Message Delay in MANET
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(2) INSTITUT NATIONAL DE RECHERCHE EN INFORMATIQUE ET EN AUTOMATIQUE. Message Delay in MANET Robin Groenevelt — Philippe Nain — Ger Koole. N° 5372 November 2004. ISSN 0249-6399. ISRN INRIA/RR--5372--FR+ENG. Thème COM. apport de recherche.
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(7) YgEHJb/WP_-^H¤I J{";J+b/f¦WP]i`Y[W \]cbcnHWUV`JIKWMnX]/Y[]iY[Yg`EH\JJ©XIKnoJ+b/W[_oWgb/Wg`~/]cJ?_b 3Y]/f Y[EHb/J?fg~cYg`JIK_HJ?JYZYgEH]cJfgXIKW+J+¦W[WgDFb/EH~/JJ Y[`ab/WMEH{HJ pXJjJ+Y¡Y[¤`]Ky/]iJ§³b/f[EH\+J+]/\p n@^Hfd`afgb WMJ+Y[biJbzaU?UKWg]binH_o\fg]cJppXIYg`a\´noEHY¤^XJ Y[Yg^oJ+EHp®_HJPfgn"JWZJDFY[fgfgEH3`a]/J+\´f[WgY[I¦JCJ+pbi\_ob/f[\J\Jab^H]/UCbinHYg{£`f[]/]/]i_oYgY[E?WM]X\fgbi]cJ+f[J W+bzFUK\b/µjnHfgJ+f[f[]iWg`JnHY[p]X`±Y[\J ]c]c^X`W;Y[Y Wj33]/Wg]cf`f IKbMY¡nH¤_-`a^o]\I `Y¡fgUc{"J+¬£JbzfYgU=EH]iJ nofgIKIK]/]XYg]/]XpH{H\J`]/`±aYZ`WU W ¬ IKWgEH]X] pXJ+_CWK{V9UCmWb/`_oI pX^H]caIRb Yg`¶]c_obzWUV n"]/`_cY¬®mbi_opH]/I¨µ`f[J+\´Y[`]c_=b/_opmb/_opX]/IR¶·bi/Jf O]/{H` `±YZUO]XpHJaW[¬®b/W ¸@¹-º0»g¼M½ z¾¥c O]/{o`J8b/pCEo]X\_HJYZ]cfg0¿Hm]/^HYg`_H~KnHf[]iY[]X\]c&¿HO@b/fg/] yB\[Eob/`_®¿oO]c{H`` YZUBIK]XpHJ. ∗ †. ÀoÀoÁÁÂdÂdÃÅÃÅÄaÄaƣƣÇgÇgÈ&Ø´É´É+ÊzÉ´ÃÅÄÅËcÎgÔÌ Í´×È&ÎZÉÙ6ÎZËzÌ Ï+ÎZÚ-ÏÎÌ ËzÄ Ð¡ÄÑÑÇgÒzÓzÃÅÄ ÃÅÒzÒiÎ´Ì Ë ÂdÃÅË/ÑdÔ¥ÕÉ´ÒÓzÃÖÂÌ ÃÅËzÈ&ÃÖÂÌ ×È. Unité de recherche INRIA Sophia Antipolis 2004, route des Lucioles, BP 93, 06902 Sophia Antipolis Cedex (France) Téléphone : +33 4 92 38 77 77 — Télécopie : +33 4 92 38 77 65.
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(17)
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(21) . TU? (θ) := E[e−θTU ]. TU.
(22)
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(27) b YgJnHf[]V]/3]/f z³§´&wc¤bi_op·9uc¡§d9}- N +1 1, 2, . . . , N. N +1. . . . (Ν−1)λ 1. (Ν−2)λ 2. λ. (Ν−3)λ 3. 2λ. . 3λ. 2λ N−2. (Ν−1)λ. (Ν−2)λ. 3λ. λ N−1. N. Νλ. N+1. /]¯¥$`~/\^H]/f[nHJ `J+cW D]/§EH]cn?f[JabzU"6Y[f[b/_oWg`±Y[`]c_@pX`b/~/fdbiI ]i$Y[EHJMO*bif[/]zyC\[E£bi`_f[JnofgJWJ_-Yg`_H~KYgEHJ _V^HI {"Jf `Y[aW¤fg`~/{H`DF^Xy/Y[EHJJ+_CJWFYg`\_?fd]/binH¯¥_o`Wg`J+~/`WYg^H`]/]/f[J®_¦Y[cpXEHt`aJbim~/IKJ+fdbi\J+b/W[IqW["bi]i~/Y[0EoJYgb YgEHY]J^H_HO@_o]XpXbipXJ+f[Jf¤/W;] YgY[y EHEoJ\[b E£YYZbi\`]i]/_¦§³IKEH\]/Jj]/nf[fgJf[`Y[JWEHn"abz`]/UK_?_onHpX`±Ydf[`W_H]i~Y[Ygf[]HYgb/\]8_o]/W"YgIKEo]/J_H`aWgYZWgUK`]/Yg]i_BEo§³EHJPfd]/biWgnK_H]/~c^Hf[J/Jfd\a$bJjDFU _HEHno]XJfgpXf[]/JJPYg3]X]cpX\fg`a]/WZJc§ ¬ 3f[]/EHEHI`aJ+\d_*E=YgpXEoY[]C]Jf[_HJ]ibiYf[E£J ¬ by/]cJ\f]c]/b nH_o`\JJM]/Wn-]/`U6_*Y[U/Y[EHJEHJ+Y+JWg¬0J _HJEoYZ\`\d]/]/EnHf[`0]XJ+W\+¬£\Y[fg^HEHJfdJb/Wj_=\dEHb JJ+Yj`WYgYgEHEoY[J+EHJMfPJKfdb b pHY[J+_HJ WJYgq`_ob \Y[]/`n-]cU_`a_HW]HbiWgpX_oJJ/_-p@¬ YjYgY[f[]Eo`~/`~/Yg\dEHE=J+f[J W]X\+bB\^HYgfdfdbiW_£b _HWY`]XYgY[pX`]/EHJ_ JW fd¯¥b `Y[~/J ^Hf[J cb/ _op@Ygf[`~/~/J+f[WjbBY[f[b/_oW`Y[`]c_@3f[]/I Y[] DFEH`aWJ©XnHabi`_oWYgEoJMYgfdbi_oWg`Yg`]/_=pX`b/~/fdbiI `_ DFEHJYg¬;fdb/bi_o_oWgp `±Y[Yg`]cEH_MJC3Ygf[fd]/biI _£W`Yg3`fg]/]c_ I 3f[]/I ]X\+Yg]\^HfdW$`YgEK]XY[EH\+\J^HnHfdf[W ]/{£bi`±Y[{HE`` Y[YZEHU J\]cInoJ+IJ+_cYdbif[UnHf[]/{ob/{H`` YZU D3YgEHJ FWEH^HJ;I Wg]i]ihZ]/^HY[f[f[b/__oYgW`IK`Yg`J ]/_?fd`b _ YgJWZWFYdb ]/Y[^XJ Y]i$WY[biYgJ ´O]c`WfgJ+J]z©Xy/n"J+]/f _HJ+_cY[`b/UpX`WY[b/fgfg`{HJP^XI Y[J+^XpPYg^ob/`YgE U ``_c_oY[pXJJ+_on"WgJ`±YZ_oU pXJ+_cY fdbi_opX]cU@I \]/y _£bipXf[`a`±biY[`{H]c_HJ+W+`_H ~B]c_Y[EHJWZYdb Y[J ]itYgEHJKO@b/fg/] y?\[Eob/`_¦hZ^oWY{"J3]cfgJ `±YdWJ+_cY[JfdWjWZYdb YgJ ¬0]/f !JpXcJ^oWZY[``y _£b/b JYg`_c]/Y[_U¬X{-J8U Eo\b]/y/_£J pX`±Y[`]/_H`_H~*]c_·YgEoJ _-^oI {"JfM]/\]cnH`J+W hZ^oWYM{"J3]/f[J¦YgEHJCIKJ+W[Wgb/~/J¦EH`YdW `Y[W ). . i. i λi. i+1. λ(N − i). i. i. . i 1 − i/N iλ/((N − i))λ + iλ) = i/N Si. N +1. i+1. i. N −i. (N − i))λ/((N − i))λ + iλ) =. N +1. i = 1, 2, . . . , N i. λN. S1 , . . . , S N. . N +1. N2. N X T2? (θ) = IE[e−θT2 | N2 = i]P (N2 = i) i=1. N Pi X = IE[e−θ j=1 Sj | N2 = i]P (N2 = i). i=1. 00. Ë. 213547698. s/.
