Asymptotics and Meta-Distribution of the Signal-to-Interference Ratio in Wireless Networks

Transcription

1 Asymptotics and Meta-Distribution of the Signal-to-Interference Ratio in Wireless Networks Part II Martin Haenggi University of Notre Dame, IN, USA EPFL, Switzerland 25 Simons Conference on Networks and Stochastic Geometry May 25 Work supported by U.S. NSF (CCF 2647) M. Haenggi (ND & EPFL) Asymptotics of SIR Distributions May 25 / 47

2 Overview Part II Overview Cellular networks and the HIP model Standard analysis of some transmission techniques for the PPP Non-Poisson network analysis using ASAPPP a The idea of the horizontal shift (gain) of SIR distributions The relative distance process The MISR b and the EFIR c Asymptotic gains at and Examples Concluding remarks a Approximate SIR Analysis based on the PPP or simply "as a PPP" b Mean interference-to-signal ratio c Expected fading-to-interference ratio M. Haenggi (ND & EPFL) Asymptotics of SIR Distributions May 25 2 / 47

3 Cellular networks models Setup From bipolar to cellular networks From yesterday: A generic cellular network (downlink) Base stations form a stationary and ergodic point process and all transmit at equal power. Assume a user is located at o. Its serving base station is the nearest one (strongest on average). The other base stations are interferers (frequency reuse ) M. Haenggi (ND & EPFL) Asymptotics of SIR Distributions May 25 3 / 47

4 Cellular networks models Standard BS association Single-tier cellular networks with reuse SIR with strongest-base station (BS) association 4 3 SIR S I S = h x α I = h x x α x Φ\{x } Φ: point process of BSs x : serving BS h, (h x ): iid fading M. Haenggi (ND & EPFL) Asymptotics of SIR Distributions May 25 4 / 47

5 Cellular networks models Standard BS association The SIR walk and coverage at db SIR (db) SIR (no fading) x M. Haenggi (ND & EPFL) Asymptotics of SIR Distributions May 25 5 / 47

6 Cellular networks models Standard BS association SIR distribution 5 SIR (with fading) SIR (db) x The fraction of a long curve (or large region) that is above the threshold θ is the ccdf of the SIR at θ: p s (θ) F SIR (θ) P(SIR > θ) It is the fraction of the users with SIR > θ for each realization of the BS and user processes. M. Haenggi (ND & EPFL) Asymptotics of SIR Distributions May 25 6 / 47

7 Cellular networks models SIR analysis Fact on SIR distributions Only the PPP is tractable exactly in some cases If the base stations form a homogeneous Poisson point process (PPP): p s (θ) F SIR (θ) =, δ 2/α. 2F (, δ; δ; θ) For δ = /2, p s (θ) = ( + θ arctan θ). If the fading is not Rayleigh or if the point process is not Poisson, it gets hard very quickly. So let us enjoy the beauty of Poissonia a little longer. Poissonia M. Haenggi (ND & EPFL) Asymptotics of SIR Distributions May 25 7 / 47

8 Cellular networks models The HIP model The HIP baseline model for HetNets The HIP (homogeneous independent Poisson) model a a Dhillon et al., Modeling and Analysis of K-Tier Downlink Heterogeneous Cellular Networks Start with a homogeneous PPP. Here λ = Choose a number of tiers and intensities for each tier, say λ =, λ 2 = 2, and λ 3 = 3. Then randomly color the BSs according to the intensities to assign them to the different tiers: P(tier = i) = λ i /λ M. Haenggi (ND & EPFL) Asymptotics of SIR Distributions May 25 8 / 47

9 Cellular networks models The HIP model The HIP (homogeneous independent Poisson) model Here λ i =,2,3. Assign power levels P i to each tier. This model is doubly independent and thus highly tractable. M. Haenggi (ND & EPFL) Asymptotics of SIR Distributions May 25 9 / 47

10 Cellular networks models Equivalence to single-tier model Equivalence of all HIP models From the perspective of the typical user, this network is completely equivalent to a single-tier Poisson model with unit power and unit density. Hence for all HIP models (with Rayleigh fading and power law path loss), the SIR distribution is p s (θ) F SIR (θ) = In particular, for δ = /2: 2F (, δ; δ; θ). P(SIR>θ) SIR CCDF p s () = 2.% The typical user is not impressed with this performance. 2 2 θ (db) α = 4 (δ = /2). M. Haenggi (ND & EPFL) Asymptotics of SIR Distributions May 25 / 47

