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On a Coverage Process Ranging from the Boolean Model to the Poisson Voronoi Tessellation With Applications to Wireless Communications

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(1)On a Coverage Process Ranging from the Boolean Model to the Poisson Voronoi Tessellation With Applications to Wireless Communications François Baccelli, Bartlomiej Blaszczyszyn. To cite this version: François Baccelli, Bartlomiej Blaszczyszyn. On a Coverage Process Ranging from the Boolean Model to the Poisson Voronoi Tessellation With Applications to Wireless Communications. RR-4019, INRIA. 2000. �inria-00072622�. HAL Id: inria-00072622 https://hal.inria.fr/inria-00072622 Submitted on 24 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. On a Coverage Process Ranging from the Boolean Model to the Poisson Voronoi Tessellation With Applications to Wireless Communications. François Baccelli — Bartłomiej Błaszczyszyn. N° 4019 Octobre 2000. ISSN 0249-6399. ISRN INRIA/RR--4019--FR+ENG. THÈME 1. apport de recherche.

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(71) ÆǒŒ‹,¾ÈŸ‹C’,Ÿ–C° Ž] *† ¤w’×Sr]t,ÆNj =Ô ‹ ԑÓ¢1¾È˜Ÿ‹Œ† ¡M‹Œ†1’,ž]–Œ˜aÂRÆÇ¡]† ¡ mK€ & 7ÆNj,¾È‹C˜â=Ô ‹ Ô ¢ ˜ ¤w’ŒÆȤgŽ]˜ Ž=‹AÆǤ! âæq¾Ç˜·ËK† –‹Œ†1 *Æ¿Å}¢ ˜ ¤w’”ÆǤgŽ]Æ¿’;“ÓÆNj‹ŒŽ=ߢ1ÆȆ1¤g’£ç-Ô ™  o  µ"¶;š {› µ  ¸  ³³ º †

(72) ËÆǾȾc¾È‹C˜–C† ÙgŽ]ÆǖŒ†'œâ’”˜¬i†Z7–”¤]¡]˜  )¢1¾È˜Ÿ‹Œ† ¡|‹Œ†-’¼æ̤]˜Ÿ’Œ†’Œƒ¦’—’”ƒ=†¢1˜ Ž]¤w’£¬]¾È† Ž]¤]Æǘ ¤Ø˜ÃH¢ ¾Ç˜ ‹C† ¡Ø‹C†1’Œ‹A¤]†1† ¡Ø¤=˜Ÿ’¬\†¢ ¾Ç˜ ‹Œ†1¡cç-ÔÜ;¤‘ÃqŸ¢-’ˆËÆȾǾ–Œ† ÙgŽ]ÆǖŒ†7’”ƒ=†‹[’”–Œ˜Ÿ¤]° † –ž]–C˜ ž\†1–C’;“’Œƒ¦’Ã̘ – Ÿ¤w“‘° ÆÇŸ† ¤Ó¬\˜ Ž=¤]¡]† ¡Ø‹Œ†1’ZÆǤ ‘æÌËÆ¿’”ƒÓ¢ ˜  *ž¦Ÿ¢-’¼¢1¾È˜ ‹CŽ]–Œ†ç

(73) ’Œƒ]†¤RŽ= ,¬i† –A˜ÃÊ¢ †1¾È¾Ç‹Z’Œƒ¦’ƒ¦ †¤]˜ ¤NÅ}† *ž=’;“ ÆȤw’Œ† –Œ‹C† ¢-’”Æǘ ¤äËÆ¿’”ƒMÆǒtÆNjtŸ¾Ç *˜ ‹C’,‹ŒŽ]–C† ¾¿“M½¦¤=ÆǒŒ†ŸÔ $Z¤à† ÙgŽ]ƿŸ¾È†1¤g’*‹[’£’Œ† *† ¤w’*ÆNj,’”ƒ]’,’Œƒ]† £ë"õöõÐð -íqôxë @ë £ð-õöõ Æȋ¼=Ô ‹1ԍm‰Z¡]˜ ¤œži˜ ÆǤg’¼ † Ÿ‹ŒŽ=–Œ†t˜ ¤œ’Œƒ]† ‹Œž]Ÿ¢ †t˜ŸÃ—¢1¾È˜Ÿ‹Œ† ¡Ú‹C†1’”‹1×鋌˜m’Œƒ¦’«Æ¿’«¢ Ÿ¤œ¬i†,’”–C†’Œ† ¡ÚŸ‹« ž\˜ŸÆȤw’ž]–Œ˜R¢ †1‹Œ‹ † J ’¡ € – æx­NÔ §gç ˜ ¤œ’”ƒ=† ‹Œž¦¢ †t˜ŸÃ—¢1¾È˜ ‹C† ¡Ú‹Œ†-’”‹1Ô‚