(28) > 6!@2MU@ 9@BA 3 C[P\M '. . . . . 6I6[AD@. kjWY[W biYgIKJ J+_cYg`]/_o9J+fgp JWJ+nbi f[`J+f+¬ 6DFEHJ+fgJ3]f[cJ+fgWgJcn®¬ 8`W YgEHJCnHfg]c{obi{H` `±YZU=]ithZ^HIKnH`_o~*3fg]cIWY[biYgJ gY ] c EH`a¶\dE?EHJ+JW_ Y[b/`{H_C` WZWgYdEHb J+YgW8J w/´ ¬-YgYgJ+EHp JPO*`YgbiECf[/`_c]zY[yJ_o\dEoWg`±biYZ`U _ \+bi_BJ+`±Y[EHJf¤J_-¬VYg]/JfFf¤WJ+Y[_cb YgY[J+J fFWY[biYgJ biYgJ+f¤b8b Y[`YIK[JJf g Y o E i b ¤ Y a ` ¤ W J X © " n / ] o _ J c _ [ Y ` / b ¦ U X p a ` W g Y [ f ` H { H ^ Y[bBEoYgb `Y IKJ ¬ `_£pXJn"J_£pXJ_cY8;]iO ]cfgJ] ¬ y/b/J+_of p*J©Xn"]/_oJ_cY[`b/U@pX`aWZY[fg`{H^XY[J+p`±Y[E=`_-YgJ_£W`Y¡U MNj{oWgJf[y/J 1 − j/N N +1. j+1. j/N. P (N2 = i) =. j. i−1 i Y i (N − 1)! j , = i 1− N j=1 N N (N −i)!. j = 1, 2, . . . , N. j +1. Sj,1. Sj,2 Sj = min{Sj,1 , Sj,2 }. (N + 1 − j)λ. Sj,1. N +1 jλ. P [Sj,1 < x | Sj,1 < Sj,2 ] =P [Sj,2 < x | Sj,1 > Sj,2 ]. v/. =P (Sj < x) =1 − exp(−λN ). bcWFbK\]/_oWgJI-^HJ+_o\JP]iY[EHJ8J©Xn"]/_HJ+_cY[`b/®pX`WYgf[`{H^XYg`]/_tDFEoJf[J3]/f[J/¬ IE[e−θ. Pi. j=1. Sj. [f J+¯obcfgpX]c`I°U¦¢ovi_op ¬ cbi_op?YgEHJ 9b/\´YY[Eob YY[EHJ f[yVW '. IE[e−θ. Pi. j=1. Pi−1. c b/fgJ8I ^XYg^£biU?`_opHJn"J_opHJ_cY¬£J c. | N2 = i] =IE[e−θ( j=1 Sj,1 +Si,2 ) | Sj,1 < Sj,2 , . . . , Si−1,1 < Si−1,2 , Si,1 > Si,2 ]. Sj. {Sj,k }j=1,...,N,k=1,2. | N2 = i] =. et^HYYg`_H~&sc´¬ -¤b/_op -;Y[]/~cJYgEoJfU-`JapHW. i Y. IE[e−θSj ] =. j=1. . λN λN + θ. i. '. . .. . T2? (θ) =. i N X (N − 1)! λ i , (N − i)! λN + θ i=1. EH`a\dE?> 6InH6 f[]z6ycJ+WP ´C9 MUR
(29)
(30) \`aW ]/obiDF`aW;]zEHpXJ¤`aJWgYgnHpfdbibzYg_£U/]*WJ+`pKpXYg``a]/`WZ_B_Y[fg¯¥`pH{H``^Xb/~/~/Y[^HJ¦fdf[biJ`I Y8wHY[]i]*(£Ygb?_oEopHJ_HJ]Hf;O@pXYgb/JKEHfg`a/W;] EonHy `fg\d]/\[E EoYg]Hb/pX\`]V_K]/Jbc¬/WPWgJ+_HWgb/]X]/\d\Y E¦`ab Eo_HY[bz]XJ+y/p pXJ¦Jb@`YgEH\E]c`a\dYgn-E¦EHU=JEobi^Hbc_oW;_Hpf[b J+W\Yg]/EHf[nV`a`a\[\´UMEYgJ]ip 0\]/Y[f[IKEHJJaJ+bW8IKU J+nHW[f[`WgYg]ib/EHYg~/`]/_ J § `b Y[W8_HJY[f[qb/_o\Wg]/In-U`aW[W`a`W]/\_f[J+fdbibiYg_HJ~cp*J/Cb YjDFY[EHEHJ+JMfgJfd3b ]/Y[f[J J/¬EoJ_·YgEHJ+fg3YgJ fdbibi_of[JWg`Yg`]/\_@]/nH3`f[J+]/W8I ]/¤Y[YgEH] JBIKJ+W[Wgb/b/~/_oJKp*`_·]/_HYgJMEHJB]i6_oY[JEHYZJ+Wg]/J f[ ]c\]/_?nH¯¥`DFJ+`EHWF~/J8^Hf[J+f[\dJ bcEo\[b/wXEo` J+_MW¤hZYg^oEHIJ n£pXWFJ+3Wf[Yg]/`_oI biYgW`Y[]/b _Y[J _H]XpXYgJ ]Bb WZYYdb YgY[EoJ J8fdb Y[J `YgYE[f[nHb/f[_o]/W{£`Ygbi`{H]c`_C` YZ3Ufg]cI Yg] ´¬£bcWpXJ+nHbi`_o\pCYgJ`p Y hZ^HIKnoWj3fg]cIWY[biYgJ Yg]CWY[biYgJ `YgEnHf[]/{obi{o``Y¡U DFEHJWg] hZ]c^Hf[_Yg`IKJ `_ . . . . i. λi(N − i). i. i N +1. i+1. i. λi. i+1 i N +1. i. (N − i)/(N + 1 − i) 1/(N + 1 − i) S˜i. 0.
