Stochastic geometry and communication networks,

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1 Stochastic geometry and communication networks Bartek Błaszczyszyn tutorial lecture Performance conference Juan-les-Pins, France, October 3-7, 25 OUTLINE I Panorama of some stochastic-geometry models in action, II Basic geometric models, III Signal-to-Interference-and-Noise ratio (SINR) coverage model, IV Modeling ad-hoc networks, V Power control in CDMA: from static to dynamic modeling a spatial Erlang formula. INRIA-ENS 2 PANORAMA... I PANORAMA The mathematical principle of the Voronoi tessellation is widely used as a simple idealization of many complex real partitions of the plane (cells in cellular communication, Gupta & Kumar protocol model of ad-hoc networks; it takes into account only locations of antennas and ignores all other physical aspects of the communication technology as a.g. additive interference) The dual Delaunay graph can be used as a protocol model of neighbuorhood in networks topology for routing Boolean model is a first model of coverage of wireless network (it does not take into account interference). As the underlying model for the study of continuum percolation it can be used to address the questions of connectivity of ad-hoc networks in the absence of interference. Mathematical representation of interferences based on (Poisson) shot noise processes a variety of results on coverage, connectivity and capacity of large interference-limited networks. 3 4

2 BASIC MODELS... Poisson Point Process II BASIC GEOMETRIC MODELS Poisson point process, Voronoi tessellation and Delaunay graph, Boolean model, Shot-Noise model, References. Planar Poisson point process (p.p.) Φ of intensity λ: Number of Points Φ(B) of Φ in subset B of the plane is Poisson random variable with parameter λ B, where is the Lebesgue measure on the plane; i.e., λ B (λ B )k P{ Φ(B) = k } = e, k! Numbers of points of Φ in disjoint sets are independent. Laplace transform of the Poisson p.p. L Φ (h) = E[e R h(x) Φ(dx) ] = e λ R ( e h(x) ),dx, where h( ) is a real function on the plane and h(x) Φ(dx) = X i Φ h(x i). 5 6 BASIC MODELS/Poisson p.p.... BASIC MODELS... Poisson p.p. is the basis of the stochastic-geometry modeling of communication networks. This modeling consist in treating the given architecture of the network as a snapshot of a (homogeneous) random model, and analyzing it in a statistical way. In this approach the physical meaning of the network elements is preserved and reflected in the model, but their geographical locations are no longer fixed but modeled by random points of, typically, homogeneous planar Poisson point processes. Consequently, any particular detailed pattern of locations is no longer of interest. Instead, the method allows for catching the essential spatial characteristics of the network performance basically through the densities of these point processes (i.e., the densities of the network devices). Voronoi Tessellation (VT) and Delaunay graph Given a collection of points Φ = {X i } on the plane and a given point x, we define the Voronoi cell of this point C x = C x (Φ) as the subset of the plane of all locations that are closer to x than to any point of Φ; i.e., C x (Φ) = {y R 2 : y x y X i X i Φ}. When Φ = {X i } is a Poisson p.p. we call the (random) collection of cells {C Xi (Φ)} the Poisson-Voronoi tessellation (PVT). Edges of the Delaunay graph connect nuclei of the adjacent cells. Borders of Voronoi Cells 7 8

3 BASIC MODELS/VT... BASIC MODELS... Boolean Model (BM) VT is a frequently used generic model of tessellation of the plane. Let Φ = {(X i, G i )} be a marked Poisson p.p., where {X i } are points and {G i } are iid random closed stets (grains). We define the Boolean Model (BM) as the union Ξ = X i G i where x G = {x + y : y G}. i Points denote locations of various structural elements (devices) of the network (base station antennas and/or network controllers in cellular networks, concentrators in fixed telephony, access nodes in ad hoc networks, etc.). Cells denote mutually disjoint regions of the plane served in some sense by these devices. Known: Poisson distribution of the number of grains intersecting any given set. Asymptotic results (λ ) for the probability of complete covering of a given set BM with spherical grains of random radii 9 BASIC MODELS/BM... BASIC MODELS... Shot-Noise (SN) model BM is a generic coverage model. Let Φ = {(X i, S i )} be a marked p.p., where {X i } are points and {S i } are iid random variables. Given a real response function L( ) of the distance on the plane we define the Shot-Noise field Points denote locations of various structural elements (devices) of the network. Granis denote independent regions of the plane served these devices. I Φ(y) = i S i L(y X i ). In wireless networks it is a simplified model (it does not take into account interference) for the study of coverage and connectivity. When Φ is a marked Poisson p.p. then we call I Φ the Poisson SN. For the Poisson SN, the Laplace transform of the vector (I Φ (y ),..., I Φ (y n )) is known for any y,..., y n R 2 (via Laplace transform of the Poisson p.p.). 2

