セルラネットワークの確率幾何解析の基礎 / IEICE RCS 201705

セルラネットワークにおける干渉や信号対干渉雑音電力比の解析手法である確率幾何解析を文献に基づき紹介する．特に，モンテカルロ法との関係，ポアソン点過程，キャンベルの定理や確率母汎関数，ラプラス変換が用いられる理由を解説する．

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1. な゚゘ぼひぷゞがぜ〣֬཰زԿղੳ〣جૅ ࢁຊ ߴࢸ ژ౎େֶେֶӃ৘ใֶݚڀՊ 2017-08-28 CQ ݚڀձಛผট଴ߨԋ 2017-07-21 RCS/RCC/NS/SR/ASN ݚڀձ 2017-05-12 RCS ݚڀձґཔߨԋ
2. . . ࣗݾ঺հ I — ୅දత〟のがだひぷ (ぎじぶゎひぜد〿) ゙ぬがと੍ޚ〣֬཰زԿղੳ (Prof. Seong-Lyun Kim (Yonsei Univ.), 2016ʙ) じゐ゘ը૾〣ػցֶश〠〽぀ゎ゙೾੍ޚ (੢ඌཧࢤઌੜ, 2013ʙ) ແઢిྗ఻ૹ (ࣰݪਅؽઌੜ, 2011ʙ) M2M, IEEE 802.11ah (क૔ਖ਼തઌੜ, 2011ʙ) や゚ぶゔゆ゛ひぜと (Ӌాউ೭ઌੜ, Dr. Taneli Riihonen, Mr. Sathya N. Venkatasubramanian (Aalto Univ.), Prof. Mikko Valkama, Dr. Ville Syrj¨ al¨ a, Dr. Dani Korpi (Tampere Univ. Tech.) 2010ʙ) ぢそぺふくゅແઢɼແઢ゙ぬがと੍ޚ (Prof. Jens Zander (KTH), 2008ʙ) ろ゚ばりひゆな゚゘ɼろ゚ばりひゆɾぎへりひぜ (٢ాਐઌੜ, ଜాӳҰઌੜ, 2001ʙ) http://bit.ly/RCS201705 2 / 38
3. . . まぐゅ゙ひへや゚ぶゔゆ゛ひぜと〣ఏҊ〝࠷ద゙ぬがとׂ౰ . . . . . . Phase 1 τ1 . Phase 2 τ2 . Phase 3 τ3 . . . . . . . . . . CSR2 . CRD2 . CSR1 . CRD3 . . . . . . S to R . R to D . τ1 . τ2 . τ3 . . . . . CSR1 . CSR2 . CRD2 . CRD3 . Optimal resource allocation problem for FDR (linear programming problem) . . . . . . . . maximize {τ1,τ2,τ3} t subject to t ≤ τ1CSR1 + τ2CSR2, t ≤ τ2CRD2 + τ3CRD3, τ1 ≥ 0, τ2 ≥ 0, τ3 ≥ 0, τ1 + τ2 + τ3 ≤ 1 K. Yamamoto, et al., “Optimal transmission scheduling for a hybrid of full- and half-duplex relaying,” IEEE Commun. Lett., March 2011. http://bit.ly/RCS201705 3 / 38
4. . . ࣗݾ঺հ II . ୅දత〟ゆ゜でこぜぷ (ڞಉ/डୗݚڀ) . . . . . . . . ૯຿ল ి೾ར༻֦େ〣〔〶〣ݚڀ։ൃʮෳ਺Ҡಈ௨৴໢〣࠷దར༻ぇ࣮ݱ『぀੍ ޚج൫ٕज़〠ؔ『぀ݚڀ։ൃʯNICT, ATR, KDDI, NEC, ࡕେ, FY2015-FY2018. . ࠷ۙ〣ぎじぶゎぎ〭〣ߩݙ . . . . . . . . Editor, IEEE Wireless Communications Letters Track Chair, APCC 2017 and CCNC 2018 IEEE ComSoc SPCE TC Member http://bit.ly/RCS201705 4 / 38
5. 1. ֬཰زԿղੳ〣୅දత〟੒Ռ〤〞⿸⿶⿸〷〣⿾ʁ 2. ֬཰زԿղੳ〤〞〣ఔ౓Ұൠత〠࢖いぁ〛⿶぀〣⿾ʁ 3. 〟】ࢁຊ〤֬཰زԿղੳぇ࢖⿸〽⿸〠〟〘〔〣⿾ʁ http://bit.ly/RCS201705 5 / 38
6. . . ֬཰زԿ⿿޿。࢖いぁ぀ܖػ〝〟〘〔 [Andrews 2011] 〣ॏཁ〟݁Ռ . . . . な゚゘ج஍ہ〣Ґஔぇ . . . れぎぬア఺աఔ ゕがづ〤࠷ۙ๣ج஍ہ〠઀ଓ (Poisson-Voronoi cells) ड৴ిྗมಈ〤ࢦ਺෼෍〠ै⿸ (゛ぐ゙がやこがでアそ) ڑ཭ݮਰఆ਺ α Ҏ্〣Ծఆ〜ɼSIR 〣໘త෼෍ (cdf, ccdf) ⿿௚઀ٻ〳〿 α = 4 〜ดܗࣜ (〔〕「ɼຊと゘ぐへશ෦〣ཧղඞཁ) . . . . -20 -10 0 10 20 30 w/o fading . . . . P(SIR > θ) = · · · = 1 1 + 2θ α−2 2F1 1, 1 − 2 α ; 2 − 2 α ; −θ = 1 1 + √ θ arctan √ θ (α = 4) 〈〣࿦จ〠〽〿ɼな゚゘ぼひぷゞがぜ〠 ֬཰زԿղੳ⿿ద༻Մೳ〝ࣔ《ぁ〔 . . . . 0 1 20 10 0 10 20 30 P(SIR  ✓) ✓ (dB) 20 10 0 10 20 30 with fading http://bit.ly/RCS201705 6 / 38