(31) -@2N$NBC J@ ] @BADC[EP\M . . . (Ν−1)λ. 2(Ν−2)λ. 1. 2. λ. 3. 2λ. . 3(Ν−3)λ. (Ν−3)3λ. (Ν−2)2λ N−2. (Ν−2)λ. 3λ. (Ν−1)λ N. N−1. (Ν−1)λ. Νλ. +1. N+1. "{¯¥J`~/f^H]if[$Jj\wH]/ nH(`J+_HW f[J+WYgf[`\YgJ+p¦f[JabU0Ygfdbi_£W`Yg`]/_BpH`b/~/fdbiI ]i YgEoJO*bif[/]zy\dEobi`_BfgJ+nHfgJWJ+_cYg`_H~8Y[EHJj_-^HI§ Y[WY[f[bib/Yg_oJ WUC`Yg`\`a]/W¤]c_?_oJ©Xf[pXbin"`YgYg]/`J+]/_oWF_HJ~/`_c_H]cY[~K``_ob/]/~_CUB]/Y[^XpXEHY`aWZJ8Y[WZYdfg_-b `^H{HY[I ^XJ Y[{"J+´J+p f]/$`Yg\E?]c`nH_-`YgJJ+W _oW`YZU ¬H¤JPEobzy/J 9]/{XYdbi`_HJ+pCb/WYgEoJPWg^HI ]iYgEHJ i. λi(N + 1 − i). i. . NU. N Pi X ˜ IE[e−θ j=1 Sj | NU = i]P (NU = i) TU? (θ) =. `YgE. i=1. P (NU = i) =. EH`aSV\dE?`IKnH`f[ b/]zfgycJ+U¦WPY[9]@}-´ -¤JPEobzy/J. i−1 Y N −j 1 1 = , N + 1 − i j=1 N + 1 − j N. . Wg]MY[Eob Y. IE[e−θ. Pi. j=1. ˜j S. | NU = i] =. TU? (θ) =. i Y. j=1. ˜. IE[e−θSj ] =. i Y. λj(N + 1 − j) , λj(N + 1 − j) + θ j=1. N i 1 X Y λj(N + 1 − j) , N i=1 j=1 λj(N + 1 − j) + θ. EH`a\dE?nHf[]zycJ+WP&uc´ W`g_cEHY[] Jetfg§WtfgIK]cY[n"JEoJ]cb YgWgY``±_HY[3`]/~]cf_YgJ+`IKbcwX\[ÅJ wEB~cnH`f[y/]iJY[W ]X\Yg]cEHXJCY[EHJ©XJjn£JJ©X\´n"Y[J+J+\p YgJ+IKpBJIWgW[JbiWg~cW[JBbi~cpHJJapHbJUab3U]/fM`aWJb b/\d`E _HJ+f[b/Jf6abz3U=^H_onH\f[Yg]i`]/Y[]X_ \]/]c0&=Y[EHDFJjEHJ`a©XW n"f[J+J+\WgYg^HJp Y 1/λ. 00. Ë. 213547698.
(32) |. > 6!@2MU@ 9@BA 3 C[P\M. . . i ¾ +)( ¾ 3d @ B<&½@$tHB3G½ @$R X½@ON= NBC @( R0! @BADC9E 6P'¹ N [P ¹ 9@BM 2E¹ ¹ ¹ º *, . . . . . . #. -. . . N 1 X i2 (N − 1)! λN i=1 (N −i)! N i r 1 1 π = . +O λ 2N N. +|- z . . M_R0@B>*3Q:&@K&M > @2NO3%>OPQHB3;@=R> @2ADC[E*<F> 693G6IH=69A . 3Q:&@ @ B<&@$HB3;@=R X @2N=NBC @(R0@BADC9E P'N [P 9@BM 2E . . . #. wc +u-. N 1 X1 λN i=1 i 1 1 , log(N ) + γ + O = λN N. . IE[TU ] =. . 5 :&@2> @ γ ≈ 0.57721 P'N K&AD@B> NH=69M#NO3GC9MF3 0 . 6I6[AD@. MUR0@B> 3%: @3%576[8;: 6=< > @2A C9E*<F> 6[3;6IH=69A . 3Q:&@. IE[T2 ] =. -. . . ½®½'/ SV`_o\J ¬ \+bi_?{"J pXJ+fg`y/J+pb Y]/_o\J3f[]/I° z´;¯o]/fWgb//J]/ \b/fgSV`Y¡`U¦IKY[`EHb/J8fgnHUcf[¬X]V]iJ¢o]/F_op? {VzUC`pXWF`D1"3]/J+f[fg¤J_-b/Ygf[`apXb JY[p¦`_oY[~@]9Y[u-EHJ 8bi nHfn"Y J+_o¬opXbi`±© _opCYgEHJ_{VUCWJYY[`_H~ ¬XYgEobiY . IE[T2 ] = − . IE[TU ] =. =.
(33). dT2? (θ)
(34) dθ
(35). θ=0. . IE[T2 ]. θ. θ=0. i N 1 XX 1 λN i=1 j=1 j(N + 1 − j). N X i X 1 1 1 + λN (N + 1) i=1 j=1 j N +1−j. N N X X 1 1 1 = + λN (N + 1) j=1 i=j j N +1−j N X 1 1 1 + (N + 1 − j) = λN (N + 1) j=1 j N +1−j =. N 1 X1 , λN j=1 j. J©VEHn£`abi\dE²_oWg`a`W]c_C `Wwc´uXDF¬Hn®EH`aWab/WY¦Wg^HIKI¦b Y[`]c_²`WKV_H]z_ bcWMYgEHJE£bif[I]c_H`a\ _-^oI {"JfdW ³YdWKbcWUVIKnXY[]iY[`\ . . . *' . N X 1 j=1. j. = log(N ) + γ + O. 1 , N. 0.