(74) ƒ=Æȋ«ÆNj7|’;“Rž]ÆÇ¢Ÿ¾Ê‹Œ‹ŒŽ= žN’”Æǘ ¤âÃ̘ –A¢1˜·ÂŸ† –ŒŸ° †tž]–C˜N¢1† ‹Œ‹C† ‹|æqÆǤ➦Ÿ–[’”ÆÇ¢ Ž]¾ÈŸ– Ã̘ –’”ƒ=†ÎK˜N˜ ¾Ç†¤‘ *˜R¡]† ¾qצ‹Œ†1††Ô °=Ôq@LJ>Ax×]†1ÙcÔeæq®=ÔLJ Ôȇaç-×]žéÔ =Ÿ©=Ô ç-Ô IH:]9 FÊ:=9[JDQ FWV ?AB[Q2L CED ¶¦³nq¸  ¶  › :  ‰˜ Ž]°Ÿƒ]¾Ç“m‹Œži†ŸÖgÆǤ]°]×ct‹Œ¤¦ž]‹Œƒ]˜’A˜ŸÃ tËÆȖC† ¾Ç† ‹C‹Z¢1˜  * ,Ž]¤]ÆÇ¢"’”Æǘ ¤Ø¤]†-’;˘Ÿ–ŒÖÆNj¢£ƒ]Ÿ–[Å Ÿ¢1’Œ† –CÆÈÀ1† ¡Ø¬w“’Œƒ]†«Ã̘ ¾È¾Ç˜aËÆȤ=°°Ÿ† ˜  *†1’Œ–ŒÆÇ¢,¤]¡’”† ¢£ƒ=¤]˜ ¾Ç˜ ° ÆÇ¢Ÿ¾ž¦Ÿ–ŒŸ *†1’Œ† –Œ‹1ê J}êʒ”ƒ=†õÐë ŒîŸíqôxë 6ë î ¦í;ð  iî Œù UJ%

(75)  edêʒ”ƒ]†Añ¦ë ð-ò ë «í (¦ð -ô¿ï iîŸõN†1 *ÆǒC’”† ¡&Ã̖Œ˜Ÿ J;ù mJS  d ꗒ”ƒ]† -ô¿ï iîŸõ2í;ë Gë"ô að«òŒîŸíqôx뫒Œƒ]–Œ†1‹Œƒ]˜ ¾Ç¡Ø–C† ÙgŽ]ÆǖŒ†1¡ØÃ̘ –