11 Cellular networks models Equivalence to single-tier model Explanation for equivalence For a single tier with unit transmit power, let Ξ = {ξ i } {x Φ: x α /h x }. The received powers from the nodes in Φ are {ξ }. If Φ R 2 is Poisson with intensity λ, then Ξ is Poisson with intensity function µ(r) = λπδr δ E(h δ ). For multiple independent Poisson tiers with transmit power P k, the union Ξ = {ξ i } = {x Φ k : x α /(P k h x )}. k [K] is a PPP with intensity function µ(r) = k [K] πλ k δp δ k rδ E(h δ ). In any case, µ(r) r δ. The pre-constant does not matter for the SIR. M. Haenggi (ND & EPFL) Asymptotics of SIR Distributions May 25 / 47

12 Cellular networks models Equivalence to single-tier model Path loss process The point process Ξ = {ξ i } R + where {ξ i } are the received powers (with or without fading) is called the path loss process or propagation process. It is a key ingredient in many proofs of many results for cellular networks a and HetNets b. The equivalence also holds for advanced transmission techniques, such as BS cooperation and silencing. Let us have a look at some of these advanced techniques. a Blaszczyszyn, Karray, and Keeler, Using Poisson Processes to Model Lattice Cellular Networks. 23. b Zhang and Haenggi, A Stochastic Geometry Analysis of Inter-cell Interference Coordination and Intra-cell Diversity. 24; Nigam, Minero, and Haenggi, Coordinated Multipoint Joint Transmission in Heterogeneous Networks. 24. M. Haenggi (ND & EPFL) Asymptotics of SIR Distributions May 25 2 / 47

13 Advanced techniques for downlink BS silencing BS silencing: neutralize nearby foes The strongest BS (on average) is the serving BS, while the n next-strongest ones are silenced. The model may include shadowing (which stays constant over time). M. Haenggi (ND & EPFL) Asymptotics of SIR Distributions May 25 3 / 47

14 Advanced techniques for downlink BS silencing SIR distribution for silencing (ICIC) With BS silencing (or inter-cell interference coordination, ICIC) of n BSs, the SIR distribution is a ( ) p s (!n) power from serving BS P power from BSs beyond the n th > θ = (n )δ where C (s,m) = 2 F (m, δ; δ; s). ( x δ ) n 2 x δ (C (θx,)) n dx, This result does not depend on the shadowing distribution as long as its δ-th moment is finite. a Zhang and Haenggi, A Stochastic Geometry Analysis of Inter-cell Interference Coordination and Intra-cell Diversity. 24. M. Haenggi (ND & EPFL) Asymptotics of SIR Distributions May 25 4 / 47

15 Advanced techniques for downlink Intra-cell diversity Intra-cell diversity from multiple resource blocks Transmission over M resource blocks Here, all base stations interfere, but the serving one uses M resource blocks (with independent fading) to serve the user. The success probability is the probability that the SIR in at least one of them exceeds θ: ( M p ( M) s P m= S m ) For the joint success probability, we have ( M ) P S m = p ( M) s m=, where S m = {SIR m > θ}. C (θ,m) = 2F (M, δ; δ; θ). p ( M) s follows from inclusion/exclusion. M. Haenggi (ND & EPFL) Asymptotics of SIR Distributions May 25 5 / 47

16 Advanced techniques for downlink Comparison of ICIC and ICD n-bs silencing (ICIC) vs. transmission over M RBs (ICD) ps(n) n =, 2, 3, 4, 5 ps(m) M = M = 5.2. κ = n κ θ (db) n-bs ICIC. α = θ (db) M-RB ICD. α = 4. Observation ICIC provides a gain in the SIR but no diversity. ICD has a diversity gain of M. As a result, ICD is superior at small values of θ (θ < 5 db). M. Haenggi (ND & EPFL) Asymptotics of SIR Distributions May 25 6 / 47

17 Advanced techniques for downlink BS cooperation (CoMP) Cooperation by joint transmission BS cooperation: turn nearby foes into friends M. Haenggi (ND & EPFL) Asymptotics of SIR Distributions May 25 7 / 47

