Relaying in Mobile Ad Hoc Networks: The Brownian Motion Mobility Model
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(2) INSTITUT NATIONAL DE RECHERCHE EN INFORMATIQUE ET EN AUTOMATIQUE. Relaying in Mobile Ad Hoc Networks: The Brownian Motion Mobility Model Eitan Altman — Robin Groenevelt — Philippe Nain. N° 5311 Septembre 2004. ISSN 0249-6399. ISRN INRIA/RR--5311--FR+ENG. Thème COM. apport de recherche.
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(395) .
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(405) vxr √ u= L−r 2. v=0. Z. L−r √ 2. u=0. Z. u. h(u, v)dvdu = v=0. =. Ìh
(406) }yhZ¶ tzjvxcDv. Z. L−r √ 2. u=0. Z. Z Z. L−r √ 2. u=0 L−r √ 2. u=0. Z Z. v=0. u. h(v, u)dvdu = v=0 L−r √ 2. L−r √ 2. Z. L−r √ 2. v=0. Z. u. h(u, v)dvdu.. °Nbdcf~tl{~¥³{h8t. u=0. h(u, v) = h(v, u). Z. v=0. L−r √ 2. h(v, u)dudv. u=v. h(u, v)dvdu.. v=u. u. h(u, v)dvdu = v=0. IE[TL,r ] =. D6D %Ù EGFHJII. √ u= L−r 2. Z. 64(L − r)2 Dπ 4 L2. Z. 1 2. Z. L−r √ 2. u=0. L−r √ 2. L−r √ 2. 0. Z. Z. L−r √ 2. h(u, v)dvdu. v=0. h(u, v)dvdu.. 0. ¿ {À.
(407) . % $ = . 0. L. 2L. (I−2)L. (I−1)L. IL. Í)~¥{|h N yxc{~¤}=z{ª3h8£¥Jrs~¥}fjg zZf~¤£¥h8t8° sp ~¥}yh vcf√h fz{|ff£¥h t®h8~¥h8t~¤} h(u, v) xh|}f~µªBzZg £¥r :z{|f}fh8 P ~¤} vxcfh ³D{~ £¤ht ¿B~¤v9t $t®zZ£¤|uvxh@³D{£¤|fh~¥t:z{|}uh8=ªBxz{g{:zJ³{h@Är À¨¶ u, v ∈ [0, 2(L − r)] ¦,hdg<r~¥}ij{zZ§{hvxcfhd:zZ|f}uh8´y
(408) z{}ij{h8Zh
(409) }y
(410) hNvxcfh
(411) zZh8g vxz²~¥}Zvxh
(412) 9yxc{}f{h,vchd(~¤}Ävh8{9 1/k £u }´) t®|f=g πg<D/36vx~¤zZ} t~¤Z}td~¤}ÿ {À¨°Nbdcf~tl{~¥³{h8t 2 2. k≥1. ∞ 64(L − r)2 X IE[TL,r ] = Dπ 4 L2. =. 128(L − r) Dπ 6 L2. ∞ X. m≥1 nn≥1 m ∞ ∞ 4 X X. 1 mn(m2 + n2 ). m≥1 nn≥1 m. Z. sin(muβ)du. u=0. 1 m2 n2 (m2. L−r √ 2. + n2 ). Z. 4. L−r √ 2. sin(nvβ)dv. v=0. .. bdcfh@£{t®vl£¤~¥}fh5ªBz{£¥£¤zD¦td:h8y
(413) {|th cos( ) = 0 ª zZ j zuf¹° . ' 8 &3
(414)
(415) l* ÁahVyz{}$t®~uh
(416) ´vcfhVt~µvx|Dvx~¤zZ} fh
(417) f~y¨vxh8 ~¤} Í3~¤Z|fhVf° bdcfh
(418) xh={h ZJ®{y
(419) h
