V 8I<-*K= 8«,G13$)'N";= *9$ " $)!+Y,A;; "; !)3$E9V F"" H, *3+",<; L.K=AO'Cr-L./ N"+"A"Y +"!":%,<" "88YE;$)*#f, %01!$)"I=%,-,!3 Y-C L &>A$/H Fj" %33J1" R 8Q";;-8"4 t@ ./8Y- 0-8"U-*0*%1 %,H2="s" RC 46t:,8j8,2L%,-"2="1 = $)#-HC ;8&=$)'B $)! B>,E-*0*B1 ( N2=*$Ü=;8&C $)'4(SF,("$) !$)"IJ,E$) B0Ð[ -AG,(-*=C 0*Y" ,?P@&=;VE=;8&=$)'A2=!-Y,2 ,R= $)#-%$)%0,Y¬RZB;+ u"uÃ"-&(, "$) R, N"2=*$b;8&$)$)%%,e¬RZ;+4 t@u"R,8H,f*3 f""0&-8".""3U$-N,-*&H%0, 16 kanalysis k actual 14 12 k 10 8 6 4 2 0 0.5 1 1.5 2 2.5 c−−> 3 3.5 4 4.5 # %$2f~Lxj?'&gH~"K82«*O<A,+B==~VÂ1ªþC*r2~~LU U[ fQ%r* 2~" fsK f<=JEX . g|ò±=Íq:hBmGòb±=Í ±"½DhB2QÔ^ ==~=Cr* ~L[ v hifA: A wjv A h A(A(A\. $(3!*( 30I'3;;V" +*3= u-K$Ô,7"3 7 306-8" 9V(38G\ "0Cr!0)";;-8*H1" H $)!+Y,E;;= "C ;= !A3$(9V T"" 9VA=;8&'8[IU+"./G,( 8 9VY-L./ JI7,!+\ N"3("R," Ij"2@,=C 0 8Aj" (!0$)*47xQ2!0&'I/,%";; /-,)[2;V2=' ;+H;= *;V*A!E*3 6¬RZB-,$)$"/Q,' & -C "@N"|a1e3 YO'Cr-L./ N"+"Lc +&=CrÈA-'E2@+*,'C j+*,')9V-"3,Y GA 8M'30 "'&W' N"-" %0,>,0 R+*,9V 4 10. ACKNOWLEDGMENTS 5,)"3=, Bj*3K<!/(,2\ST,:S% " N?2xy?L Ë dQ* H "$´,ADF,e'N(F./ P&'I? /""@Ï!# "$ K *$ j$)"F./ 0&'IFÏFºt@' Ê :K *$ 3+ z&=3 z"?T./ 0P&'I2*3=,{zH 8=I2"2"*'&=$)*3 1 R" ,0 .""329F-"$)$)'j2)192-)">8 T= NL ,R;";V 84 11. REFERENCES {qH(4ST"?"S 4Ï(4=;V- 84 X ¶ ä £#¢Pd¹Pdâp¦ â1Î pâwÆ|: ¶ ¢8Ä"4 *",>tÇS & }Ô="I2"±*±"±=4 { 8BS 4S% " NE2R"p4 Ê F!,F"2[6S«%0 R" Y" %N +"T-*UI-!"0»2-*JI[" N"-N+45f-N,-8"J ;V" 8I2DH=RC#jZ#=¬RuCqLE "±"zCP5¬Ù'q*I DF,'NT./ 0P&'I2±*±"z=4 { zL T4 Ó # 842DF, y>$E3$F=H+" B"2 j*-!.0&"s(tÇ0 t y>30,*;>FP" V4Z £ 8 ¢ N Æ Ä 1 â " § ( Î # ¢ ã R è ~ | < | ¢ P d â i H¢8ÆI2;"+"Rà*±CP²q*I Ê "3" ä ^ %0É K2[I["±"±/'4 { ½ Ó 4 Ó **9"4 K ¹"VÄ*¢"~ · £¹¥ ¶ Î4["$(9 !