(76) ’”ƒ]†7° ˜R˜R¡Ó–C† ¢ †1ž=’”Æǘ ¤Ó˜ŸÃe’Œƒ]†‹CÆÈ°Ÿ¤¦Ÿ¾ † *Æ¿’Œ’Œ† ¡Ø¬w“m’Œƒ]ÆNjZ¤g’Œ† ¤]¤]=ù JŸ  d êʒŒƒ]†mð « íEð-ò iîŸõ Gë"ô aðÆǤ&’”ƒ]†«ÂRÆÈ¢1ÆȤ=Æǒ;“&˜ŸÃeŸ¤w’”†1¤]¤¦ ŒÔ y9  ^ b  d ×]˃]ÆÇ¢£ƒØ° Æ¿Â †1‹’Œƒ]†|îŸíkí;ð ¦ì]îŸíqôxë *Ã̎]¤]¢1’ŒÆȘŸ¤Ó˜ –ñ=îŸí ( 3;õÐë  ë 7í (¦ð -ô¿ï iîŸõ÷Ô º ÆǒŒƒ]ÆǤœ’”ƒ]ÆNj¼‹Œ†-’Œ’ŒÆȤ]°=×\ËK† ËÆȾǾ ‹Œ“Ø’”ƒ]’¼|ž\˜ ÆǤw’c &˜ŸÃH’Œƒ]† ž]¾ÈŸ¤]† ¢ Ÿ¤œ–C† ¢ †1ÆÇŸ† ’”ƒ]†,‹ŒÆÇ° ¤¦Ÿ¾˜ŸÃ—‹C’”’”Æǘ ¤â¨ æq˜ –W† ÙgŽ]ƿŸ¾Ç† ¤w’”¾¿“Ò¬i† ¾Ç˜ ¤]°Ÿ‹W’Œ˜’”ƒ=†Z¢1† ¾Ç¾G’C’£Ÿ¢£ƒ=† ¡Ò’Œ˜’Œƒ]†A¤g’Œ† ¤]¤],¾È˜R¢"’”† ¡Ò’ t£ç—Æ¿ÃG’Œƒ]†A‹CÆÈ° ¤]Ÿ¾c† *ÆǒC’”†1¡ ¬w“U’Œƒ]ÆNj‹[’£’ŒÆȘ ¤6Æȋ–Œ†1¢ †1ÆÇŸ† ¡z’ži˜ ÆȤw’ ڍ’‘¾Ç†1Ÿ† ¾ T5t &  4t ¼Ëƒ=ÆÈ¢£ƒÝÆNj’,¾Ç†‹CP’ t’”ÆÇ *† ‹¬]ÆÈ°Ÿ° † – ’”ƒ¦¤m’”ƒ]†’”˜Ÿ’”Ÿ¾\¤=˜ ÆȋC†Ÿ×R˃]ÆÈ¢£ƒ&ÆNjƒ]†1–Œ†¼¡]†-½¦¤]†1¡&Ÿ‹’”ƒ=†A‹ŒŽ= Á˜ŸÃ tŸ¤]¡Ò˜ŸÃŸ¾Ç¾\˜Ÿ’Œƒ]† –K‹ŒÆÇ° ¤¦Ÿ¾Ç‹–C† ¢ †1ÆÇŸ† ¡‘’ \×N’”ƒ¦’ÆNj r]t]   z TMt GpP  5t ]} 8t &K€ p   T!t &P  5t q 5t [ „ º †Ó–Œ†1¢ ˜ ° ¤=ÆÈÀ1†œ˜ Ž]–° † ¤=† –ŒÆÇ¢Ø *˜R¡]† ¾ËÆ¿’”ƒV  d ×®/` N9f 7   f  è×IX3t (c ta×®Z[t tŸ¤]¡ \‚t* t tÔ Ìý CED. . . . .

(77). . . . . . . . . . . . . . . . . . . =.

(78) ¢npš l ¶ :

(79) ´  n l ´  µ   µ ¤l ¸  µ"¶N´n  ›˜ ¤]‹CÆÈ¡]†1–¼¤]˜aË’”ƒ=†¢ Ÿ‹Œ†Ëƒ]† –C†,’”ƒ]†‹ŒÆÇ° ¤¦Ÿ¾†1 Æ¿’Œ’Œ† ¡â¬w“‘‹C’”’”Æǘ ¤ h ÆȤw’Œ† –CÃ̆1–Œ†1‹ËÆǒŒƒm’”ƒ]†¼¢1Ž] ,Ž=¾Ð’Œ† ¡‹CÆÈ° ¤]Ÿ¾È‹† *ÆǒC’”†1¡&Ã̖C˜  ‹C’”’”Æǘ ¤]‹ ® 8 h ÆȤҋŒŽ]¢£ƒ‘Ë“Ò’”ƒ]’˜ ¤=¾Ç“‹Œ˜Ÿ † ž]–Œ˜Ÿž\˜ –[’”Æǘ ¤  t˜Ã钔ƒ]†1‹Œ†¼¢1Ž] ,Ž=¾Ð’Œ† ¡‹CÆÈ° ¤]Ÿ¾È‹K‹Œƒ]˜ŸŽ]¾È¡Ò¬\†Z¢ ˜ ¤]‹CÆÈ¡=† –Œ†1¡&Ÿ‹¤=˜ ÆȋC†ŸùNÆȤҒŒƒ]†‹Œž]–C†Ÿ¡‹Cž\†1¢1’”–CŽ] › #«š $»æq¢1˜N¡=† ¡]Æ¿ÂNÆNjŒÆǘ ¤Ú ,Ž=¾Ç’ŒÆȞ]¾Ç†*Ÿ¢1¢ † ‹C‹£ç¼¢ ˜Ÿ¤g’Œ†-R’×钌ƒ]ÆNj«ÆȤw’”†1–CÃ̆1–Œ† ¤=¢ †*‹C’Œ† *‹¼Ã̖Œ˜  ’”ƒ=†,ÃqŸ¢1’A’”ƒ¦’A’”ƒ]†. ¡]Æ¿ÞG†1–Œ† ¤w’