7. . . ICC 2017 〣 1172 ݅த (୯७ずがゞがへݕࡧ) Keyword Number of papers i.i.d. 199 stochastic geometry 132 convex optimization 106 Poisson point process 104 Shannon 93 Fourier 81 machine learning 57 game theory 49 IoT 167 802.11 127 millimeter/mmWave 112 5G & latency 108 massive MIMO 106 full duplex/full-duplex 75 UAV 26 http://bit.ly/RCS201705 7 / 38
8. . . େ༰ྔ௨৴ぇ࣮ݱ『぀〔〶〠〤 てをぽア〣௨৴࿏༰ྔ W (1) log2 (1 + SNR (2) ) bit/s (/channel) (3) . . . . . (1) ଳҬ෯ W ぇ֦େ . てアる゚゛がぷ૿େ ろ゚ばずを゙ぎ఻ૹ ゎ゙೾ प೾਺ڞ༻ ˠ゙ぬがと੍ޚඞཁ . . . . . (2) SNR ぇ૿େ . దԠมௐɾଟ஋มௐ ߴརಘぎアふべ もがわやさがゎアそ ఻ൖଛ〣খ《⿶ばをぼ゚ தܧ〠〽぀ෆײ஍ଳରࡦ . . . . . (3) ௨৴࿏〣਺ぇ૿େ . MIMO ௨৴࿏ (Massive MIMO) ᜚ີج஍ہઃஔ ۭؒతप೾਺࠶ར༻ ˠ゙ぬがと੍ޚඞཁ http://bit.ly/RCS201705 8 / 38
9. . . ແઢ゙ぬがと੍ޚ〣ຊ࣭త〟೉「《 . ׯবଘࡏ࣌〣௨৴࿏༰ྔ〣૯࿨〣࠷େԽ sum rate maximization problem . . . . . . . . max (pi)i∈N i∈N ln[1 + SINRi((pi)i∈N )] nat/s/Hz = max (pi)i∈N i∈N ln 1 + Giipi σi 2 + j∈N \{i} Gijpj nat/s/Hz ໨తؔ਺〤ඇತؔ਺ (ょひなߦྻ⿿ਖ਼ఆ஋〜〟⿶) NP ࠔ೉ [Raniwala et al., 2004, Hayashi and Luo, 2009] جຊత〟ૹ৴ిྗઃఆ໰୊〜『〾 NP ࠔ೉ ˠ ແઢ゙ぬがと੍ޚ〤めゔが゙とふくひぜ⿿Ұൠత ֤࿦తٞ࿦ぇආ々ɼ〜　぀ݶ〿Ұൠతٞ࿦ぇߦ⿸〔〶 ૹ৴஍఺〣゘アはわੑ〠ىҼ『぀ׯব〣֬཰෼෍〠〤֬཰زԿղੳ ׯবぇ૬ޓ࡞༻〝ଊ⿺ぁ〥だがわཧ࿦ ⿿༗༻ (ࢁຊ〣ҙݟ) http://bit.ly/RCS201705 9 / 38
10. . . ろ゚ばゕがづとたでゔが゙アそ〣ղੳ〭〣Ԡ༻ྫ (α = 4) . ゘げアへ゜もアとたでゔが゙アそ [Andrews 2011] . . . . . . . . P(SIR > θ) = 1 1 + √ θ arctan √ θ . ਖ਼نԽ SNR とたでゔが゙アそ [Ohto 2017] (PFS 〜゛がぷ〣୅い〿〠 SNR 〠「〔〷〣) . . . . . . . . P(SIR > θ) ≃ ∞ n=0 fN (n) n+1 k=1 n+1 k (−1)k+1 1 + √ kθ arctan √ kθ , fN (n) = Γ(n + c + 1)(λu/cλb )n Γ(n + 1)Γ(c + 1)(λu/cλb + 1)n+c+1 λu : density of users, λb : density of base stations, c = 3.5 T. Ohto, K. Yamamoto, S.-L. Kim, et al., “Stochastic geometry analysis of normalized SNR scheduling in downlink cellular networks,” IEEE Wireless Commun. Lett., 2017. (৴ֶٕใ RCS2016-313 2017 ೥ 3 ݄) http://bit.ly/RCS201705 10 / 38
11. . . ฉ⿶〛௖。্〜〣໰୊ҙࣝ ֬཰زԿղੳҎલ〤〞〣〽⿸〟ධՁ⿿《ぁ〛　〔⿾ʁ ゑアふじ゚゜๏ (てゎゔ゛がてゖア) 〝֬཰زԿղੳ〣ؔ܎〝〤ʁ 〽。Ծఆ《ぁ぀れぎぬア఺աఔ (PPP) 〝〤Կ⿾ʁ Ұൠత〟Ծఆ〝〣ؔ܎〤ʁ Campbell 〣ఆཧɼpgfl for PPP ⿿〟】ඞཁ⿾ʁ 〞⿸⿶⿸৚݅〜࢖⿺぀⿾ʁ ゘ゆ゘とม׵⿿〟】ग़〛。぀〣⿾ʁ ⿶〙〜〷࢖⿺぀〣⿾ʁ ICC 2016 〜 960 ݅த 80 ݅ぇӽ⿺぀࿦จ⿿֬཰زԿղੳぇ༻⿶〛⿶぀ཧ༝〤 Կ⿾ʁ [Andrews et al., 2011] 〣Ҿ༻݅਺⿿ 1000 ݅ぇ௒⿺぀ཧ༝〤Կ⿾ʁ . . . . . . . جຊ《⿺෼⿾〘〛⿶ぁ〥Ԡ༻〷ཧղ〜　぀〝ࢥ⿸〣〜ɼຊと゘ぐへ〤 [Andrews et al., 2011] 〣جૅ〕々〤ཧղ「〛௖。〈〝ぇ໨త〝「〛⿶〳『 Ԡ༻〠ؔ「〛〤つがよぐ࿦จ [ElSawy et al., 2013, ElSawy et al., 2017] http://bit.ly/RCS201705 11 / 38