18 Advanced techniques for downlink BS cooperation (CoMP) SIR distribution with BS cooperation In the HIP model, let the users receive combined signals from the n strongest (on average) BSs, denoted by C. Channels are Rayleigh fading, and BSs use non-coherent joint transmission. The amplitude fading coefficients (g x ) are zero-mean unit-variance complex Gaussian, and the signal power is S = g x Px x α/2 2. x C S is exponentially distributed with mean P x x α. The interference stems from Φ\C. M. Haenggi (ND & EPFL) Asymptotics of SIR Distributions May 25 8 / 47

19 Advanced techniques for downlink BS cooperation (CoMP) BS cooperation with non-coherent JT Let u = (u,...,u n ) ~u = (u n /u,...,u n /u n ) Z(u) = ~u α/2 θ δ F(x) = x r +r αdr The success probability is independent of power levels and densities a p s (θ) = <u <...<u n< exp P(SIR>θ) ( u n ( α = 4 Single BS.3 JT from 2 BSs JT from 3 BSs θ (db) +2 F( Z(u)) Z(u) a Nigam, Minero, and Haenggi, Coordinated Multipoint Joint Transmission in Heterogeneous Networks. 24. )) du. M. Haenggi (ND & EPFL) Asymptotics of SIR Distributions May 25 9 / 47

20 Non-Poisson networks What is possible outside Poissonia? Ginibre point process (GPP) For GPP with Rayleigh fading a : p s (θ) = [ e v s j e s ][ ( ds v i s i e s ) ] ds dv j! +θ(v/s) α/2 +θ(v/s) α/2 j= v a Miyoshi and Shirai, A Cellular Network Model with Ginibre Configured Base Stations. 24. i= v M. Haenggi (ND & EPFL) Asymptotics of SIR Distributions May 25 2 / 47

21 Shape of SIR distributions Observation on SIR distributions Shape of SIR distributions In many cellular papers, we find figures like this: SIR CCDF.8 P(SIR>θ) Scheme Scheme 2 Scheme 3 Scheme θ (db) It appears that: The curves all have the same shape they are merely shifted horizontally! M. Haenggi (ND & EPFL) Asymptotics of SIR Distributions May 25 2 / 47

22 Shape of SIR distributions Different BS point processes Coverage Probability Experimental data PPP Poisson hard core process Strauss process SINR Threshold (db) Indeed visually, the curves are shifts of each other. Since the shift (or gain) is due to the deployment, we call it deployment gain a. a Guo and Haenggi, Asymptotic Deployment Gain: A Simple Approach to Characterize the SINR Distribution in General Cellular Networks. 25. M. Haenggi (ND & EPFL) Asymptotics of SIR Distributions May / 47

23 Horizontal gap (gain) ASAPPP: Approximate SIR analysis based on the PPP SIR CCDF.8 G G 2 P(SIR>θ).6.4 G 3.2 PPP Better scheme Better scheme 2 Better scheme θ (db) If the SIR ccdfs were indeed just shifted: p s,ppp (θ) P(SIR PPP > θ) p s (θ) = p s,ppp (θ/g). G is the SIR shift (in db) or the SIR gain or gap. M. Haenggi (ND & EPFL) Asymptotics of SIR Distributions May / 47

24 Horizontal gap (gain) Horizontal gap and asymptotics The shift at threshold θ is G(θ) F hence we have p s (θ) = p s,ppp (θ/g(θ)). The asymptotic gains are So (if G and G exist), SIR (p s,ppp(θ)) θ G lim θ G(θ); G lim θ G(θ). p s (θ) p s,ppp (θ/g ), θ ; p s (θ) p s,ppp (θ/g ), θ. Observation: G(θ) G for all θ, i.e., a shift by G results in an approximation that is quite accurate over the entire distribution., M. Haenggi (ND & EPFL) Asymptotics of SIR Distributions May / 47