(36) -@2N$NBC J@ ] @BADC[EP\M . . . EHJ+DFfgJ EHJ@_H`aWJ©VltYB^Hf[JJ+f!Wg ^HWY¦\]c~/_o`WZy/YdJ+biW_-Y+Y[tEHJ@DFEHJ©X`aWn"~/J+`\´y/Y[J+J+W8p _-^Huc¤I bi{"_oJ+pfB\]i]/8_£\\]c^onHpX`JJ+WKW¤]iY[EHJ8YgEHnHJ*f[]VIK]iZJ+ WgW[bi~cJb Y¦Y[EHJYg`IKJY[EHJ IKJ+W[W[bi~/J`aWpXJ+`y/Jf[J+p® 3%: ½ @ 3%½ 57d 6[d3H8;: º6=< > @2 "ADC[!E*<F> 6693G¹ 6I/H=69¹A ¾ P'N 0P @BM O¹ E ½H ' 0 ½ ¹ *, :&@(@ B<&@=H23G@$R M K&X $@2>6 H=6=<FPQ@2NK&MUR0@B> }V . γ. . . . 2@ MUH$@ H NB63%: C[3. #. IE[N2 ] =. . 0. . . N 1 X i2 N! . N i=1 N i (N −i)!. . IE[N2 ] = λN IE[T2 ], r. πN +O(1). 2. : @ @ B<&@=HB3;@=RMFK&X @B>6 H$6O<FPQ@2NK&M_RJ@2>*3Q:&@K&MF> @ONO3%>2P HB3G@$R> @2A C9E*<F> 6[3;6!H$69A P'N H IE[N2 ] =. %#. . . %#. . . s/. . . 0.
(37) . N +1 . 2. \+binXY[DF^HEHfgJKJpBf[JY[aEHb f[Y[]/`y/^HJK~cE n"JYgfgEo3J8]/f[f[IKbiYgb/`_o]cW\J]/YgEHJKYZ]/§EH]cbi_£n=p f[JabUbi_£p=^H_Hf[J+WYg~cf[``y/\J+Yg_CJ+p={VU=f[Ja9b\´ZU*£nHetf[f[]i]/Y[n"]X]c\Wg]c`YgWP`]/\_bi_wXÅw/{" J IE[NU ] =. IE[TU ]/IE[T2 ]. b/_op9\Z"L]/f[]/abifgU wX . . IE[NU ]/IE[N2 ]. PN N i=1 1i IE[TU ] = PN i 2 N ! IE[T2 ] i=1 i N. (N −i)!. IE[NU ] N (N + 1) = PN i 2 N ! , IE[N2 ] 2 i=1 N i (N −i)!. f[J+Wgn"J+U\´Y[^o`ycWgJ`_HU/~KtYgxEH]iJ Y[Jb/WgY[UVEoIKb YnXYg{"]/]/YgYg`a\jE?Jfd©Xb noY[`bi]-_oWFWgbi`]/f[_oJWP`_o pXzJ+n£´¬®J+_o +pXucJ+F_cYbi_o]ip s/¬XJ WJ+JYgEobiY . . λ. . . IE[TU ] log(N ) ≈ √ IE[T2 ] N. r. 2 π. /b _op 3f[]cJfFabUb/fgnH~cf[J ]iY[]X\ ]/¡o_C`aW]/b/YgEHnHJ+nHfFf[]©X]/`I¦fdpHb W+Y[¬cJ3]cU fFabif[~/J YgEHJPJ©Vn"JYg`\´IKY[J+J+p W¤IKWI¦J+W[b/W[biJ~/f;JYgpXEoJ+b/_BbzU¦^H_o^opH_oJpXfJfFYgEHYgJEHJPYZ^H]i_o§³fgEHJ]cWZn¦Y[fgf[`a\´JY[aJ+bpU nHf[]iY[¯H]X]/\f]c`&_o¬VWY[b/EH_o` \JJ/Y[¬XEH3J8]/f J©Xn"J+\YgJ+p?_-Yg^HEHI J+_ {"J+f]i$\]cnH`J+WF`WbinonHfg]z©Xbi`I¦_£p biYgJU Y[`IKJ+Wabif[~/J+f+ N. IE[NU ] ≈ IE[N2 ]. . Ë. 213547698. N , 2π. Np log(N )/ πN/2. N = 103. 00. r. IE[TU ]/IE[T2 ] ≈ 0.17. p. N/2π IE[NU ]/IE[N2 ] ≈ 12.6.
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(40) J \b pHY[u ``³W]cMYg_Bf[k`]i{oa^XWg]ob Yg`¬ ]/_H_®]HpXJj¡_`W¤_on£JIKJ+bipofg]/YgWjf[`aJj\Yg^H]?\a]/bi{"_ofJ¦\¬ J`±WZ_cY Y[Y[fgEof[`abi\´b/Y[YgWPJU@p¦{"Jn"_HJ]-J+_Wb/`fYg]c`{oYgy/EHWJMJ+JPfgYg\y/]?J+J_cJ+p@Y[_oJYgWf¤E£^Hb f[]/JY Y[Eob YYgEHJ bzy/Jfdbi~/JPWn"J+J+pC] y/Jf¤Y[`IKJ pX]XJ+WF_H]/Y~/]Yg] Jf[] ³ ¾ ½ 3 ¹ / ½ c ½ d 3 º c ½ ¾ ¹ d ¡Wg_?n"JYgJ+EHp J mbi_£pX]/I µ`f[J+\´Y[`]cb/_@_opObB]c{H¢o`_H` `YZYgU?JMOYgfdbz]Xy/pXJJ+ YgÅ`wXI¬"Jv J+ bc\[DFE?EH_oJM]X_HpX]HJPpX`aJ WYgb/EHW[J+W`_~c_HYgJ+fdbpCy/b/J+_?W``_o_`±Y[Yg`EHb/JKpXpX``fgfgJJ\´\´YgYg``]/]/_ _ ¬ . . . '. 2. . S. S (vmin , vmax ). . . v. )min . . . S ∈ [vmin , vmax ]. *. τ. θ θ. 0.