12. . . な゚゘てとふわઃܭ — ྫɿな゚܁〿ฦ「਺ܾఆ . . . . [ࡾළ 2002, pp. 373–374] 1. BER-Eb /N0 ಛੑぇܭࢉػてゎゔ゛がてゖア〹ཧ ࿦ղੳ〠〽〿ٻ〶぀ (ม෮ௐɼූ߸ɼやこがでアそิঈ๏ɼやこがで アそ〟〞〠ґଘ) 2. ڐ༰ BER ぇຬ〔『 Eb /N0 〽〿 SIR 〣ᮢ஋ θ ぇ ٻ〶぀ . . . . 10 5 10 4 10 3 10 2 10 1 0 10 ✓ Allowable BER BER Eb/N0 (dB) . . . . 3. SIR 〣 cdf ぇむ゘ゐがの (な゚܁〿ฦ「਺〟〞) ຖ〠ٻ〶぀ ྼԽ཰ P(SIR ≤ θ) ⿿ڐ༰஋ (ྫ⿺〥 10%) ぇຬ ଍『぀む゘ゐがのぇٻ〶぀ む゘ゐがの L 〣ࡍ〣 SIR ぇ SIR(L) 〝ද『〝ɼ ࣍ࣜぇຬ〔『 L ぇݟ〙々぀໰୊ P(SIR(L) ≤ θ) ≤ 0.1 . . . . 0 1 20 10 0 10 20 30 P(SIR  ✓) ✓ (dB) [ࡾළ 2002] ࡾළ, ぶくでの゚ゞぐん゛と఻ૹٕज़, 2002. http://bit.ly/RCS201705 12 / 38 . . . SIR 〣 cdf 〣ٻ〶ํʁ
13. . . Լ〿ճઢ SIR 〣 cdf 〣ٻ〶ํɾ֬཰زԿղੳ〣લ [ࡾළ 2002, pp. 373–374] . . . . i ൪໨〣܁〿ฦ「〠⿼⿶〛ɼҎԼ〣࣮ݱ஋ぇఆ〶぀ ج஍ہ〝〣ڑ཭〣࣮ݱ஋ ri ॴ๬৴߸〣やこがでアそɾてをへげぐアそ〣ిྗ རಘ〣࣮ݱ஋ hi ׯবిྗ〣࣮ݱ஋ Ii SIR 〣࣮ݱ஋ SIR(ri, hi, Ii) ぇٻ〶぀ . . . . . . . . . . . Ii . ج஍ہ . ج஍ہ . user . ri, hi SIR 〣ܦݧ cdf ぇٻ〶぀ 1 M M i=1 (SIR(ri, hi, Ii) ≤ θ) a.s. M→∞ − − − − − − − → P(SIR ≤ θ) = FSIR(θ) ࣮ݱ஋ realization: X ⿿ [0, 1] 〣Ұ༷෼෍〟〾ɼ࣮ݱ஋ྫ〤 x1 = 0.2429, x2 = 0.3271, . . . http://bit.ly/RCS201705 13 / 38 . . . ܦݧ cdf 〣ఆٛ〝ɼcdf 〠֓ऩଋ『぀ࠜڌ
14. . . ܦݧ cdf (ecdf: empirical cdf) . . . . M ݸ〣࣮ݱ஋ {x1, x2, . . . , xM } 〠ର『぀ܦݧ cdf ˆ FX (x) := 1 M M i=1 (xi ≤ x) . . . . . . x . (0 ≤ x) . 1 . 0 {x1, . . . , xM } ぇঢॱ〠ฒ〮௚「〔 {x′ 1 , . . . , x′ M } 〠ର「〛 {(1/M, x′ 1 ), . . . , (i/M, x′ i ), . . . , (M/M, x′ M )} ぇ֊ஈঢ়〠݁え〕〷〣 . . . . ྫɿҰ༷෼෍ xi ∼ U(0, 1) 0 1 0 1 M = 20 M = 1000 ecdf ˆ FX ( x ) x . . . . େ਺〣๏ଇ〠〽〿֤఺ x ∈ R 〜 cdf FX (x) 〠֓ऩଋ ˆ FX (x) a.s. M→∞ − − − − − − − → FX (x) = P(X ≤ x) (x): ࢦࣔؔ਺, indicator function, which is one when x is true and is zero otherwise http://bit.ly/RCS201705 14 / 38 . . . 〟】 cdf ぇ௚઀ٻ〶〟⿶〣⿾ʁ
15. . . SIR 〣 cdf ぇղੳత〠ٻ〶〽⿸〝『぀〣〤ࠔ೉〝ࢥいぁ〛⿶〔 . . . . . . . . . . . I . ج஍ہ . CCI ج஍ہ . user . r, h . . . . ֬཰ม਺ SIR 〤 r, h, I 〠ґଘ ج஍ہ〝〣ڑ཭ r 〣 pdf ぇ fr(r) ॴ๬৴߸〣やこがでアそɾてをへげぐアそ〣ిྗ རಘ h 〣 pdf ぇ fh (h) ׯবిྗ I 〣 pdf ぇ fI (I) ج஍ہ〣ૹ৴ిྗぇ Pb 〝『぀〝 P(SIR ≤ θ) = ESIR[ (SIR ≤ θ)] = Er,h,I [ (SIR ≤ θ)] = hr−αPb I ≤ θ fr,h,I (r, h, I) dr dh dI (a) = hr−αPb I ≤ θ fr(r)fh (h)fI (I) dr dh dI 〜〤⿴぀ ((a) r, h, I ⿿ಠཱ〜⿴ぁ〥) 〔〕ɼ〒〷〒〷֤ pdf ⿿ղੳత〠ಘ〾ぁ぀⿾ෆ໌〕「ɼಘ〾ぁ〔〝〈あ〜਺஋ੵ෼〠པ 〾》぀ぇಘ〟⿶〈〝⿿ଟ⿶ 〒ぁ〟〾ゑアふじ゚゜てゎゔ゛がてゖア〜 ecdf ぇٻ〶〛「〳⿺〥⿶⿶〽⿸〟... http://bit.ly/RCS201705 15 / 38