+"HF!.* 0& ds I[±*±=q"4 { °8 u *9V8H4[dQ8j F$2+*$)'%2Y92 !4 èj¥/¥V¦ âwƹ" pâw¢I sH¢" PkÎ t-èRæ¹"âp¦P¹ d¦! ¹" ¥ W c¡7¡7 ¡ M PdN¢ ¡ MKÆN¢ · VÅ¥*¥=¢"£, 2¹¥/¥22¢" P]Î M ¶ p·EIs"±"±"½4 ¶ p 0? { ¯L 4 Ë *JI Ê 4VyF0 [IV"E4 Ë # !J42xQCr+" N"8 Y$)#&=-,= *É8L*>3+E - 9 **=-8"#4Z ä £¢8ÆNÄ"â1'§"ÎE¢ãH ¶ b â ã ¶ V¨·F¥=¢ÎNâÅ=·i¢" Q¥£¹" pâ1 § V¨"Î P· T ÎNâ§"\¹"VÄ ` ·T¥V¦!· r¹" pâw¢"C V T`G I ; "+"Hq½/ÙLCq8¯"z=I Ó "#*JILy?S I2"±"±/'4 2 { Ù8([42y" "pI2¬B4zH3$ 8I2y<4 Ó 4V .*#N8.*4 5!$)+"Cr#&=-;= "'-"U1 R Fj" 4ZP3è ^ | V[ΨÎI2;2+*Bq8z"àCq½*²=I Ê *RST+*IVS I2"±"±*z=4 { àL ds42yF3;=NE2>dQ4¬ 4 zH3$L 84u 0!-"Q;V8 R" "#&=$);!-F-*-.0&Y>%0 RP" 4Z ` ZZ³Z DL ¶ ^ ¢"8ã8£VÆNE¢" T ÆâÎNâp¢"<¹"VÄ ^ ¢" p£#¢*¦ I2;"+" q*q8±"¯Cq*q"q8±IJ5Q"$);=Ix Ê IJq8²"²*à=4 { ²LBE4 Ê 4=Ï"f"2 4 Ê 2"L 4 B¹VPÄ d¢8¢L¤¢ ã |GÅ=¦ pâ1Î΢£ T ¹" r¹ b ÅÎâw¢"4Vj¬RÇd7 IV"±"±=q*4 {0q8±L ds4Ïw4 ` p£#¢8Ä"ÅÆ pâw¢"< r¢ ¶ X ¶ ¢"£N¨>¢ã ^ ¢æ"£¹8§' 4 **,tÇS & }`=*IUq²*à"à=4 ä £#¢8ÆÎÎÎ? {0q*q 5F4ÏTB"2YT"p4 Ë +"&CrÈ)-'R3 .*!!-H#&=#$ 3+E%0 T RP" 4Z ä £#¢8ÆÄ"â1'§"Î(¢#ãRè ^ | |<P ¢ dâ1ΨÎNI Ó *#"U I y?S I"±"±"½4 {0qL8 y<4ÏT" *JI(4 Ë 4Vj3! 8 I zA4d6# 8( I 42ÏT!pI ¬ 4=É-É&VI">S 4=t@'=425,H-*$)$) -!"É8*e $)!- *" R$)"IZc V[΢£ÎIV./"3$)(q²=I4½I ;2"+"%½/±LCP½*à=I[SR; U"±"±/4 {0q8zL(T4ÏT32"+)"Yß(45+425,H-L./ N"+*B;= *9$µ> % !F" RP [4Z ä £#¢8ÆNNÄ"â§"Î(¢# ã V)sRèFI ;2"+"Hq*q8°LCqL'q*IJ="?+"IVS I2"±*±"z=4 {0q½7 4ÏT3p I s4¬TUI" Ó 4Ï(D 4 z *+",U42=' &Cr92"Y;V8 $"2+*$)!% !T" TP =4Z ä £#¢8ÆÄ"â1'§"Î(¢#ã `#ä V)sI;2+*%½'°àCr½'Ù"I[dsST0I2S I "±*±"z=4 {0qL°8 (4 Ê pIVdQ4tWUI=ßE4tW+)2>T4ßp4x2"30R5f* N ;8&=$)'%">5f";V**+&Gj" "ftª ! 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