(41) +u 3Wg]cn"fPJJ+bCp pXbi^H_£fdpKb YgYg`fd]/bz_ y/JoYgbi`_oIKpJbib/YPfgJjWgn"\[JEoJ]cp WgJ_ Pb Y¤¶fdEobi_oJ_pX]cYgIEoJ`_£_HpX]XJpXn"J JY[_£f[pXbzJy/_cJY[U¦Yg`IK]iJb/Eo"b/nHWjf[JJ©XyVnH`]/`^of[J+WppX`b fgJ_H\´J+Y[q`]c_opXW`¬VfgJWg\´n"YgJ`J+]/po_W ¬ b/_ofdp*biDFnoYgWEHfdJBbbiy/f[WJ+]/Y[^HbiY[_oYg``pCIK]/_oWgJ+b/]W+fgMYgUEo¶bipXY`aEoWZJY[YgEHfg_·`J8{HbC^X_HY[_H]X`]XpH]cpX_oJPJW8fgJf[]/biJ+¤nHbcYgn"\[EHEHJJBJbiWfd]XWb \]/bi{"_CYg]c`Y[]/^HEH_·_oJ8pHbi]ib/_oY[fgp·EHU@JpH`±fY8`f[Wg`a`J+WpH\JYgJ``]/Y[Åw _EH³J E£fPbf[y/J J¦£J+{"\´JY[J+J+_ p WgEHi¬]zw 6_]/Y[fj]@Y[EH{"J¦J¦bi^Hf[_HJ+`b § 3{"]c]cfg^HI _opH/bi¬f[`J+W6¥{"3J+]c`f_o~ b/fgf[{HJ `oYgJfdbi\´Ygf[`U?_H~ pX`]/fgfJ\´Y[`fd]cbi_®n ¬0biWgf[n"]/J^HJ_£pp bi¥_oDFp@EoYg`fdWb`ay/W;J+`_CYg`\IK]/_-JKYgfdpXb/`WWYgY;f[`{H`^XYgEBYg`]/Y[_£EHJW¬0m`fgb/f[_oJ+pXWgn"]/J+I \Yg¶·`y/J bUV]/n"t]cY[`EH_-JY kDFOEH_H]cJ ]i{HY[`WEHn"` JY¡J+UCfMJ+ppXO`T10\]Hb/JpX_f[JJ {£_oJ \J¦EHJIJ-`af[WP^oJb/Y[_HEo ]Xb YgpXY ] JYg+WFEoJbiJBf[]¦f[IJjWIK``_o_H]/\`I JPf[Jj^HY[EHI `J /J+Wg_Hn"U¦]XJpXYgJ+J8]Kp I¦{"JPb/`\_-]/Y[_ob/pX\`_o]VJ_cJWW8Y[b¦f[_Hbi\Yg]iJJ+Y fgp Y[Eobi_Hb`_J+y/b/JKWgfn"Y[Y[]@JEHJJ8{"p?J\]/J+_HWZ_cY[YgfgUBJ+`afF\´3Y[]/]/fU=bKYgn"EH]cJ8` WgI`±biY[`f[`Ygy/J+JboJcp b/I]c^H_cY]iYg`IKJ/ &¼ ½o» Q ¹ ½ # d ¾ ½ q# d ¹ c ½ ¾ ¹ d ¡]c_8_*Y[b¦EHJ;YZY¡¤]i§¡]ipX§¡pX`I`IKJ+_oJ+_oWg`W]c`_o]/_£bibicW$m-^ob/bi_of[pXJ ]/I biY¶Yg`a\b/Jcj/JDFf¥EHO@J ]/Yg{H`I`` JMYZU `aWjOpX]X`aW[pX\Jf[-JJ+Ygb/JM\dbiE _£_Hp@]XpXbiYjJ;IKJ+bc] \[y/E@J+WYg`b/IKWJbWZf[Y[b/J_on*pX]/Jb/I \dE@¤b/_H]H/pXJ+Jf Eo]/bcW`±YdbCWMnH\^Hf[]/f[fg{oJ+b/_c{HY `n"`Y¡]cU@Wg`]iYg`]/_® ³]iYgEHEo]/J nH_Hno]X`_HpX~CJBYg`aW ]n"bC]-n"W`]cYgWg`]/`±Y[_o`]cJ+_=p·bi]/{"_] b@y/J/{"¬0]/{"^oJ+_o] pH8bif[¬0UcYg¬]CYgY[EoEHJJ_`J_o YWZY[¬®J+]/bcfjp·Y[]C]iYgEHEH]cJKnHfgnH`~/`_HE-~ Y ]0¤1 b/_oY[pXEHJJ*f[`a_Hb ~KYY[bi`\YJ@bK`Y\]/EH_o]cWY[nob/W¦_cY{obcWg\[n"JJY[p ]·3fgY[]cEHIJWg`b/_-IKYgJJ@fdWgJ+WY[\´biY[`Yg]cJc_C(YgDF]KEo`_c`WBY[JIKfdW]zJy/\´Y[J+`I]c_ J+_cY[YCEHf[\]/b/^H_ ~cE?{"JbKWg\J`±YZJUc_¬HbcWBEHJWf[]cJ8IKbiJ]/_o]/J Y[WY[EHbiJMYg`W]/Yg_of[Jb/JfgYdUjWpXb/`afgWJ Ygf[IJ`{H-^H^oYgb/`]/_8U?]/WgnoVYgbcEH\J;J+p@]Xb/\_obiYgp`]/n"_PJ+fg]/n"HJ+bF_oY¡pX¤`a\]i^H§¡pXabi`IKfY[J+]B_oWJ`]/b/_o\dE@b/i]ifdY[biEH_oJpHf]/&I(O@¤b/_HbiEo/b JYgfY[b/]/_@_ b_HJW$Y¡c¤^£]/bif[f[XJ$ab YgDFYg`aEH\JJ `afdWtbi_o^opX_H]c`±3I ]cfgI ¤b/] y//JJ+ftfYg\+EHbiJ_?b/{"fgJ JbH\¥]c_oDFWEHYg`f[W6^onH\´Y[f[J+]/p n"J+3ffgY[]c`I JW$Y¡`aW6b8]¦\`_o]/pX_oJWgn"IJ -J+^H_oJ+pX_oJ\_-JY]i]/"_HY[JEH§³pHJ`9IKb/\´JY6_oWgY[`Eo]/b _oYb/b0fdYZbi_o]/pH§³pX]/I`IKJ¤_ob/Wg`/]/J+_of[b/W+¬ b/3Y[_obip/J YgE£b YjYgEoJWZ`_CYdb ¯oY[`]/]cf[_oI bi^Hf[UCab ]H9uo\b zY[s/`]c6_@`_ ]isXb ¬Xn®WgU-tIKucs-IKv J£Y[Y[fg]M`a\ ]/fd{HbiY[_obipH`_B]/IYgEHJP¤WbiY[bi?Yg``]/__o]/b/_HfgUKJpXpX`a`WIKYgf[J+`_o{HW^H`Yg]/`_]/_C`Wjb/^H_o_Hp¦`3Yg]/EHf[J+I _ WgJY -@2N$NBC J@ ] @BADC[EP\M . . . τ. S. . . . / *. . $5. . vmin. )-. . . 1/4. +. . n→∞ p = q = 21. . *. $ &
(42)