16. ྫ 1 ྫ 2 (な゚゘) TX: HPPP, RX: 1 ఺ Poisson-Voronoi cell Nearest BS association やこがでアそ〟「 ࢦ਺やこがでアそ (゛ぐ゙が) ظ଴஋ E[I(ϕi)] ccdf P(SIR > θ) ৽「。ಋೖ『぀ࣄฑ Laplace transform . . . HPPP pgfl for HPPP Campbell’s theorem จݙ [Haenggi 2012, §5.1] [Andrews 2011] [Haenggi 2012, §13.4.2] http://bit.ly/RCS201705 16 / 38
17. . . (Homogeneous) Poisson point process (PPP) Φ with intensity λ . . . . . . . ಠཱ〟఺〣ີ౓ (ڧ౓) ⿿ λ 〣࣌ ྖҬ B 〠ؚ〳ぁ぀఺〣਺⿿ै⿸෼෍〤ظ଴஋ λ|B| 〣れぎぬア෼෍ ఺ࣗମ〣෼෍ぇれぎぬア఺աఔ〝ݺ〫 . λ = 1, |B| = 1 . . . . . . . . . . . . . . . . 0 0 . 1 0 . 2 0 . 3 0 . 4 0 1 2 3 4 5 6 P( X = x ) x ظ଴஋ 1 〣れぎぬア෼෍〣 pmf P(X = x) |B| 〤ྖҬ B 〣゚よがそଌ౓ (໘ੵ) http://bit.ly/RCS201705 17 / 38
18. . . PPP 〣࣮ݱ஋〣࡞〿ํ . Ұൠ࿦ . . . . . . . . 1. ྖҬ B 〠ର「〛ɼظ଴஋ λ|B| 〣れぎぬア֬཰ม਺〣࣮ݱ஋ n ぇఆ〶぀ R 〜⿴ぁ〥 rpois(10,lambda=100) 〝『぀〝ظ଴஋ 100 〣ɼ10 ݸ〣れぎぬアཚ ਺ྻ⿿ग़ྗ《ぁ぀ 117 86 93 102 101 96 85 88 94 113 2. ྖҬ B 〣த〠 n ݸ〣఺ぇҰ༷〠ஔ。 . . . . . . . . . spatstat むひたがで [Baddeley et al., 2015] ぇ࢖⿸ɽલらがで〣ਤ〤 P <- rpoispp(lambda=1, win=owin(xrange=c(0,10), yrange=c(0,4))) plot(P) http://bit.ly/RCS201705 18 / 38
19. . . PPP 〣ࣜ〜〣ѻ⿶ ྖҬ B 〠ର「〛 Φ ⿿ڧ౓ λ 〣 PPP 〣৔߹ P(Φ(B) = n) = e−λ|B|(λ|B|)n n! , n = 0, 1, . . . Φ(B): ྖҬ B ಺〣఺աఔ Φ 〣఺〣਺ぇද『֬཰ม਺ . 〈〈⿾〾『〃〠ܭࢉ〜　぀〈〝 . . . . . . . . ఺〣਺〣ظ଴஋ EΦ[Φ(B)] = ∞ n=0 nP(Φ(B) = n) = λ|B| ఺⿿〟⿶֬཰ P(Φ(B) = 0) = e−λ|B| গ〟。〝〷 1 〙఺⿿⿴぀֬཰ P(Φ(B) ≥ 1) = 1 − e−λ|B| http://bit.ly/RCS201705 19 / 38
20. . . ୈ n ۙ๣ڑ཭ [Haenggi 2012, §2.9.1] Φ ⿿ڧ౓ λ 〣 HPPP 〜⿴ぁ〥 P(Φ(B) = k) = e−λ|B| (λ|B|)k k! . ୈ n ۙ๣ڑ཭〣 cdf, pdf . . . . . . . . F(n)(r) := 1 − P(Φ(b(o, r)) < n) = 1 − e−λπr2 n−1 k=0 (λπr2)k k! f(n)(r) := dF(n)(r) dr = 2(λπ)nr2n−1 (n − 1)! e−λπr2 b(o, r) 〤த৺ oɼ൒ܘ r 〣ԁɽ|b(o, r)| = πr2 . ࠷ۙ๣ڑ཭〣 pdf . . . . . . . . f(1)(r) = 2λπre−λπr2 (1) . . . ఺աఔ〟え〛Ұൠ〠࢖いぁ぀ʁ http://bit.ly/RCS201705 20 / 38
21. . . ແઢ LAN IEEE 802.11ax Simulation Scenarios ແઢ LAN 〜〷ແઢہ〣Ґஔ〣゘アはわੑ〠഑ྀ「〔ධՁ⿿ߦいぁ぀ TGax Simulation Scenarios, IEEE 802.11-14/0980r16 Residential Scenario: “In each apartment, place AP and STA in random xy-locations (uniform distribution)” . . . 3m . . . 10m . 10m http://bit.ly/RCS201705 21 / 38 ࢀߟ . . . ఺աఔ〝〞⿸⿶⿸ؔ܎⿿⿴぀ʁ