4 BASIC MODELS/SN... BASIC MODELS... References SN is a good model for interference in wireless networks.. Stoyan & Kendal & Mekke (995) Stochastic Geometry and its Applications. Marks S i correspond to emitted powers. Wiley, Chichester Response function correspond to attenuation function. 2. Okabe & Boots & Sugihara (2) Spatial Tessellations: Concepts and Applications of Voronoi Diagrams. Willey, Chichester. 3 4 III SINR COVERAGE MODEL SINR COVERAGE MODEL... Φ = {X i, (S i, T i )} marked point process (Poisson) {X i } points of the p.p. on R 2 antenna locations, (S i, T i ) (R + ) 2 possibly random mark of point X i (power,threshold) { cell attached to point X i : C i (Φ, W ) = y : S } il(y X i ) W + κi Φ (y) T i where I φ (y) = i S il(y X i ) shot noise process, κ interference factor, W external noise, l( ) attenuation (response) function. C i is the region where the SINR from X i is bigger than the threshold T i. Coverage PROCESS: Ξ(Φ; W ) = i N C i (Φ, W ). Motivations, Snapshots and qualitative results, Typical cell study (coverage probability for the point is some distance to the antenna, simultaneous coverage of several points, mean area of the cell), Handoff study (overlapping of cells, coverage probability for a typical point, distance to different handoff states), Macroeconomic optimization example, References. 5 6

5 SINR COVERAGE MODEL... Motivation I: CDMA handoff cells x a point in R 2 (location of an antenna), s and t (pilot signal power of the antenna and SINR threshold (bit energy-tonoise spectral power density E b /N O ) for the pilot signal), φ = {x i, (s i, t i )} pattern of antennas, w external noise, κ orthogonality factor, l( ) attenuation function (x, (s, t )) w y (x, (s, t )) (x, (s, t )) (x 3, (s 3, t 3 )) (x 4, (s 4, t 4 )) s l(y x ) w + I φ (y) t 7 SINR COVERAGE MODEL / CDMA motivation... Parameter values Intensity of Poisson process of base stations λ BS.2 BS/km 2. Pilot signal power s 3 mw SINR threshold (bit energy-to-noise spectral power density E b /N O ) for the pilot t 4 db External noise w 5 db Interference factor for pilots from different BS s κ = Attenuation function l(x) = A max( x, r ) α or l(x) = ( + A x ) α with α SINR COVERAGE MODEL / CDMA motivation... SINR COVERAGE MODEL / Motivations II This is a relatively simple model, which takes into account only locations of the Base Stations, their pilot signal powers and SIR s for the pilots. In particular there is no any pattern of mobiles assumed yet and it does not take into account power control issues. Gupta & Kummar physical model for ad-hoc networks (see Modeling of ad-hoc networks) 9 2

6 SINR COVERAGE MODEL... SINR COVERAGE MODEL / Snapshots... Snapshots and qualitative results κ =.5 κ =. κ =. κ = Constant emitted powers S i, e i, T =.4 and interference factor κ Constant emitted powers S i, e i, T =.4, W =, l(r) = (Ar) β and attenuation exponent β. Small interference factor allows one to approximate SINR cells by a Boolean model (via perturbation methods) SIR cells tend to Voronoi cells whenever attenuation is stronger, e.g. in urban areas SINR COVERAGE MODEL / Typical cell study... SINR COVERAGE MODEL / Typical cell study... Probability for a typical cell to cover a point Given: Φ marked Poisson point process representing antennas in R 2, (, (S, T )) additional antenna located at fixed point with random (S, T ) distributed as any mark of Φ, independent of it (thus Φ {(, (S, T )} has Poisson Palm distribution), y location (of a mobile) in R 2. Probability for C to cover a given point y located at the distance R to the origin: ( ) p R = P y C ( S(/T )l(r) W I Φ (y) > = P ). Res. For M/G case (general distribution of (S, T )) the coverage probability p R can be given via Laplace transforms of S(/T ), W and the Laplace transform of I Φ (y) that is [ E[exp( ξi Φ (y))] = exp where L S (ξ) = E[e ξs ] is the Laplace transform of S. R d ( ) ] L S (ξl(y z)) µ(dz), Cor. Fourier transform of the Poisson shot-noise variable I φ (y) Rieman Boundary Problem probability of coverage by the typical cell