(43) _HDF]HEHpXJJ+_VWF^HIKI] J+y/fgJ`a\`b/_@Xf[bKJ+WgW ^H- ^oY[b/W$fgnoJfg]iJ+$WgJWg_-` YgJJ pM4EHVIJ+fg×4Jb YgVJ+I?f6¬Hbif[J¤`Yg{oEHbc]cW^XJYpMno]/bi_K^£WgW``I _H~o^H ab Yg`]/_nofg]c~/fdbiIKW¥`_KEH`\dEKIK]/{H` J . . . . ¶·¶·bbUVUVk n"n"W]c]c``IK_-_-Y YJ_-OOYg`]c]/]/{H{H_o``J+ `p`±Y¡YZUKU`_ O@O@]X]XzpXwpXJJoYgnHEHf[J+¡]/_fgn"J]cYgEHbiWgJ+f[`aWPJ?pM`Wg_ J]/y/f[Jzfdw bi¥¤3JnHE£``YYgb9Eoy/bi]/Ja^XWM^oYWY[noJ]bip*^obzYgWgy/EH`]c_HJK`~-p`¥InoEHEHJJ+`a\d_IE¦J+W\_c`]/I Yd_ob ^HWgYga`a`b WZ]/YdYg_W$`_H]/]i~=;W[YgYgbiEHEHIKJ@J¦nHmm`b/b/_H_o_o~PpXpXY[]/]/EHII J `b/_H_o`pCYg`abi]HF\Wb n"Y[J+`]cJ+pH_oWMWFb/b/fg_oJPp W[bi]XIK\+b nHY[J+`]cp _o3W8f[]/3fgI ]cI°YgEHY[J8EH^HJ_o`f`±3W]cY[fgb I Y[`]/pX_o`aWZbiY[f[fgU`{HpX^X`aY[WZ`Y[]cfg_®`{H ^XY[`]c_oW=DFEoJ_®¬6Wg^H{oWgJI-^HJ_-YWgn"JJ+poW µjpX]X`f[J+J+SVWF\`_HYg_£`]i\]/YJ¤_Y[n"EHO@]cJWg]/JPWZ{HYdb/`b _-`YgYZUC`U·]/_£pXObi` Bf[]XU8\pX^HpXJ+±`aYZWUcYgbif[_o`{Hp^HYgYgEH`]/J _m]ibi£_oY[EHpXJ]cI°]X\+¶b Y[b/`]c/_ J+fM]/"Ob]/_H{H]H`pX `±JFYZU·`W$O^o_H]X`±pH3]cJfg¬IqY[EH`_J`{"f ]i`Y[IKEMnHYgEHJIKJmJ+_cb/Y[_obipXYg`]/]/I _ bKWgn"¯HJ]/J+fp·Y[3EH`J _m-I bi_o pXE£]cI ¤bc¶·Wb\[UVEHn"]-]cWJ+`_-_CYj^HO_o`±]/3{H]c`fg IK`±YZU?UBO@`_ ]XpXJbi_opCY[EHJ mbi_opX]cI µj`f[J+\Yg`]/_*O]/{o``Y¡UO]XpXJ+&¬ -. *. . *. !. . 00. Ë. 213547698. [vmin , vmax ] = [4, 10].
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(48) +. . K3I¯o^H]/J_of¤_-\J+YYgbc`]i]/\[-_ E YZIK&SV]]cJ_H{H\´]HYg``pX`]/Y¡J+_ UBW®b/IKuo_o uo]XppXwcEoJbz]/y/biJ$_£Jp¦Y[EHWZY[3J¦]/`I¦fF`_cbiy YgYgbiJ+J+f[fp`§³]/IKYg^oEHJWJ;J\YgpX]c`_H`aIKWZ~?Y[I fgYg`{H^H`IK^X_HY[`aJ+\`W ]cb _¦Y[{"`J]/YZSV_ J+f[\´JbcY[J+`pX_]/`_8&Yg¬VEHuoJ uoJWJK zEo"YZbzy/biJ_£]pjW_o`YgI ]XEHpX^HJ;J+abiW+b ^HBY[J+YgDF]ip¦§¡EH\Y[J¦EH] y Jf[biJ+IKfgWg`a^H] biy/_oY[J\W+§J¬ Y[{ofd`bibc]c_HW_·J~cp J3]/]/]if _$Yg\+EH]c|cJ IK|HIK¬I |/|/J^HJ|_HYg`a]c`\_H{ob ~*WgY[J`Yg]cf[`y _CIKb fdY[JCb/`]cpHW_oJ!`W¥c ^obiJ_o_opM\J+W EH`a`a\dW EKyibibif[`J¤pfgJ+3]/n"f]/fgbiYgJ6p Y[EH{"fgJJ+] JB8IK¬/]/WgEH{H]z` `±YZUYgEoIKbiY$]XY[pXEHJ+JWMe¥bi]/_o`aWgpWg]/3_]cfMbcWgb@Wg^HabiIKf[~/nXJ§ ¹ $» c ¹V¹ ( & Q ¹ 3+ 3 ® ½ ¯¥³L¤`~/µj^H¯tf[JuP]/$pX`aY[WgEHnHJ bz`UX_cWY[J¬/fg]/§_KIKbPJ+JYg]c`~i_H§¡~BWg\+Ygbi`IJ;JM3]/{"ftJYgYZEoJ J+J_@b ©XYZ`aW]¬ Y[_HEH]XJpX`J_-Wy/J+3f[]cWgfJJLb/^H\dE@I IK^Ha]/b {HY[``y/` JYZU?µjIK`WYg]Xf[pX`{HJ+^X¥Ygbi`_o]/_¦p¯o3]/^Hf_oYg\EoYg`fg]/J+_ J 3pXpX]cD``afB10WYgJf[Jf[`b/J{H\d_c^HEY@Yg`\]/\]c]/_IKIK3`I I Jc^H^H-_H_Hb `a`a\\Wb biYgfdY[Ygbi``]c]/`~/__ E-fdY6fdb/b/pX`pX_H``^oJKWB]/ _BJ*b Eo]cbz~iy/§¡J*Wg\+bib/¬ J¤Wg] 3]/nHf;Yg]iEHYg¬ YgJ J+pb Yg©XEH`JWd´`¬c_V¯oy/]/EoJf?fd]cWWgJJJb/`L¤\d_-E µjYgJ¯IK_£W]c`]iY¡{H U@`bi`9Y¡_`&U J/JVIK©XWn"]X]/]/pHn"_oJJzJ8 _cbiY[_o`b/`p W Y[\+EHb/J WgJ+`W_-y/YgEoJ+f[biWgYPJ8]iJ$YgE£EHbJ y/JJ©X`_Vn"y/J+J+\WYgYgJ`p?~cbi`_cYgJ+Y[pJfg§3IKYgEHJ+f[JJYgJK`_HI~¦]cYg{H`IK`` J Y¡U*]c{XIKY[]Xb/`pX_HJ+JpWPbibc_£\f[p@]cW[Y[WFEHfgb/J+JKWgpX``TI 10^HJf[biJ_-Yg`Y8]/_o\W+]/FIK¯HI ]c^Hf_Hb/`a\bi_HYg``]/_H_ J Y[fd\+EHb/b/pXJBWgJK`J©Vn"EH]cJ+J¦_HfgJ\Jb/_-_=Yg`abi]/F{£WL¤J+µjfgy/¯Jb/&pH\+_=b/bi_·WgJEH©XJ+\p·JJBJ+_H`_c_H]iY JzY[´JBbi~cWg]/fgNjJ+IK_HJIKJBUJpX_-3`D]/1"Yf J+{"fgYgJ+JEH_oY¡JC¤\J+JmW Jbi_={"_oJY[pHYZEH]/JKI°J+JJWZ_·¶·Y[`YgbiI¦EH/biJBJYgfMJ+Jp WZOY[`L¤I¦]cµ{Hbi`¯qYgJ+` Y¡pU 9Wg]/L¤Oµj` p=]X¯ pXJ+`b/_H;_oJz`_·p=b/Y[Yg_oEoEHpJJ J©Vn"DF]c_HEHJJ_-9Ygb/`abi\´Y;L¤Y[Eoµb ¯Y¬cbc3]/W¤fYgJ+EHb/J8\dE¦`_cIKY[J]/fg§{oIK`J`±YZJUYg`IK_H~]XpXYg`J+IKoYgJ8EH`J8_£\L¤f[µjJ+b/¯WgJ+]/W+0 