25 Examples for gains Example : Deployment gain of square lattice.8 SIR CCDFS for α= 3. PPP Square lattice Shift by G.8 SIR CCDFS for α= 4. PPP Square lattice Shift by G P(SIR>θ).6.4 P(SIR>θ) θ (db) For the square lattice: θ (db) G = 3.9 db for α = 3 and G = 3.4 db for α = 4. So applying a gain of 2 yields an accurate approximation. For α = 4, p sq s (θ) (+ θ/2arctan θ/2). M. Haenggi (ND & EPFL) Asymptotics of SIR Distributions May / 47

26 Examples for gains Example 2: Gain of joint transmission P(SIR>θ) JT from 2 BSs.3 Single BS MISR based gain θ (db) Again the ccdf for the cases without and with cooperation are very similar in shape. The shift here is G = 2/(4 π) M. Haenggi (ND & EPFL) Asymptotics of SIR Distributions May / 47

27 MISR The ISR and the MISR Definition (ISR) The interference-to-average-signal ratio is ISR I E h (S), where E h (S) is the desired signal power averaged over the fading. Remarks The ISR is a random variable due to the random positions of BSs and users. Its mean MISR is a function of the network geometry only. If the interferers are located at distances R k, ( MISR E(ISR) = E R α ) h k R α k = ( ) R α E. R k M. Haenggi (ND & EPFL) Asymptotics of SIR Distributions May / 47

28 MISR Relevance of the MISR for Rayleigh fading p out (θ) = P(hR α < θi) = P(h < θ ISR) Since h is exponential, letting θ, P(h < θ ISR ISR) θ ISR P(h < θ ISR) θ MISR. So the asymptotic gain at is the ratio of the two MISRs a : G = MISR PPP MISR The MISR for the PPP is easily calculated to be MISR PPP = 2 α 2 = δ δ = δ +δ2 +δ a Haenggi, The Mean Interference-to-Signal Ratio and its Key Role in Cellular and Amorphous Networks. 24. M. Haenggi (ND & EPFL) Asymptotics of SIR Distributions May / 47

29 MISR ASAPPP The method of approximating the SIR ccdf by shifting the PPP s ccdf is called ASAPPP "Approximate SIR analysis based on the PPP". Can we explain the unreasonable effectiveness of ASAPPP? Can we calculate G and G? How close are they? Can we show that the shape of the SIR distributions are similar by comparing the asymptotics? How sensitive are the gains to the path loss exponent and the fading model? Some of these question are addressed in (very) recent work with Radha K. Ganti a. a Ganti and Haenggi, Asymptotics and Approximation of the SIR Distribution in General Cellular Networks. 25, arxiv. M. Haenggi (ND & EPFL) Asymptotics of SIR Distributions May / 47

30 RDP RDP and MISR Definition (The relative distance process (RDP)) For a stationary point process Φ with x = arg min{x Φ: x }, let R {x Φ\{x }: x / x } (,). MISR using the RDP We have ISR = y R h y y α and MISR = E y Ry α = r α Λ(dr). For the stationary PPP, Λ(dr) = 2r 3 dr. M. Haenggi (ND & EPFL) Asymptotics of SIR Distributions May 25 3 / 47

31 RDP Pgfl and moment densities of the RDP of the PPP For the PPP, the probability generating functional (pgfl) of the RDP is G R [f] E x Rf(x) = +2 ( f(x))x 3 dx, and the moment densities are Pgfl for general BS processes ρ (n) (t,t 2,...,t n ) = n!2 n n i= t 3 i. For a general stationary process Φ, the pgfl can be expressed as [ ( ) ] x G R [f] = λ f ( +x b(o, x ) c ) dx. +x R 2 G! o M. Haenggi (ND & EPFL) Asymptotics of SIR Distributions May 25 3 / 47

32 Generalized MISR Generalized MISR We define MISR n (E(ISR n )) /n. For a Poisson cellular network with arbitrary fading, E(ISR n ) = n k= ( δ k!b n,k δ,..., δe(h n k+ ) ), n k + δ where B n,k are the Bell polynomials. A good lower bound on MISR n is obtained by only considering the term n = k in the sum: MISR n MISR (n!) /n = δ δ (n!)/n The bound does not depend on the fading. For δ (α 2), it is asymptotically tight. M. Haenggi (ND & EPFL) Asymptotics of SIR Distributions May / 47

33 Generalized MISR Generalized MISR for PPP with Rayleigh fading 8 Exact Lower bound Lower bound 2 MISR n n=5 n=2 n= α M. Haenggi (ND & EPFL) Asymptotics of SIR Distributions May / 47