22. . . てゎゔ゛がてゖア〜〽。⿴぀Ծఆ〤ɼೋ߲఺աఔ〝ݺ〮〫　〷〣 . . . . ೋ߲఺աఔ (BPP: binomial point process) 2 む゘ゐがの (ご゙ぎɾ఺〣਺) . . . . れぎぬア఺աఔ (PPP) 1 む゘ゐがの (఺〣ີ౓ λ) . . . . てゎゔ゛がてゖア〜〽。Ծఆ . . . . ֬཰زԿղੳ〜〽。Ծఆ ࠷ۙ〤てゎゔ゛がてゖア〜〷ݟ⿾々぀ . . . . ⿴぀ご゙ぎ〣த〠ɼ⿴぀਺〣ૹ৴఺ぇ Ұ༷゘アはわ〠഑ஔ . . . . PPP 〤ޙड़〣 Campbell’s theorem 〹 pgfl ⿿஌〾ぁ〛⿼〿ɼղੳత〠ѻ⿶〹『⿶ http://bit.ly/RCS201705 22 / 38
23. . . 〒〣ଞ〣఺աఔ Cluster point process: (attractive) Mat´ ern hard-core point process type II: (repulsive) CSMA 〣ゑぶ゚〝「〛࢖い ぁ぀ Ginibre point process: (repulsive) ౦޻େ ࡾ޷ઌੜ⿿な゚゘〠ઌۦత〠Ԡ༻ http://bit.ly/RCS201705 23 / 38 . . . てゎゔ゛がてゖア〤ɼ਺ֶత〠Կぇ〹〘〛⿶぀〝ଊ⿺〾ぁ぀⿾ʁ
24. . . ఺〣゘アはわੑ〠ؔ『぀ゑアふじ゚゜てゎゔ゛がてゖア〣਺ֶతදݱ ఺ (ૹ৴ہ〟〞) 〣゘アはわ〟࠲ඪ〣ू߹ (ʹ֬཰఺աఔ) Φ = {xi } 〣࣮ݱ஋ ϕ1, ϕ2, . . . , ϕM ぇੜ੒ . . . . . . . . . ࣮ݱ஋ ϕ1 . ࣮ݱ஋ ϕ2 . ࣮ݱ஋ ϕ3 · · · . . ࣮ݱ஋ ϕM ֤࣮ݱ஋ ϕi 〠ର「ɼԿ〾⿾〣ಛੑ f(ϕi) ぇܭࢉ . ྫ 1ɿݪ఺ o 〠⿼々぀ׯবిྗ (やこがでアそ〟「) . . . . . . . . f(ϕi) = I(ϕi) := x∈ϕi ∥x − o∥−α, α: ఻ൖ܎਺ ܦݧظ଴஋ 1 M M i=1 f(ϕi) 〹ܦݧ cdf 1 M M i=1 (f(ϕi) ≤ x) ぇ݁Ռ〝『぀ ܦݧظ଴஋ sample/empirical mean/average/expectation http://bit.ly/RCS201705 24 / 38 . . . ٻ〶〽⿸〝「〔〷〣〝Ұக「〛⿶぀〣⿾ʁ
25. . . ྫ 1 〜ٻ〶〽⿸〝「〛⿶〔〷〣 . . . . EI [I] = EΦ[I(Φ)] = N I(ϕ)P(dϕ) (2) I 〤֬཰ม਺ɽ〒〣 source of randomness (〣Ұ〙) ⿿ɼ఺〣࠲ඪ〣 ゘アはわੑ EΦ ぇ〞⿸〹〘〛ແ。『⿾⿿ॏཁɽ〜 　ぁ〥ดܗࣜぇٻ〶〔⿶ɽ〈ぁ⿿ ࢒〘〛⿶぀〝ɼmathematical expression 〝ݺ〮〟⿶ . . . . . . . . (2) 〣ҙຯぇઆ໌「〕『〝　〿⿿ແ⿶ 〣〜ɼ ֬཰ม਺〣ظ଴஋〣࣍〣දݱ〝 〣ؔ࿈〕々ࢦఠ EX [X] = Ω X(ω)P(dω) X 〤֬཰ۭؒ (Ω, A, P) ্〣֬཰ม਺ ࡉ⿾⿶આ໌〤 [Haenggi2012, §2.5.4] . (2) ぇٻ〶぀〔〶〣 2 〙〣ぎゆ゜がば . . . . . . . . 1. ゑアふじ゚゜๏ N I(ϕ)P(dϕ) ぇ M i=1 I(ϕi) 1 M 〠ஔ　׵⿺぀ 2. ֬཰زԿղੳ http://bit.ly/RCS201705 25 / 38
26. . . . . . 1. ゑアふじ゚゜๏ . 1 M M i=1 I(ϕi) M→∞ − − − − → EI [I]? 〔。《え ϕi ぇੜ੒「〛ɼಛੑ I(ϕi) ぇ ٻ〶〛ɼฏۉԽ「《⿺『ぁ〥〽⿶ (େ਺ 〣๏ଇ⿿੒〿ཱ〛〥) Pros. 〕⿶〔⿶〞え〟ؔ਺ I(·) 〜〷Մ (I(ϕi) = x∈ϕi ∥x∥−α 〤େ਺〣๏ ଇ⿿੒〿ཱ〔〟⿶ྫ) Cons. 1 ܻ〣ਫ਼౓޲্〠ɼ2 ܻ〣 M ૿Ճ ඞཁ 104 ճ〠Ұ౓「⿾ى〈〾〟⿶৔߹ɼ M = 106 〤ඞཁ ݁Ռ〣ଥ౰ੑݕূ ゆ゜そ゘わゎと ぶがの፻଄ . . . . . 2. ֬཰زԿ . EI [I] = EΦ x∈Φ ∥x∥−α . . . . . Campbell’s theorem for HPPP EΦ x∈Φ f(x) = λ R2 f(x) dx . . EI [I] = λ R2 ∥x∥−α dx ਺ࣜදݱ⿿〜　〔ɽ〈〈⿾〾ࣜมܗぇਐ 〶〾ぁ぀ [Haenggi, 2012, §5.1] 〔〕「ɼ〈〣৔߹ɼੵ෼〣஋〤ൃࢄ http://bit.ly/RCS201705 26 / 38