YgEHJj`_-YgJfg§³IJ+JY[`_H~MpX`aWZY[fg`{H^XY[`]c_¦`aW;J±§ fdb/Wg^Hb/nHpXIKnH`fgnX]zYg©X3`]/`IK_£`YgWFbiE YgnHJf[p=J+bcWg\n"{-J+UJ+pC\´Y8b/]/_·_?Yg]@J+Jb/©XYg\dEHn"ECJ]/_HIKWgJ+` ]/+_cJK{HY[``b/]/ `±;¤YZUBpXYgEH`aIKWZJ Y[]Xfgb/`pH{HfgJJ+^Xb-Y[ `¡]c_o¢o_®pX_o¬$J+pHb J+W Y p `¬HY[J+_HW b/]XWfgpXY8]XJ]i3W¤]cY[WPf IK`Wg] _IKy/JY[b/EH`6_oJBpXYgy ]*Jbin"f[IKJ+`]/_o]X^opXpXWPJJ_-fd`_oYgb Y[pXU JJ+n"]/Y[Jf[b/_oJ+_opHbcWg\[JIEC_o\`a]/W[J YgWEH`bc]/J+WZ_ f§ b/_op3^XYg^ofgJjpX`f[J+\´Y[`]c_oW;bi_op¦Wgn"JJpHW&bi_opKYgEHJ+fgJ3]/f[J]X\biYg`]/_oWd¥]ib8_H]XpXJb/fgJ`_opHJn"J_opHJ_cY¤]i n£b/WY . . . . y. R = 50m 100m 250m y. λ. R = 250m. -. . 0.
(49) s. -@2N$NBC J@ ] @BADC[EP\M . . . 0. 0. 10. Random Waypoint Mobility Model. −2. R=0.05km λ=0.075217 R=0.1km λ=0.14944. −3. Random Direction Mobility Model. −1. 10. 10. Simulation Exp(λ). 10. Inverse CDF. −1. 10. Inverse CDF. 10. Simulation Exp(λ). −2. 10. R=0.05km λ=0.057694 R=0.1km λ=0.11574. −3. 10. R=0.25km λ=0.29869. R=0.25km λ=0.39248 −4. 10. −4. 0. 20. 40. 60. 80. 100. 10. 120. 0. 20. 40. Inter−meeting time (hours). 60. 80. 100. 120. 140. 160. Inter−meeting time (hours). 0. 10. Simulation Exp(λ). −1. Inverse CDF. 10. Random Walker. −2. 10. R=0.05km λ=0.00013795 R=0.1km λ=0.00026638. −3. 10. −4. R=0.25km λ=0.00043182 1 2. ¯¥O`]c~/{H^H`f[J` Y¡uoU? O¡_-]Hy/pXJJfdWg¬VJY[L¤EHJ µm¯bi]/_££pXY[]/EHI J`_cµYg`J+fgfJ§³\´IKY[`J]cJ_@Y[`O_H~P]/{HYg``IK `±YZJU?]iO"YZ]XpX]8J+®_Hb/]X_opHpJ+W$Yg3EH]/J ftmY[EHbiJ_oYgpXEH]cJI m¶·b/_obipX/]cJIqf¶·O]XbUVpHn"J]c `_-Y 10. 0. 3. 4. 5. Inter−meeting time (hours). 6. 7. 4. x 10. . [Y[YpXf[``]cb/f[_oJ+_o\´biWY[f[IK`UK]c`a_oWgY[WgW `IK`]/b/_Jc_o¬Xpf[Ygb/EHW_HJ+n"~/_?JJMJYgpH]iEH¤W J+fg]/]iJP_H`aJKYgWEHb/b`a_HWMnH]i_HY[f[EH]/]X{oJpXf´biJc{o `{"³`JY¡3U ]/f[JJMY[YgnHEoEH`ab \dJKY=_HY[YZJEH©-J]Y U \dIEo`]cbi {H_oIK`~/JJ J+J]i_HY ¤]H3pXpX`J+`_?fgWMJYg\´b EHY[YMJ `]cfdWg_biJ_o_£]/pXWJP]cJI]i`Yg$EHbi{"J+YMJfP`Wg_H_o]/~K]XIKpXJCJ/`YgWZEHYdk¤`b _ Y§ Y[nHEHf[JB]X\_HJJWg©VW;YMf[J\dn"EoJbib _HY[~cWJ¦`Y[]iWgJpX`b/f[_oJ+\´p¦Y[`Yg]cEH_®J+¬fgJj{"`aJ+WF\+bib ^onoWgfgJ¦]c]i{o¤bi{HY[EH` JB`±YZU `_opXJYgn"E£J+b _oY¤pXYgJEH_-JYWJb/_HW[W]X^HpXIKJWnXY[`]c`_o0W8IKfgJJJ\YFb/{"JJ+3p·]/f[biJ{"Y[]zEHycJJ/¬®_HY[JEH©VJY Y[\d\d`EoEo]cbibi__H_H~c~c{"JJ+JWF3]/]/]if[$J8pXpX`{"f[`J+f[]i\J+Y[Yg\´E@`Y[]/`_H]c_®]X_CtpXDF`aJW WEHZ`IKW`_oJUVJ+J`biJ+Y+f[pHU DFW¤jEHb f[J J~/aJJb ©V]cY[n"IKJ+]cp J_HYgYgJf[]`a_-\YgY[`apHEHbi`JPWYgYgpXf[``aIK`WZ{HY[JP^Xfg`YgYg{H`fd]/^Xb_¦Y[yc`]cJ3]c_J+f;p n"YgEH]c{"JjnoJW]_Vc^H^HfgJI n@Y[{"{"EHJJ+J8f¤\+bi_H]i^o]XWgpX\dJPEoJWFbiY[EH_HIKJ ~cJJ+_-JW;^oY]iI {"pXJ`fgfJ]/\´§ W{HgJ`f[`±y/YZDFU?J+pEHOJ?¬0yV]H9`bcpXbC\´JY¦®W`]/I YgEo_^HbibKaY¦b Y[W``no_c]cY[EH_oJJW+fgf[¬£§J IK3]cJ+fjJ} YgYg³`EH _HJ ~¤Ygf[`IK]zJ+WB_H`abibif[_=J?OJ©V]/n"{o]c`_H`±YZJU_-YgO`abi]XpXUJ+$pXb/`aW_oYgp@f[`{oYg^XEHYgJKJ+pmb/Eo_obcpXW¦]/IRbifg¶Jb/bzpXUVU n"{"]/`J+_cJY8_ O]c{X]/§§ nob/f[b/¶IKEHJJYg_=J+fJJWZ©XY[`EHI¦`{obi`±YgY[`J+_Hp·~ 4b YgEocJ^£`b/_-pXYgfdJ+b _oY[W``\YZU@pXJ+]/n"6JYg_oEopXJJ+`_o_-\YgJ¦J+f]c§³IK_JY[JEHY[JB`_o\~ ]cIYg`IKI J^o_H`\+¬ b Y[`J]c_=Eobzfdy/b/JMpX`]/^o{£W WJf[y/¬Jbcp?WPY[pXEoJ+b nHYP`a\´YgYgEHJ`ap W . q. q. . / . λ. R. 00. Ë. 213547698.