7 SINR COVERAGE MODEL / Typical cell study... Example Fourier transform F IΦ (ξ) of the homogeneous Poisson (intensity λ) shot noise with exponential S (parameter m) and attenuation l(x) = A max( x, r ) 4 F IΦ (ξ) = [ ] E e iξi Φ [ ( ) iaξ m = exp λπ m arctan r 2 λ iaξ iaξ 2 π2 m + λπr 2 r 4 iaξ rm 4 ] iaξ + r 4m, for ξ R, where the branch of the complex square root function is chosen with positive real part. SINR COVERAGE MODEL / Typical cell study... Special M/M case Res. [Baccelli&BB&Muhlethaler (24)]: Assume that {S i } are exponential r.vs. with par. µ and T i = T are constant. Then the probability for C to cover a given point located at the distance R: is equal { p R = exp 2πλ } u + l(r)/(t l(u)) du. proof: Say the emitter is at the origin and consider the corresp. Palm distribution P; p R = P(S T I Φ /l(r)) = e µst/l(r) dp(i Φ s) = ψ IΦ (µt/l(r)), where L IΦ ( ) is the Laplace transform of the value of the hom. Poisson SN I Φ SINR COVERAGE MODEL / Typical cell study / M/M case... SINR COVERAGE MODEL / Typical cell study / M/M case... Some optimizations One can study the following optimization problems for the expected effective transmission range r p r : Cor. For the attenuation function l(u) = (Au) β given the density of stations λ find the targeted range r that optimizes the p R (λ) = e λr2 T 2/βC, ( ) where C = C(β) = 2πΓ(2/β)Γ( 2/β) /β. expected effective transmission range ρ = ρ(p) = max {rp r(p)} = r T /β 2λpC r max = r max (p) = argmax r {rp r (λ)} = T /β 2λC 27 28

8 SINR COVERAGE MODEL / Typical cell study / M/M case optimization SINR COVERAGE MODEL / Typical cell study... Probability for a typical cell to cover two points given the targeted range R find the density of emitters λ that optimize the expected effective transmission range R p R : λ max = λ max (R) =argmax λ {Rp R (λ)} = max λ {Rp R(λ)} = R 2 T 2/β C R 2 T 2/β ec (y, y 2 ) two point to be covered by a given cell C (Φ, W ) under Palm distribution of Φ {(, (S, T )} We need the joint Laplace transform of (I Φ (y ), I Φ (y 2 )) that is given by [ ( ) ] E exp ξ I Φ (y ) ξ 2 I Φ (y 2 ) [ ( ) = exp L S (ξ l(y z) + ξ 2 l(y 2 z)) R d ] µ(dz) SINR COVERAGE MODEL / Typical cell study... SINR COVERAGE MODEL / Typical cell study / Perturbation of Boolean model... Denote p (κ) R C (κ) = Coverage probability via perturbation of Boolean model valid for small interference factor κ = P(x C(κ) { } y R 2 : Sl(y) κi Φ (y) + W ), where x = R and Assume F (u) = P((Sl(x) W ) u) admits Taylor approximation at : F (u) = F () + and R (u) = o(u h ) u. h k= F (k). () u k + R (u) k! Res. p (κ) R = value for the Boolean model {( }} ){ P Sl(x) W provided E[(I Φ (x)) 2h ] <. correcting terms {}}{ h κ k F (k) () [ E (I Φ (y)) k] + k! k= error {}}{ o(κ h ), 3 32

