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Coverage and Connectivity of Ad-Hoc Networks in Presence of Channel Randomness

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(1)Coverage and Connectivity of Ad-Hoc Networks in Presence of Channel Randomness Daniele Miorandi, Eitan Altman. To cite this version: Daniele Miorandi, Eitan Altman. Coverage and Connectivity of Ad-Hoc Networks in Presence of Channel Randomness. RR-5377, INRIA. 2004, pp.29. �inria-00070626�. HAL Id: inria-00070626 https://hal.inria.fr/inria-00070626 Submitted on 19 May 2006. HAL is a multi-disciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers.. L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés..

(2) INSTITUT NATIONAL DE RECHERCHE EN INFORMATIQUE ET EN AUTOMATIQUE. Coverage and Connectivity of Ad–Hoc Networks in Presence of Channel Randomness Daniele Miorandi — Eitan Altman. N° 5377. ISSN 0249-6399. ISRN INRIA/RR--5377--FR+ENG. ISRN INRIA/RR--5377--FR+ENG. November 2004 Thème COM. apport de recherche.

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(53) &&. hkv]*lonpx¬npvmn‚x§w]–. x=. ln a−ln Kρ−α σ. =.  α ln aρ K σ. E[R2 ] =. ¯]£3`·–{`0nF. +∞ Z dρ2ρ 0. ln. +∞ Z. $4*N‹-6. x2 1 dx √ e− 2 . 2π. α. Ψρ ( WPtx K ). hkx§w r0`·np[]`,x°wFn‚`q–u}z¦ylr0t{wFu`_}p–u`·z l‚tu¦§vmnp`_¦§ju¯m£3`·^zNj™zu„]„]¦§j™¶ v]]x§w]x  lnp[]`_t{}‚`q^¯]–u`0n‚n‚x§w]–LF σ. 2. E[R ] =. +∞ Z. dx. . −∞. eσx Ptx K ΨW. Z. 1. α. 0. =. x2 1 dρ2ρ √ e− 2 = 2π. . +∞  σx 2 Z e Ptx K α 1 − x2 √ e 2 = dx ΨW 2π. −∞. +∞  2 Z  2  √ 2 2σ Ptx K α Ptx K α 1 − x2 + 2σx 2 α √ = e e α , dx ΨW ΨW 2π. $4*556. £[]`q}‚`·£3`,v l‚`q†™np[]`·¨ªtu¦§¦°tU£x§w]–•x§w{np`_–{}pzu¦ (ő”7, F. −∞. $4*q”56. +∞ Z β2 x2 1 dx √ e− 2 e±βx = e 2 . 2π. `_w r0`·np[]`,w]tm†m`·xyl‚tu¦yzUn‚x§tuw„ }‚t{ z]x§¦§x¬noj©x§l–ux§u`_wFjF −∞. 1. −λπ (. Ptx K ΨW. 2. )α e. √. 2σ α. 2. $G‘’56. |R¦§`qz{l¤`·w tn‚`·n‚[ znn‚[]xyl}‚`l¤v ¦¬n£(z{lx°^#„]¦§xyr0x°n‚¦§j†m`q}‚x§u`q†x§w (+*qŠ,D³ Z[]xyl.l‚v]–u–{`ql¤nplÂn‚[*zUn.n‚[]`(„]}p`ql‚`_w r0`3t¨ ¦§tu–uw tu}p^#zu¦{l¤[*zu†mtU£x°w –0 ,!7