34 Generalized MISR Generalized MISR for PPP with Nakagami-m fading 8 m= m=2 m= 2 m= m=2 m= Lower bound α=4 MISR n 6 4 n=5 MISR n 8 6 n=2 4 2 n= α MISR n as a function of α n MISR n as a function of n for α = 4 For the PPP, MISR n is essentially proportional to n. For Rayleigh fading, MISR n (n/e)misr, n. M. Haenggi (ND & EPFL) Asymptotics of SIR Distributions May / 47

35 Generalized MISR Gain G for general fading If F h (x) c m x m, x, p s (θ) c m E[(θ ISR) m ], θ, and thus G (m) = ( E(ISR m PPP ) E(ISR m ) ) /m = MISR m,ppp MISR m. The ASAPPP approximation follows as p s (m) (θ) p (m) s,ppp (θ/g(m) ). This applies more generally to any transmission scheme with diversity m. If MISR m grows roughly in proportion to MISR, G (m) G, and G is insensitive to the fading statistics. M. Haenggi (ND & EPFL) Asymptotics of SIR Distributions May / 47

36 Generalized MISR Deployment gain for lattice with Nakagami fading.9.8 SIR CCDF, α=4, Nakagami fading, m= 2 SIR CCDF, α=4, Nakagami fading, m= P(SIR>θ).7.6 P(SIR>θ) PPP.4 square lattice ISR based approx θ (db) PPP.4 square lattice ISR based approx θ (db) Here the gain for m = (Rayleigh fading) is applied, which is 3 db. Indeed G (m) G in this case. How about G? Is it close to G? M. Haenggi (ND & EPFL) Asymptotics of SIR Distributions May / 47

37 EFIR EFIR Definition (Expected fading-to-interference ratio (EFIR)) Let I x Φ h x x α and let h be a fading random variable independent of all (h x ). The expected fading-to-interference ratio (EFIR) is defined as ( [ ( ) ]) h δ /δ EFIR λπe! o, δ 2/α, where E! o is the expectation w.r.t. the reduced Palm measure of Φ. EFIR properties I The EFIR does not depend on λ, since E! o(i δ ) /λ. It does not depend on the distribution of the distance to the serving BS, either. For the PPP with arbitrary fading: EFIR PPP = (sincδ) /δ = (sinc(2/α)) α/2 δ. M. Haenggi (ND & EPFL) Asymptotics of SIR Distributions May / 47

38 EFIR SIR tail and G Theorem (SIR tail) For all stationary BS point processes Φ, where the typical user is served by the nearest BS, with arbitrary fading, Corollary p s (θ) ( ) θ δ, θ. EFIR G = EFIR ( λπe! = o (I δ ) /δ )E(hδ ). EFIR PPP sincδ Implication on tail of SIR distribution The asymptotic behavior p s (θ) = Θ(θ δ ) is unavoidable for the singular path loss law and stationary BS deployment. M. Haenggi (ND & EPFL) Asymptotics of SIR Distributions May / 47

39 Example: Square lattice Scaled success probability p s (θ)θ δ for square lattice Square lattice with Rayleigh fading for α= Simulation EFIR based Asymptote Analytical upper bound Analytical lower bound P(SIR>θ)*θ δ θ (db) The curve approaches EFIR δ. The EFIR is bounded as (πγ(+δ)) /δ Z(2/δ) EFIR sq ( π ) /δ sincδ Z(2/δ), where Z is the Epstein zeta function. The asymptote is at EFIR.9. M. Haenggi (ND & EPFL) Asymptotics of SIR Distributions May / 47

40 Summary Summary: MISR and EFIR For θ and Rayleigh fading: p s (θ) θmisr; G = MISR PPP MISR For θ and arbitrary fading: p s (θ) ( ) θ δ ; G = EFIR EFIR EFIR PPP The reference MISR and EFIR for the PPP have very simple expressions: MISR PPP = δ δ ; EFIR PPP = (sincδ) /δ They are efficiently obtained by simulation for arbitrary point processes. M. Haenggi (ND & EPFL) Asymptotics of SIR Distributions May 25 4 / 47