9 SINR COVERAGE MODEL / Typical cell study... SINR COVERAGE MODEL / Typical cell study... Numerical examples Mean cell area formula p (κ) () p (.) (y) v (κ) () Denote the mean area of the cell of the BS located at by v = E[ C ] Recall that p R is the coverage probability for location at distance R Then v = R d p y dy κ y Probability of coverage as a Probability of coverage as a κ Mean cell area as a function of the interference function of the distance. function of the interference coefficient coefficient SINR COVERAGE MODEL / Handoff study... SINR COVERAGE MODEL / Handoff study / Overlapping of cells... Overlapping of cells Deterministic scenario: given n cells C(x i, s i, t i ; φ, w), i =,..., n Res. The inequality n i= t i/( + t i ) < is a necessary condition for n i= C(x i, s i, t i ; φ, w) Random scenario: Cor. If the distribution of the ratio T is such that T τ for some τ >, then the number K y of cells of the coverage process Ξ covering any given point y is a.s. bounded K y < + τ τ (Given point cannot be covered by ( + τ)/τ or more cells, no matter how close they are located and how their signal is strong pole handoff number.). Example: For the maximal pilot s bit energy-to-noise spectral power density τ = E b /N O = 4 db the pole handoff number (theoretical maximal handoff number) K

10 SINR COVERAGE MODEL / Handoff study... SINR COVERAGE MODEL / Handoff study... Moment expansion of the number of cells K y covering y Res. The factorial moment of K y is given by E[K (n) y ] = E[K y (K y )... (K y n + ) + ] ( n ( n ) ) = y C x k, S k, T k ; Φ + ε (xi,(s i,t i )), W (R d ) n P k= i= i k µ(dx )... µ(dx n ). Little law In particular, for a homogeneous Poisson point process with intensity λ E[K ] = λe[ C ], where C is the area of the typical cell. Moreover, in this case the volume fraction p (fraction of the space covered by Ξ) is given by ( ) k+ p = E[(K ) (k) ]. k! k= SINR COVERAGE MODEL / Handoff study... SINR COVERAGE MODEL / Handoff study / Contact distribution functions... Example of contact d.f. s estimation Contact distribution functions Conditional distribution of the model Two finite sets of points: z,..., z n and z,..., z p. Density Density L R Histograms of linear L and spherical R contact d.f. given the point is not covered. EL varl ER varr.423 km.9 km 2.2 km.3 km 2 Condition: points z i are covered by at least n i cells and points z i are covered by at most n i cells, for some given numbers n,..., n n and n,..., n p. This type of conditions allows one to consider cases where the exact number of cells covering a point is specified. 39 4

11 SINR COVERAGE MODEL / Handoff study / Contact distribution functions... SINR COVERAGE MODEL / Handoff study / Contact distribution functions... Almost exact simulation of the shot-noise For a given size of observation window (radius R) one selects a larger influence window (radius R ) in order to get good estimate of the shot-noise term I φ in the smaller observation window. Th. If the attenuation functions is of the form l(x, y) < C/ x y β for some constants C >, β > and if the distribution of S has finite moment E[S /(β/2 δ) ] < for some δ [, β/2], then one can show that for any R, ε, α >, there exists R > such that ) P (sup y <R X i >R S i l(y, X i ) < ε > α. Perfect simulation in the observation window One constructs a Markov process ( Z t ) of patterns of points that has for its stationary distribution the conditional distribution. Points are generated at exponential periods and located in the window but only if their presence does not violate conditions of maximal coverage of the points z i. Points located in the window stay there for exponential times and are removed, but only if their absence does not violate the conditions of maximal coverage of the points z i. If a particular removal would lead to the violation, then the point are exponentially perpetuated. The exact stationary distribution of the Markov process ( Z t ) is obtained using backward simulation (coupling from the past) similar to that proposed by Kendall SINR COVERAGE MODEL... Macroeconomic optimization example: densification / magnification SINR COVERAGE MODEL... Increase the mean power m of existing antennas or increase the density λ of antennas? C total budget of an operator per km 2, C λ cost of one antenna, C m cost of increasing the power of one antenna by W. constraint: λc λ + C m λ/m = C. Plots of mean handoff as a functions of mean antenna power /m under budget constraint with C =, C λ = 5 and from the top: C m =, 2, 5. Solution: Plot maximum = Optimal configuration. mean handoff level EN References. Baccelli & Błaszczyszyn (2), On a coverage process ranging from the Boolean model to the Poisson Voronoi tessellation, with applications to wireless communications Adv. Appl. Probab Tournois (22), Perfect Simulation of a Stochastic Model for CDMA coverage INRIA report 4348, 3. Baccelli, Błaszczyszyn & Tournois (22), Spatial averages of downlink coverage characteristics in CDMA networks (INFOCOM ) mean antenna power /m 43 44