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(55) ’. 0 ,!)/2 */0. *.  . ¶ }‚t{^ ( uŠ ’7,⯠$G‘{‘-63xyl` Fv]x§Uzu¦°`qw{nnpt ¨ªt{}szwFj™x°w*r0}‚`zul‚x°w –<¨ªv w rn‚x§t{w. $G‘Š56. E[f (Rσ1 )] ≤ E[f (Rσ2 )], σ1 ≤ σ2 1 F f (·). ³ s`qw r0`{¯]£3`·[ zNu`en‚[ zn. E[Rσh1 ]  E[Rσh2 ], $4* 56. %)/2. $D‘U“:6. σ1 ≤ σ2 , h ≥ 0.. Z.z®kx§w]– zw*†I}p`qrqz¦§¦°x§w]– q“ 0¯Â£3`#}p`0np}‚x§`_u`bnp[ zUn,n‚[ `#w tm†m`#x§l‚tu¦yzUnpx°t{w²„]}‚t{ z]x§¦§x¬nojI†m`qr_}‚`zul‚`ql ^#tuw tn‚t{w]hxyr_=z¦§¦§2j£x¬np[n‚[]`,¦§tu–{w]tu}p^z¦Ml‚„]}p`qzu† σ ³   wnp[]`¦§x°^#x°n σ → +∞ n‚[]``q[ zkx§tu}t¨.np[]`}‚`l¤v]¦°npx°w]–w]`0no£3t{}‚®™£x§¦°¦2}p`ql‚`_^<]¦§`enp[ zUnt¨‰z#}zw †]tu^ –{}pzu„][ ( GŠ *,D¯Âx§wInp[ zUn·n‚[ `#l‚„ zn‚xyz¦‰r0t{^•„t{w]`_wFn·£x°¦§¦‰`qr0t{^#`•w]`_–u¦§x§–ux§]¦§`u³   w±l¤v*rp[²¦§x§^•x°n‚x§w]–@}p`_–ux§^#`u¯ np[]`b„]}ptu zu]x§¦°x°nµjn‚[ zn·znojk„]x§rqz¦.w]tm†]`<xyler0tuw]w `qrnp`q†Œn‚tzwkjtun‚[]`q}´w]tm†m`<n‚`qw †]l´n‚t ³   w †]`_`q†Œnp[]` „]}ptu zu]x§¦°x°nojtu¨ r0t{w]w]`qr0n‚x§tuwn‚t©z•w]tm†m`,zns†mxyl¤npzw*r0` ρ x§l F 1 2. . . WΨ |r = ρ = P l≥ Ptx. +∞ Z. 1 √ dae 2πσa. − 21. ln. a Kρ−α σ. !2. =. +∞ Z. α. x2 1 dx √ e− 2 = Q 2π. Ψρ ln W Ptx K. σ. !. $G‘u+ˆ 6 £[]xyr[¡¨ªtu}zwkj $ w]x°n‚` ρ n‚`qw †]l©n‚t Q(0) = zul σ → +∞ ³ Z[ `€„]}ptu zu]x§¦°x°nµj¥np[ zUn¯–ux§u`_w n tun‚[]`q}#w]tm†]`ql#x°wYnp[]`Œw]`0no£3tu}p®M¯Rn‚[ `nojk„]x§rqz¦w]tm†]`xyl#xyl¤t{¦§zn‚`†¢xyl zul ³ hmx°w r_` x§w²tuv]}·v]wFt{v]w †m`†I}‚`q–ux§tuwŒnp[]`_}p`•x§le|‰­kz ³ lq³zuwIx§w3$ w]x°n‚`#wkv]^<`_}·tu¨w →tm†m`q0lq¯¸tuv]}·n w]→`0no£3+∞ tu}p®@£x§¦§¦ | ­ z ³ lq³Rw]tn„]}‚`l¤`qw{nzwkj™x§l‚tu¦yzUnp`q†w]t]†m`u¯ £[]x§r[@„]}ptUFxy†m`lzw@x°wm¨ªt{}‚^zu¦k~ov l¤n‚&x $*rqzUn‚x§tuwŒt¨.n‚[]`,¨Tzur0nn‚[*zUn ³<c´wŒnp[]`•tn‚[ `_}e[ zw †Â¯Mx°¨ σ → 0 £3`b}p`0np}‚x§`_u`np[]`•†]`0n‚`q}‚^#x§w]xylonpx§rb„ zUnp[k­F¦§t{lpl^•t]†m`_¦ lim P = 0 „]}p`_kx§tuv l‚¦§j«zw z¦§j '_`†¸³   nbxyl<x§w{np`_}p`ql¤n‚x§w]–€n‚tIw]tun‚`©np[ zUn•x°wYl¤v r[¢¦§x°^#x°n‚x§w]–²rqzul‚`u¯.n‚[]`r_tuw]w]`rnpx°kx°nµj „]}ptu„`q}¤npx°`l·z}p`#†]}‚x§u`_w«tuw ¦°jIFj€np[]`#¨ª`qzn‚v]}p`ql·tu¨3n‚[ `v]w †m`_}p¦§jkx°w]–Œl‚„ zUnpx§zu¦‰„]}ptmr0`l‚l x°w«np[]x§l<l¤`_w*l¤`{¯ £3`©^zNjI}p`_–Fz}†€np[]`©}p`ql‚v]¦°n‚x§w]–Œw]`0no£3t{}‚®«zulzŒ–u`_t{^#`0n‚}pxyr•}zw †mt{^'–u}z„][ ( ŠF‘ , $ªn‚[]xyl<}‚`q¦§zn‚x§tuw l‚[]x§„ `0no£3`q`_w5zu†F­k[]tmr@w]`0no£3t{}‚®ml™zw †¡–{`_tu^#`_n‚}px§r@}zw †mt{^ –u}z„][ lxyl©w tn™w `_£,¯sl‚`_` ( ŠuŠ ,360³   wWl‚tu^#` l‚`_w l‚`u¯*np[]`b„ z}‚n‚xyz¦}pzuw †mtu^#w]`l‚lsx§w{np}‚t]†mv r0`†ŒFjn‚[]`b¦§tu–{w]tu}p^z¦2l‚[ zu†]tN£x§w]–©^zj@`<` "k„`rnp`q†@n‚t „]}ptm†mv r_`sz,w `0no£3tu}p®•l¤x§^#x°¦yz}Rn‚tbzl¤^z¦§¦ £3t{}‚¦y† ( Š“ ¯{ŠFˆ ,â¯{£[ x§r[xylRx§w †m`q`q†®kw]tN£w#npt„]}p`ql‚`_w{n3`0n¤np`_} r_tuw]w]`rn‚x§kx°nµj™„]}ptu„`_}‚n‚x§`ql3np[ zw@–u`qtu^#`0np}‚xyr´}zw †mt{^ –{}pzu„][ lq³ ƒ`q¦§zn‚`†I}p`ql‚v]¦¬nl·z}p`•tumnzx§w]`q†Ix§Dw ( Š /7,â¯Â£[]`q}‚`bnp[]`zv]n‚[]t{}plq¯¸Fj€v*l¤x§w]–@zr_tuwFn‚x§wFv]v]^'„`_}r0t{¦§zn‚x§tuw zu„]„]}‚tFzur[¯l¤[]tU£²n‚[]``_w]` $*r0xyz¦ux§^#„ zur0n2t¨]l¤„]}p`qz{†F­Ft{vmnÂr0t{w]w]`rn‚x§tuw*1l $ªxD³ `u³§¯Nzsl¤„ }‚`zu†F­ktuvmn†m`qw l‚x¬noj¨ªt{} x§w™t{v]}(l‚`0n‚n‚x§w]5– 6tuw©n‚[]`ew]`_nµ£3t{}‚®r0t{w]w]`rn‚x§kx¬noCj    03254)   * ] ³c´w©n‚[ `´tun‚[]`q}3[ zuw †¸¯k£3` R †m`q^•t{w l¤n‚}zUnp`q†©np[ zUnl‚tu^#`e¨ªtu}p^lt¨‰rp[ zuw]w]`_¦Â}zw †]tu^#w]`qlp@l $ª`{³ –*³§¯m¦§E[R tu–{w]tu}p^z¦Ml‚[ z{†mtN£x§w]:– 63z}p`,z]¦§` npt@x§^#„]}‚tUu`r0tuw w]`qr0n‚x§Fx°noj€¨ª`qzUnpv]}p`ql·Fj«zur0n‚v zu¦°¦§j  2L)! 4 032:4 E[R ] .>0 3(=, '3 @) 2L

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