41 Comparison of gains Asymptotic gains for lattices G and G for square and triangular lattices G square G tri G square G tri Gain α G barely depends on α, while G slightly increases. M. Haenggi (ND & EPFL) Asymptotics of SIR Distributions May 25 4 / 47

42 Comparison of gains Insensitivity of G to α Recall: MISR = rα λ(r)dr, where λ is the intensity function of the RDP. Relative density of RDP for square lattice Relative density of RDP for triangular lattice.8.8 λ(r)*r 3 /2.6.4 λ(r)*r 3 / r λ sq (r)/λ PPP (r) r λ tri (r)/λ PPP (r) Relative intensity of RDPs of square and triangular lattices. The straight line corresponds to /G,sq and /G,tri. It is essentially the average of the relative densities. M. Haenggi (ND & EPFL) Asymptotics of SIR Distributions May / 47

43 Comparison of gains The gains for the β-ginibre process Gain G G approx. G G approx. G and G for β GPP for α= β So quite exactly (and almost independently of α): G (β) +β/2; G (β) +β. The square lattice has gains of 2 and 3.5, so the -GPP falls quite exactly in between the PPP and the square lattice, both for G and G. M. Haenggi (ND & EPFL) Asymptotics of SIR Distributions May / 47

44 Gain trajectories Gain trajectories G(θ) and asymptotics for lattices Gap [db] square triangular G, G square G, G triang. Horizontal gap for lattices, α=3 Gap [db] square triangular G, G square G, G triang. Horizontal gap for lattices, α= θ [db] θ [db] The gap is relatively constant over more than 5 orders of magnitude for θ. It is not monotonic, but probably G(θ) max{g,g }. M. Haenggi (ND & EPFL) Asymptotics of SIR Distributions May / 47

45 Conclusions Conclusions The world outside Poissonia is harsh. Even for the PPP, the SIR ccdfs for advanced transmission techniques (including MIMO) are unwieldy. To explain the unreasonable effectiveness of the ASAPPP method p s (θ) p s,ppp (θ/g ), we have compared G with G, which is the gap at θ. The asymptotic gains G and G are given by the MISR and the EFIR, respectively. The MISR is closely related to the relative distance process and can be generalized for different types of fading. G and G are insensitive to fading, and G is insensitive to α. The ASAPPP method is relatively accurate over the entire range of θ and highly accurate for p s (θ) > 3/4 (or θ < ). A lot more work can and needs to be done. M. Haenggi (ND & EPFL) Asymptotics of SIR Distributions May / 47

46 References References I B. Blaszczyszyn, M. K. Karray, and H.-P. Keeler, Using Poisson Processes to Model Lattice Cellular Networks. In: INFOCOM 3. Turin, Italy, Apr. 23. H. S. Dhillon et al., Modeling and Analysis of K-Tier Downlink Heterogeneous Cellular Networks. In: IEEE Journal on Selected Areas in Communications 3.3 (Mar. 22), pp R. K. Ganti and M. Haenggi, Asymptotics and Approximation of the SIR Distribution in General Cellular Networks. ArXiv, May 25. A. Guo and M. Haenggi, Asymptotic Deployment Gain: A Simple Approach to Characterize the SINR Distribution in General Cellular Networks. In: IEEE Transactions on Communications 63.3 (Mar. 25), pp M. Haenggi, The Mean Interference-to-Signal Ratio and its Key Role in Cellular and Amorphous Networks. In: IEEE Wireless Communications Letters 3.6 (Dec. 24), pp N. Miyoshi and T. Shirai, A Cellular Network Model with Ginibre Configured Base Stations. In: Advances in Applied Probability 46.3 (Aug. 24), pp M. Haenggi (ND & EPFL) Asymptotics of SIR Distributions May / 47

47 References References II G. Nigam, P. Minero, and M. Haenggi, Coordinated Multipoint Joint Transmission in Heterogeneous Networks. In: IEEE Transactions on Communications 62. (Nov. 24), pp X. Zhang and M. Haenggi, A Stochastic Geometry Analysis of Inter-cell Interference Coordination and Intra-cell Diversity. In: IEEE Transactions on Wireless Communications 3.2 (Dec. 24), pp M. Haenggi (ND & EPFL) Asymptotics of SIR Distributions May / 47