27. . . ࿨〣ظ଴஋ɾੵ〣ظ଴஋〣ܭࢉํ๏ (ূ໌౳〤จݙࢀর) Let Φ be a HPPP on Rd with intensity λ, and f : Rd → R and v : Rd → [0, 1] be measurable functions. Then, . ࿨ɿCampbell’s theorem for sums [Haenggi, 2012, §4.2] . . . . . . . . EΦ x∈Φ f(x) = λ Rd f(x) dx ఺աఔ〣࿨〣ظ଴஋〤ɼੵ෼〠ม׵Մೳ . ੵɿProbability generating functional (pgfl) for PPP [Haenggi, 2012, §4.6] . . . . . . . . EΦ x∈Φ v(x) = exp −λ Rd (1 − v(x)) dx ఺աఔ〣ੵ〣ظ଴஋〤ɼੵ෼〠ม׵Մೳ . . . . . . . PPP 〕〝〈ぁ〾〣ੑ࣭⿿࢖⿺぀ (〣〜ɼPPP ⿿〽。Ծఆ《ぁ぀) 〒〷〒〷〣ҙਤ〤 EΦ ぇ〟。『〈〝ɽ࢒〘〛⿶぀〝ɼmathematical expression 〜 《⿺〟⿶ɽՄೳ〟〾ดܗࣜ〟〞〠「〔⿶ http://bit.ly/RCS201705 27 / 38
28. . . ゑアふじ゚゜๏〣݁Ռྫ (පత〟ྫ) 10 7 10 6 10 5 10 4 10 3 10 2 10 1 100 10 100 1000 10000 Average interference 1 M M X i=1 I('i) Number of averages M 1 M M i=1 I(ϕi) = 1 M M i=1 x∈ϕi ∥x∥−4 M ぇେ　。「〛⿶〘〛〷ɼऩଋ「〟⿶ (େ਺〣๏ଇ⿿੒〿ཱ〔〟⿶) ड৴఺〝〣ڑ཭⿿ݶ〿〟。 0 〠ۙ⿶〝ɼׯবిྗ⿿ແݶେ〠〟぀〔〶 http://bit.ly/RCS201705 28 / 38
29. ྫ 1 ྫ 2 (な゚゘) TX: HPPP, RX: 1 ఺ Poisson-Voronoi cell Nearest BS association やこがでアそ〟「 ࢦ਺やこがでアそ (゛ぐ゙が) ظ଴஋ E[I(ϕi)] ccdf P(SIR > θ) ৽「。ಋೖ『぀ࣄฑ Laplace transform . . . HPPP pgfl for HPPP Campbell’s theorem จݙ [Haenggi 2012, §5.1] [Andrews 2011] [Haenggi 2012, §13.4.2] http://bit.ly/RCS201705 29 / 38
30. . . ྫ 2: な゚゘てとふわԼ〿゙アぜ — てとふわゑぶ゚ . . . . ج஍ہ〤 HPPP Φb with intensity λb ゕがづ〣Ґஔ〤〞〈〜〷⿶⿶〔〶ɼݪ఺ o 〝「〛〷Ұ ൠੑぇࣦい〟⿶ ݪ఺ o 〣ゕがづ〝ɼ࠷ۙ๣ج஍ہ bo = arg minx∈Φb ∥x − o∥ ؒ〣ڑ཭ぇɼ֬཰ม਺ r 〝 ද『〝ࣜ (1) 〽〿 fr(r) = 2πλb re−πλbr2 ج஍ہ〣ૹ৴ిྗ Pb ࡶԻిྗ〤 0ɽ[Andrews 2011] 〠〤 0 〜〟⿶৔߹〷 ⿴぀ ॴ๬೾⿼〽〨஍఺ x ⿾〾〣ׯব೾〣゛ぐ゙がやこがで アそిྗརಘ (ૹ৴ిྗࠐ) h, hx ∼ Exp(1/Pb ) [Andrews 2011] 〠〤ࢦ਺෼෍Ҏ֎〣৔߹〷 . . . ෮श ׯবݯ〤࠷ۙ๣ج஍ہ bo (ڑ཭ r) 〽〿ԕ。〣ج஍ہ Φb \ {bo } = Φb \ b(o, r) b(o, r): ൒ܘ r := ∥bo ∥ 〣ԁ಺ྖҬ Ir := x∈Φb\{bo} hx ∥x∥−α = x∈Φb\b(o,r) hx ∥x∥−α Φb \ b(o, r) 〤ڧ౓ؔ਺ λ(x) = λb (∥x∥ > r) 〣 (inhomogeneous) PPP . . . . bo o r ੺఺⿿ Φb \ {bo } http://bit.ly/RCS201705 30 / 38
31. . . ֬཰ม਺ɾࢦ਺෼෍ (゛ぐ゙がやこがでアそ〣ిྗ෼෍) . . . . . . . ظ଴஋⿿ 1/λ 〣֬཰ม਺ X (i.e., EX [X] = 1/λ) ⿿ࢦ਺෼෍〠ै⿸〈〝ぇ࣍ࣜ〜ද『 X ∼ Exp(λ) . . . . . . . X 〣 cdf FX (x) := P(X ≤ x) = EX [ (X ≤ x)] ぇ࣍ࣜ〝「〔〈〝〝ಉ」 FX (x) = 1 − e−λx, (x ≥ 0) . . . . . . . X 〣 ccdf ¯ FX (x) := 1 − FX (x) = P(X > x) = EX [ (X > x)] 〤 ¯ FX (x) = e−λx, (x ≥ 0) . . . . . . . x ∼ Exp(λ), Fx(x) = 1 − e−λx 〝ॻ。ྲّྀ⿴〿 ຊと゘ぐへ〜〷྆ํ〣ྲّྀぇஅ〿ແ「〠ࠞࡏ ੨৭දه〤〽。লུ《ぁ぀Օॴ cdf: ྦྷੵ෼෍ؔ਺, cumulative distribution function ccdf: ૬ิྦྷੵ෼෍ؔ਺, complementary cumulative distribution function http://bit.ly/RCS201705 31 / 38 ෮श