(80) ¢1˜N¡=† ‹

(81) Ÿ–C†Z¤=˜Ÿ’ži† –[Ã̆ ¢1’Œ¾Ç“|˜ –[’”ƒ]˜ °Ÿ˜ ¤¦Ÿ¾æ̋Œ† †A†ŸÔ °]ԗÉʃ]–Œ†1¤R¬i† –C° † –Ÿ¤]¡Õ2† ÆǬ]¤]Æ¿’”À­@ § AG˜ –—Ï—† †1–”Ÿ¾Ç¾ÈÆ †1’ZŸ¾M@¿‡a© A ç-ԗ‚

(82) ƒ]†1¤Ó˜Ÿ¤]†7° †1’Œ‹Z¤]˜Ÿ’”ƒ=† –ÆȤ=‹C’£¤]¢ †7˜ŸÃ ’”ƒ]†«° †1¤]† –CÆÈ¢ *˜N¡=† ¾éËÆ¿’”ƒ r]t]

(83)  z T!t &  !t I} 8t  t pK€ p ) T5t pP  5t  ¢ 4t [ „  ˜Ÿ’Œ†’Œƒ¦’7’”ƒ=Æȋ‹ŒÆÇ° ¤¦Ÿ¾Ê’Œ˜Ó¤]˜ŸÆȋC†|–Œ’”ÆǘӞ=–ŒÆǤ]¢ ÆǞ]¾Ç†ÒÆNj¤=˜Ÿ’,¬i˜ Ž]¤=¡U’Œ˜œ › #«š $,× Ÿ¤]¡MÆ¿’ÆȋŸ¢-’”Ž¦Ÿ¾Ç¾¿“ÚŽ]‹Œ†1¡ Ÿ‹«Ò°Ÿ† ¤]†1–ŒÆÇ¢ ž¦Ÿ–ŒŸ¡]ÆÇ°  Ã̘ –ËÆǖŒ†1¾È†1‹Œ‹7¢ ˜Ÿ  Ž]¤]ÆÇ¢’ŒÆȘŸ¤]‹ ù2‹C† † †Ô °]ÔZ’”ƒ]†ž]ƒw“N‹CÆÈ¢ Ÿ¾H *˜R¡]† ¾ ÆȤ «Ž]ž=’”mŸ¤]¡ «Ž] Ÿ–@ ¯ AkÔ ´  µ l ¶ : l  nq³  ¶¦³m¶Ýµ·¶ l    : Ü;¤Ó’”ƒ=ÆȋAŸ–CÆЍ¤g’ ×iËK†,–C† ž]¾ÈŸ¢1†,’”ƒ]††-R’”†1–Œ¤¦¾ ¤]˜ŸÆȋC†,‹Œ†1ÙgŽ]† ¤]¢1† Jg*¬w“œÒ‹Cƒ]˜Ÿ’A¤]˜ ÆNjŒ† ž]–C˜N¢1† ‹C‹«¢ –Œ† ’”†1¡œ¬w“Ó“Ÿ†1’Ÿ¤=˜Ÿ’”ƒ]†1–A Ÿ–C֟† ¡âži˜ ÆǤw’¼ž]–C˜N¢1† ‹Œ+‹  ¬ ËÆǒŒƒÚÒži˜ ‹Œ‹CÆȬ]¾¿“ ¡]Æ¿ÞG†1–Œ† ¤w’A’Œ’Œ† ¤gŽ¦’ŒÆȘŸ¤‘Ã̎]¤=¢1’”Æǘ ¤  ¬ ԗ‚

(84) ƒ]ÆȋÆNjÃ̘Ÿ–

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