(50) > 6!@2MU@ 9@BA 3 C[P\M. . . . . . 6I6[AD@. Simulation (Random Waypoint) 2 λ=0.00017736R +0.0011374R Simulation (Random Direction) λ=0.0006224R2+0.0014187R. 0.5. 0.4. λ. 0.3. 0.2. 0.1. df¯¥b/`pX~/^H`^of[WJ }£mJ+biYg`]/_oWgEH`n@{"JY¡¤JJ_*YgEHJM`_cY[Jfg§IKJJYg`_H~BYg`IKJM`_-YgJ+_oW`YZU b/_opYgEHJ\]/IKI ^H_H`a\biYg`]/_ `O_@]H¯pX`J~/F^ofg3YgJPEH}`aW3]c]/fF{oWgYgJEof[J y bimYg`bi]/_£_*pX]/biIaW]C¶\+bibzf[UVfgn"`J+]/W`_c]cYj_@OY[]]/{HYg`EH `±JKYZU?mOb/_o]HpXpX]/JIRbi¶_opCb/Yg/EHJ+J fPmObi]c_o{HpX`]c`I Y¡UµjO`fg]XJpH\´JYg`¥]/{H_*^XOY8pX]/{H^o`J `±YZYgU] Y[Y[EHEHJMJ pX-T`^o10bcJpXf[fdJb _£Yg\`aJ \K`fg_J+Wgbi\+Ygbi`]/`_o_oWg~KEH`Ygn·EHJ {"f[JJ+YZWg^HJJ+Y[_WEobzy/biJ _op _H]/Y{"pXJJJ+n"_@J_£`_opH\W ^o]cpX_Jp@Y[EH`J _@IKYgEo]/J {H`¢o`~/YZ^HU=f[JIK]XpXxJ+]i&¬tYb/WW ^Hf[WnHEHf[] `Wg`_H_·~/`_U ¯¥`~/^H¡_cf[YgJ^o}£`±Y[ `y/J+Uc¬ Y[EH`W -^ob/pHf[biYg`a\;pXJ+n"J_opHJ_o\J¤3]/ ]zW3fg]cI YgEHJ¤9b/\´Y$YgEobiY$_H]HpXJ+W$\b/_MbinonHfg]-b/\dE J+b/\dE Y[]/pXEHYgJ+EHJ n"J+J_of;_o]X3pXpXfgJ+]cJ _oI \\J8]cb/IK]/_-UKI ^Hb/_H_H]/~/`a_ \b J/Y[¬-`]c_EH`afd\[b/E pX``I^oW+nHF` DFJ+W;EHY[`aWEob b/Y¤fgJ`_cbYg`aJ+Wf§³bBIKpXJ`aJWY[?`_H]i~ 6YgWg`` IK+J J+W¤pXJ+n"¬£J_opBEH`a\d]/E@_B~/YgEH`y/JJ+Wbif[YgJ+EHb J pX-J^o¢obc_HpXJf[p¦biYg{V`aU \ Y[Eob Yk`aaWFW]j^o_HWgJ+]/p YgJ; Y[Eob YtpX^HJY[]YgEHJFb/{oWJ+_o\J]io`_cY[Jfg3Jf[J_o\JW pX]VJW_o]iY$pXJ+n"J_op ]/_ Y[EHJ¤fgJ+bzUnofg]/Yg]X\]/ ¾ ¹ 6¹ ¾ ¹ ¹*½ ' ¹ $» c ¹V¹ ( & Q ¹ Y[JªEoWZJY[b Y `YKI¦Y[b EHY[J]/fFf[3yV]/W fF{"YgJEoJ YgEHbiJ8^X`Y[_c]iYg§¡biJ+\f[f]/§³JCf[IKfgI JJaJ^Hb YgYgYg`^o`_H]/b/~M_BY[U3`^HIK_o`_oJ+\WpXYg`J+{"]/n"_JJYZ_o]/pX JJ+J+_c_CYBYZfgyX]KW+¬;~/`y/JJ_CEo_ob]Xy/JpXJ+^oW+WtJDp]KY[\dEHEHJ?J+\d3]/¦Yg]zEoJ8`_Hb/~·W[Wg^H\IKab/W[nXWYg`a`\]/b/_ 0 0. 50. 100. 150. 200. 250. Transmission radius (meters). λ. R. λ. R. . -. λ. πR2. R. λ. . . {τ (n)}n. {τ (n)}n. EHJ+fgJ. {τ (n)}n. ρm (h) =. γm (h) :=. γm (h) , γ0 (h). h ≥ 0,. m−h 1 X τ (n) − τˆ(m) τ (n + h) − τˆ(m) m n=1 Pm τˆ(m) = (1/m) n=1 τ (n). KI`aWPJ+b/b/_=_CJ+3W]/Ygf `I¦b Yg]/]c{ofWgJ]/f[;y YgbiEHYg`J¦]/_ob/W+^X Yg]/§³\]zyibifg`bi_o\J 3^H_o\Yg`]/_®¬`YgE m. YgEHJ¦W[biIKnHJ 0.
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