(420) }Zv t®h8{g h
(421) }Ävxt8¶Nh{y9cKz{ª £¥h
(422) }f{vc L ¶u{}<vch
(423) xh~¥tdjt®~¥}fZ£¤hg z{f~¥£¥h:h
(424) dth
(425) Zg h
(426) }Zv°6Á»h5uh8}fz vxhIÄr X vxcfh²g´zZf~¥£¤h²~¤}=t®h8{g h
(427) }Äv ° hv ¿ À,:h²vxcfh $ %" )d:zÄt®~¤v~¥z{}=z ªXvxcfh ¸^vxc=g z{f~¥£¤h5~¥}=~¤vxtlth
(428) Zg´h8}Zv° i Áah´{txt®0|fg ≤hxv(t)c v≤vxcfLhjfix=zuyh81,txt . .X. , I= {x (t), t ≥ 0} ~t·½"xzJ¦l}f~ }Vg i z vx~¤zZ}V¦l~¤vc¾©
(429) h8zu ~¤ª v5 } u"~ Q|t®~¥z{} y
(430) zsh ! y~¥h
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(432) }uh8}Zvfxzuyhtth8t8° {t®v8¶«¦,h Z tt|fg hvxcDvh{yxcth
(433) Zg´h8}Zvlc${tdxh¼hy¨v~¥}f :z{|f}$f x~¤htdDvlvcfh@h8}ft8° hv T = inf{t ≥ 0 : x (t) + r ≥ L + x (t)} :hvxcfh'v9 }$twªBh
(434) v~¥g hj:hvw¦,h
(435) h8}Âg zZf~¤£¥h8t X {} ¶svxcDvl~t ~t,vch¬9t®v"v~¥g h²¦lch
(436) } } {h£¥zuy8Dvxh8· vl´u~¥t®vx{}yh²£¥h8txt,vc{}zZ""h Ä|{£ X vxz r ªBzZg hT{y9c»z{vcfh88°<bdch xh
(437) £rVv~¥g Xh8t T ≤X· · · ≤ T :h
(438) v¯¦,h8h
(439) } g z{f~¥£¤ht X {} X ¶ . . . ¶ } X ¶uxh8t:h8yv~¥³{h
(440) £¥r{¶f xh²xh8y|xt~¤³{h8£¤r fh¬}fh=Är X jπ 2. i. i. i. i. 1. 1. I. 1. 2. 1. I−1. I. 2. 1. 1. 2. 2. I−1. Ti = inf{t ≥ Ti−1 : xi (t) + r ≥ L + xi+1 (t)},. 2. 3. i = 2, . . . , I − 1.. k |flzZswhy¨v~¥³{h5~¥}=vcf~tt®hy¨vx~¤zZ}~tdvz<y
(441) z{g f|uvxh IE[T ] ªBzZ i = 1, . . . , I − 1 ° bdcfzZ|f{cfzZ|uvvcf~tth8yv~¥z{}V¦"h'Ztt|fg h@vc$Dv °bdcf~tZtt|fg uv~¥z{}V~tg<{uhªBz{vch tx §{h,z ª$g vcfh8g<Dv~y
(442) {£ZvxxZy¨vx{f~¥£¤~¤vwr{°X±¯}$uh
(443) h¹¶{ªBh
(444) L¦ ≤th8ry
(445) z{≤}f2Lt)z{ªxh¼hy¨v~¥z{}'¦l~¥£¥£uyzZ}ijs~¤}$yhvxcfh"xh8{uh8 vxcDv'¦lch
(446) } L ≤ r ≤ 2L {} (x(0), y(0)) = (x , y ) vxcfh<vxx{}t®ªBh
(447) @v~¥g h }h
(448) h8uhavzVvxx{}twªBh8' g h8txtx {h\:hvw¦,h
(449) h8}5vw¦,z{J®Zyh
(450) }Äv)t®h8{g h
(451) }Ävxt«~¥t«vcfh,t{g hN{t ¶vcfhhu:h8yvh²v9 }t®ªBh
(452) vx~¤g h z{uv9 ~¥}fh8a~¤}Kpsh8yv~¥z{} ªBz{'Vt®h8{g h
(453) }Äv@z{ªd£¤h8}f vxc 2LT ¿R¦l(x~µvxca,vxycfh +L){~¥³{h
(454) }a~¥}f~¤v~ £,yzZ}u~¤v~¥z{}t9À¨° i. 0. 0. 2L,r. 0. 0. ')(3D*')+.