32. . . ֬཰ม਺ぇ༻⿶〔఻ൖゑぶ゚ . i.i.d. block Rayleigh fading with an exponential-decaying path loss model . . . . . . . . . . . . ड৴఺ぇݪ఺ o 〝『぀࠲ඪܥ〜ɼૹ৴఺࠲ඪ⿿ x ∈ R2 ૹ৴ిྗ Pb 〣ࡍ〠ɼॠ࣌ड৴ిྗぇ࣍ࣜ〝『぀ゑぶ゚ hx ∥x − o∥−α = hx ∥x∥−α, hx ∼ Exp(1/Pb ), i.e., Ehx [hx] = Pb . . . . . . . x, Pb . o hx : o 〝 x 〣ؒ〣やこがでアそిྗརಘ〝ૹ৴ిྗ〣ੵ Pb 〤ɼجຊత〠〤ૹ৴ిྗ〕⿿ɼૹड৴ぎアふべརಘ〹ɼ∥x∥ = 1 〣ࡍ〣఻ൖଛ 〟〞⿿৫〿ࠐ〳ぁ〔஋〝ଊ⿺぀〝〽⿶ ゅ゜ひぜ (४੩త) 〤ɼや゛がわૹ৴〠ඞཁ〟࣌ؒ⿿ぢめが゛アぷ࣌ؒ〽〿୹⿶, i.e., hx ⿿Ұఆ〝Ծఆ . . . . . . . hx 〝 x ∈ R2 ⿿֬཰ม਺ i.i.d.: independent and identically distributed. ಠཱಉ෼෍ (⿴぀⿶〤ಠཱಉҰ෼෍) ∥·∥: Euclidean distance. ゕがぜ゙ひへڑ཭ ∥x∥ ⿿খ《⿶〝ԕํք〠〟〾〟⿶〟〞〣ཧ༝〜ɼ∥x∥−α 〣୅い〿〠 min{1, ∥x∥−α} 〝『぀ゑぶ゚〷⿴぀ http://bit.ly/RCS201705 32 / 38 ෮श
33. . . ॏཁ〟݁Ռ〝ಋग़〣֓ཁ P(SIR > θ) = ∞ 0 P(SIR > θ | r)fr(r) dr (b) = ∞ 0 LIr (θrα/Pb | r)fr(r) dr (c) = ∞ 0 exp − 2πλb r2θ α − 2 2F1 1, 1 − 2 α ; 2 − 2 α ; −θ · 2πλb re−πλbr2 dr = 2πλb ∞ 0 exp −πλb r2 1 + 2θ α − 22F1 1, 1 − 2 α ; 2 − 2 α ; −θ dr = 1 1 + 2θ α − 22F1 1, 1 − 2 α ; 2 − 2 α ; −θ (3) = 1 1 + √ θ arctan √ θ (α = 4) (b), (c) ࣍ทҎ߱ࢀর (3) BS 〣ີ౓ λb 〠ґଘ「〟⿶ɽྼԽ཰〤ׯব੍ݶ〟〾な゚つぐど〠ґଘ「〟⿶〝⿶ ⿸ఆੑత〠஌〾ぁ〛⿶぀〈〝ぇࣜ〜ࣔ「〛⿶぀ P(SIR > θ): Coverage probability, success probability, ৔ॴ཰. SIR 〣 ccdf P(SIR ≤ θ): Outage probability, ྼԽ཰. SIR 〣 cdf http://bit.ly/RCS201705 33 / 38
34. . . (b) P(SIR > θ | r) = P hr−α Ir > θ r = P(h > θrαIr | r) ֬཰ม਺ h 〝 Ir ぇ෼々〛ܭࢉ「〔⿶ = EIr [P(h > θrαIr | r, Ir)] ৚݅෇　֬཰ɽ֬཰ม਺〤 h 〕々 = EIr [e−θrαIr/Pb | r] h ∼ Exp(1/Pb ), i.e., P(h > x) = e−x/Pb = LIr (θrα/Pb | r) ֬཰ม਺ Ir 〣 pdf 〣゘ゆ゘とม׵〣ఆٛ〤 LIr (s) = EIr [e−sIr ] = R fIr (z)e−sz dz = . . . (c) LIr (s) 〣ܭࢉํ๏ = exp − 2πλb r2θ α − 2 2F1 1, 1 − 2 α ; 2 − 2 α ; −θ . 〈〈⿾〾෼⿾぀〈〝 . . . . . . . . ॴ๬৴߸ిྗ⿿ࢦ਺෼෍〠ै⿸ (゛ぐ゙がやこがでアそ) ৔߹〠〣〴ɼ ʮSIR 〣 ccdf P(SIR > θ | r)ʯ⿿ʮׯব Ir 〣 pdf 〣゘ゆ゘とม׵ LIr (s | r)ʯ〠〟぀ ॴ๬৴߸⿿ࢦ਺෼෍〠ैい〟々ぁ〥ੵ෼⿿࢒぀ɽtractability 〣〔〶〠ࢦ਺෼෍ぇ Ծఆ『぀〈〝⿿ଟ⿶ [Andrews 2011] http://bit.ly/RCS201705 34 / 38
35. . . (c) LIr (s | r) 〣ٻ〶ํ LIr (s | r) = EIr [e−sIr ] = E Φb,{hx} ⎡ ⎣exp ⎛ ⎝−s x∈Φb\{bo} hx ∥x∥−α ⎞ ⎠ ⎤ ⎦ (d) = EΦb ⎧ ⎨ ⎩ x∈Φb\b(o,r) Eh exp −sh∥x∥−α ⎫ ⎬ ⎭ = EΦb ⎡ ⎣ x∈Φb\b(o,r) ∥x∥α ∥x∥α + sPb ⎤ ⎦ pgfl(e) = exp −λb R2\b(o,r) sPb ∥x∥α + sPb dx = exp −λb 2π 0 ∞ r sPb vα + sPb v dv dθ = exp −2πλb ∞ r sPb vα + sPb v dv (d): hx 〤 i.i.d. (e): Φ ⿿ఆৗ (ڧ౓ λ) 〜〟。ɼڧ౓⿿Ґஔ x 〠ґଘ (ڧ౓ؔ਺ λ(x)) 『぀৔߹〣 pgfl EΦ x∈Φ v(x) = exp − Rd (1 − v(x)) λ(x) dx http://bit.ly/RCS201705 35 / 38