12 AD-HOC NETWORKS... IV MODELING AD-HOC NETWORKS A random set of users distributed in Ad-hoc networks, A few sg interference aware models for ad-hoc networks, Some optimization problem in capacity and medium access control, space and sharing a common Hertzian medium. Users constitute ad-hoc network that is in charge of transmitting information far emitter receiver References. away via several hops. Users switch between emitter and receiver modes AD-HOC NETWORKS... AD-HOC NETWORKS... Multi-hop transmissions Interference and successful reception Emitter sends a packet in some given di- Emitter sends a packet emitting some rection far away via several hops. power. The packet is received by some number Transmission distance attenuates the (possibly ) of neigbouring receivers. emitted power. An optimal receiver among them is in charge of forwarding this packet in (one emitter receiver optimal receiver towards destination Emitted power causes interference at all receivers. emitter used signal receiver interference of) his next emission time-slots. Reception is successful if the Signal to In- In the case of no reception, emitter re- terference Ratio (SIR) at the receiver is emits the packet next authorized time. large enough

13 AD-HOC NETWORKS... AD-HOC NETWORKS / A few models / General settings... A few sg interference aware models General settings Nodes distributed in (a subset of) the plane according to a Poisson p.p. Φ = {X i }, Nodes X i are marked in some way (not necessarily independently) by e i = when node is emitting or when not (it is a potential receiver); Call Φ = {X i Φ : e i = } emitters and A way of choosing marks e i for nodes is called medium access control. Two possibilities can be considered: Some central authority assigns marks e i (in some optimal way) for each given configuration of nodes {e i } are dependent in some way. Each node independently switches between modes e i = and e i = {e i } are independent, given configuration of nodes. Φ = {X i Φ : e i = } potential receivers AD-HOC NETWORKS / A few models... AD-HOC NETWORKS / A few models / Protocol model... Examples of medium access Voronoi tessellation principle: Protocol Model [Gupta & Kumar (2)] When the node X i Φ transmits it can be successfully received by node X j Φ if centralized independent X j X i ( + ) X j X k X k Φ, where is some constant. X j modified Voronoi Cell C Xi (Φ ) The Protocol Model with centralized medium access was studied asymptotically (when number of nodes goes to ) by G & K (2); to be commented. 5 52

14 AD-HOC NETWORKS / A few models... AD-HOC NETWORKS / A few models / Carrier-Sense model... Hard-core principle: Carrier-Sense Model Close to some currently used protocols implemented e.g. in IEEE 82. When the node X i Φ transmits it can be successfully received by node X j Φ if X j X i R and X k X i > R cs X k Φ, where R cs > R are some constants. Each communication within the transmission range R is protected by the exclusion disc centered at the transmitter with radius R cs > R. (One can also think of the models with exclusion discs centered at the receiver and at both transmitter and receiver.) Examples of medium access Matérn hard-core model: Nodes are first independently marked by some auxiliary marks in [, ]. A node X i is selected as emitter (e i = ) if its auxiliary mark is larger than all auxiliary marks within its neighbourhood of range R cs. Otherwise the node is marked receiver. Gibbs hard-core model: The steady state distribution of a spatial birth-and-death process of nodes. When a node is born it is marked as emitter if in its neighbourhood of range R cs there is no any other emitter. Otherwise it is marked receiver. The Carrier Sense Model, has not been studied yet by means of stochastic geometry tools (to the best of our knowledge) AD-HOC NETWORKS / A few models... AD-HOC NETWORKS... SINR principle: Physical model Points X i are independently identically marked by some S i representing powers used when emitting signals. l(r) = (Ar) β signal attenuation function, T signal-to-interference ratio threshold, W > constant external noise, κ > interference factor. When the node X i Φ transmits it can be successfully received by node X j Φ if S i l( X j X i ) W + κi Φ \X i (X j ) T, See Performance 5 tutorial by P. Thiran. Connectivity for the physical model where I Φ is the shot-noise process of Φ I Φ (y) = X k Φ S k l( y X k ). Communication is successful if the signal-to-interference ratio at the receiver is bigger than the threshold T