(455) {.
(456) "!#%$'& (. bdcf~t²z{$t®h8³ Dv~¥z{}¾ £¥£¥zJ¦t|tvxz¬}» v5z{}yhjvch´hu:hy¨vhÂv9 }t®ªBh
(457) vx~¤g h $h
(458) v¯¦"h
(459) h
(460) }ag´zZf~¥£¤ht {} X ªBzZ }Är·~¤}~µvx~¥{£Xyz{}f~µvx~¤zZ}t x (0) }$ x (0) °NÁ»h5¬}. X1. ¿ ÄÀ bdcfh²u~"! y|f£¤vwr<{~th8tN¦lcfh8} vxrs~¤}fvxz¬$}<vcfh²hu:h8yvh8<v9 }$twªBh8Nvx~¤g h²:hvw¦,h
(461) h
(462) }·g zZf~¤£¥h8t X {} ªBz{ ¶:t~¥}yh@vxcfh':zÄt®~¤v~¥z{}Vz ª ¦lcfh8}vxcfhv9 }$twªBh8:h
(463) v¯¦,h8h
(464) } X {} X X v9 §{h8tdf£{y
(465) ih5=~¥tl2,}fz . v. .|f, }fI~¤ªB−z{x1g ~¤} [iL, (i + 1)L] ° X bXz zD³{h
(466) 9yz{g hvcf~tu~"!<y
(467) |f£µvwrZ¶f¦,h@Ztt|fg h²vc vdvxcfh½,xzD¦l}f~¥{}·g z{v~¥z{}t {h5{£¤£«~¤} t®vh{urĸ¯twv9Dvh· v'v~¥g h t = 0 °»bdcf~t´{txt®|fg uvx~¤zZ}Ã~¥g f£¤~¥h8t8¶ =~¤}K v~y|f£ ¶)XvxcD, .vj. v.cf, hX:zZt~¤v~¥z{}Ãz{ª h{y9cg´zZf~¥£¤h' vlvx~¤g h ~¥t|}f~µªBzZg £¥r=u~twvx~¥f|uvxh8zD³{h
(468) ~¤vxtt®h8{g h
(469) }Zvj¿R~^° h{°"vchQsª\z{ª ~¥t |f}f~¤ªBz{xgzJ³{h8 [0, L]À¨°Ntbd=ch50tx g h²cfz{£ft"z{ª)y
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(471) [0, L] ~µª t ~tl{f~¤v9 xruÀ° }fz{vcfh8,yz{}$t®h"Ä|fh8}yhlz{ª:vxcf~tZtt|fg uvx~¤zZ} ~¥t\vxcDvvch:zZt~¤v~¥z{} z{ªQg zZf~¤£¥h vNv~¥g h ¿R~^° h{°¦lcfh
(472) } X hyh
(473) ~¥³{h8t<g h8txt{{h5ªBzZg X Àl~tt®v~¥£¥£)|f}f~¤ªBz{xg´£¥ru~¥t®vx~¥f|uvhVXzJ³{h8 [0, L] °Tbdcf~t fxz{:h8®vwr<¦l~¥£¥£«$h@|th8£ vh
(474) lzZ}«° \zZ:zZt~µvx~¤zZ} f°¥d:h
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(476) } 5h8£¥Jrzuy8y|f9t8° Íz{²£Dvxh
(477) ²xhªBh8h8}yhZ¶Q¦,h t®vx vhjvcfh ht®|f£¤v5~¤}a{h8}fh
(478) 9 £)ªBz{xg=°´i,zZ}t®~uh8v¯¦,zZJ®{yh8}Zv@th
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