36. . . ௒زԿؔ਺ hypergeometric function 〠〽぀දݱ LIr (s | r) = exp −2πλb ∞ r sPb vα + sPb v dv (f) = exp − 2πλb sPb r2−α α − 2 2F1 1, 1 − 2 α ; 2 − 2 α ; − sPb rα (f): [Gradshteyn and Ryzhik, 2014, (9.111)] 〽〿ٻ〶〾ぁ぀ҎԼ〣ؔ܎ ∞ x t 1 + tα dt = x2−α α − 22F1 1, 1 − 2 α ; 2 − 2 α ; −x−α 〈〈〜ɼ2F1(α, β; γ; x) 〤すげと〣௒زԿؔ਺ Gauss hypergeometric function 2F1(α, β; γ; x) = ∞ n=0 (α)n(β)n (γ)nn! xn (α)n 〤れひりまろがه߸ Pochhammer symbol (α)n = α(α + 1) · · · (α + n − 1), n = 1, 2, . . . 0, n = 0 http://bit.ly/RCS201705 36 / 38
37. 1,000 ճ〣ゑアふじ゚゜๏〣 ecdf (੺) 40,000 ճ〣ゑアふじ゚゜๏〣 ecdf (྘) ղੳ〠〽぀ cdf P(SIR ≤ θ) = 1 − 1 1+ √ θ arctan √ θ (੨) α = 4ɽcdf 〝 40,000 ճ〣 ecdf 〤〰〱Ұக 0.0 0.2 0.4 0.6 0.8 1.0 20 10 0 10 20 30 P(SIR  ✓) ✓ (dB) http://bit.ly/RCS201705 37 / 38
38. . . Bibliography I [Andrews et al., 2011] Andrews, J. G., Baccelli, F., and Ganti, R. K. (2011). A tractable approach to coverage and rate in cellular networks. IEEE Trans. Commun., 59(11):3122–3134. [Baddeley et al., 2015] Baddeley, A., Rubak, E., and Turner, R. (2015). Spatial Point Patterns: Methodology and Applications with R. CRC Pr. [ElSawy et al., 2013] ElSawy, H., Hossain, E., and Haenggi, M. (2013). Stochastic geometry for modeling, analysis, and design of multi-tier and cognitive cellular wireless networks: A survey. IEEE Commun. Surveys Tuts., 15(3):996–1019. [ElSawy et al., 2017] ElSawy, H., Sultan-Salem, A., Alouini, M.-S., and Win, M. Z. (2017). Modeling and analysis of cellular networks using stochastic geometry: A tutorial. IEEE Commun. Surveys Tuts., 19(1):167–203. [Gradshteyn and Ryzhik, 2014] Gradshteyn, I. S. and Ryzhik, I. M. (2014). Table of Integrals, Series, and Products. Academic Pr., 8th edition. [Haenggi, 2012] Haenggi, M. (2012). Stochastic Geometry for Wireless Networks. Cambridge Univ. Press, Cambridge, U.K. [Hayashi and Luo, 2009] Hayashi, S. and Luo, Z.-Q. (2009). Spectrum management for interference-limited multiuser communication systems. IEEE Trans. Inf. Theory, 55(3):1153–1175. [Raniwala et al., 2004] Raniwala, A., Gopalan, K., and Chiueh, T.-c. (2004). Centralized channel assignment and routing algorithms for multi-channel wireless mesh networks. ACM SIGMOBILE Mobile Comp. Commun. Rev., 8(2):50. http://bit.ly/RCS201705 38 / 38