15 AD-HOC NETWORKS... Some optimization problem in capacity and medium access for the physical model AD-HOC NETWORKS / Some optimization problem... Suppose emitter X = has to send information to infinity along the real axis. Define effective distance traversed in one hop, called also progress Define also a modified progress effective distance ( successful rec. ind. {}}{{ ( }} ) {) + D = max X j Φ δ(x j,, Φ ) X j cos(arg(x j )), D = max X j Φ {{}}{ ( }} ) {) + p Xj (λp) X j cos(arg(x j )). probab. of ( successful rec. effective distance effective distance emitter Denote d(λ, p) = E[D], d(λ, p) = E[ D]. optimal receiver Search for receivers that realize the progresses, D and D can be implemented AD-HOC NETWORKS / Some optimization problem... AD-HOC NETWORKS / Some optimization problem... Fact The mean total distance traversed in one hop by all transmissions initialized in some unit area (called density of (modified) progress) is equal to λp d(λ, p) (resp. λp d(λ, p)). proof: For B R 2 of unit area, by the Campbell s formula [ ] E D i = λp (x B)E[D ] dx R 2 X i Φ B = λp d(λ, p). For z [, ], define an auxiliary function ( ze t ) G(z) = 2 arccos dt. {t:e t / 2et /z} 2et Res. For the M/M model with W = The distribution function of the modified progress is given by Fact For all λ, p > we have d(λ, p) d(λ, p). F D(z) = P( D z) = e λ( p)(r max(λp)) 2 G(z/ρ(λp)). proof follows from Jensen s inequality. 59 6

16 AD-HOC NETWORKS / Some optimization problem... AD-HOC NETWORKS / Some optimization problem... proof: D is an extremal shot-noise max Xi Φ g(x i) with the response function g(x) = x p x (cos(arg(x))) +. Its distribution function can be expressed by the Laplace transform of the (additive) shot noise [ [ ]] P( max i) z) = E exp X i Φ ln((g(x i ) z)) X i Φ and thus, for Poisson p.p. Φ with intensity λ( p) [ ] P( D z) = exp λ( p) (g(x) > z) dx. R 2 Passing to polar coordinates in the integral R... dx 2 completes the proof. Cor. For the M/M model with W = the expectation of D is equal to d(λ, p) = E[ D] = T /β 2λpeC [( exp ) G(z) ] dz, p 2T 2/β C 6 62 AD-HOC NETWORKS / Some optimization problem... AD-HOC NETWORKS / Some optimization problem... Res. The maximal density of modified progress λp d(λ, p) is attained for p satisfying ( + G(z) ) [( exp ) G(z) ] dz =. pt 2/β C p 2T 2/β C The optimization Goal: optimize the density of progress λp d(λ, p) in MAP p, for fixed λ. λp d(λ, p).6 Note that replacing d by d in the optimization problem will result in conservative.5 bound on the density of progress. Example: Modified transport capacities for the special case with T = {3, 5, 7}dB (curves from top to bottom) p 63 64

17 AD-HOC NETWORKS / Some optimization problem... AD-HOC NETWORKS... Relation to Gupta & Kumar s (2) result The transport capacity of a bounded network with Protocol and SIR (Physical) Model with centralized medium access is proportional to λ (as λ ). This law holds true even if the network architecture and operation is optimally organized in a centralized manner. Our density of progress can be seen as an instantaneous transport capacity. We show that the independent medium access gives an instantaneous transport capacity of K(p) λ. Closed form of K(p) allows for optimization of K(p) in p. References Gupta & Kumar (2), The Capacity of Wireless Networks, IEEE Inf. Theory, Dousse & Baccelli & Thiran (23) Impact of Interferences on the Connectivity of Ad Hoc Networks, IEEE INFOCOM, Baccelli & Blaszczyszyn & Muhlethaler (24), An Aloha protocol for multihop mobile ad-hoc wireless networks, Proc of 6th ITC Specialist Seminar to appear in IEEE Inf. Theory Question: Is this performance implementable and stable in the strong sense (existence of positive recurrent Markov process describing the system)? POWER CONTROL... V POWER CONTROL IN CDMA: CDMA: Interference limited radio channel FROM STATIC TO DYNAMIC MODELING CDMA Power allocation algebra, Decentralized Power Allocation Principle DPAP, Network architecture models, (Y, S ) 5 5 W (Y 2, S 2) Maximal load estimates, Feasibility probabilites, X (Y,S ) Blocking rates via a spatial Erlang formula, (Y,S ) (Y,S ) References. S l(x Y ) W + I(X) C 67 68

18 POWER CONTROL... CDMA Power allocation algebra N BS = {Y j } j : locations of base-stations (BS s) in R 2, N M = {X j i } i: locations of mobiles served by BS No. j, C j i : SINR required for mobile X j i, W j i : total non-traffic noise at X j i (from common overhead channels, thermal noise), κ j, γ: orthogonality factors, l(x, y): path-loss from y to x. Power allocation feasible if exist antenna powers S j i S j i l(y j X j i ) W j i + κ j l(y j, X j i ) i i S j i } {{ } own-cell interference + γ k j < such that l(y k, X j i ) i S k i } {{ } other-cell interference C j i all i, j. 69 POWER CONTROL / Power allocation algebra... Local power allocation feasible for each BS Local and global problem Power allocation feasible and Global power allocation feasible 7 POWER CONTROL / Power allocation algebra... Fix BS j. Local problem Fix total powers emitted on traffic channels by other BS s: S k = i S k i (k j). Power allocation is locally feasible in cell j if exist powers S j i < s.t. S j i l(y j X j i ) W j i + κ j l(y j, X j i ) i i Sj i + γ l(y k, X j i ) Si k k j i }{{ } fixed=γ k j l(y k,x j i )S k Res. Local power allocation is feasible iff κ j i C j i + κ j C j i <. (local) pole capacity condition C j i all i. 7 POWER CONTROL / Power allocation algebra... Global problem Suppose for each BS local power allocation is feasible. Define: a jk = γ i b j = i H j i l(y k, X j i ) l(y j, X j i ) for j k and a jj = i H j i W j i l(y j, X j where Hj i ), i = C j i + κ j C j. i Denote the matrix (a jk ) = A, the vector (b j ) = b and (S j ) = S. κ j H j i, Global power allocation is feasible if exist antenna powers S j < (total powers emitted on traffic channels) such that S b + A S 72

19 POWER CONTROL / Power allocation algebra... POWER CONTROL... Res. Global power allocation feasible iff the spectral radius of the matrix A is less than. The minimal solution S is equal to n A n b. The minimal solution can be obtained as the limit of the iteration Ab, A 2 b,... A sufficient condition for the spectral radius to be less than one is that A is substochastic (has row-sums less then ). Decentralized Power Allocation Principle Decentralized Power Allocation Principle (DPAP) Each BS j verifies for the pattern N j M of the mobiles it controls if κ j H j i + γ H j l(y k, X j i ) i l(y X j i N j k j M X }{{} j i N j j, X j i ) M }{{} pole capacity term other-cell-interference correcting term equivalently X j i N j M H j total path loss of user i i own-bs path loss of user }{{} i user s i weight <. <, POWER CONTROL... POWER CONTROL / Network architectures... Network architecture models Poisson-Voronoi (P-V) model ( too random ) Hexagonal (Hex) model ( too regular ) BS s {Y j } located according to hexagonal grid, p.p, with spatial density λ BS. All antenna parameters are i.i.d. marks. All mobiles form independent Poisson p.p. N M with spatial density λ M. Each mobile is served by the nearest BS. BS s {Y j } located according to Poisson p.p, with intensity λ BS. All antenna parameters are i.i.d. marks. All mobiles form independent Poisson p.p. N M with intensity λ M. Each mobile is served by the nearest BS. (Equivalently: Each BS j serves mobiles N j M in its Voronoi cell.) 75 76

20 POWER CONTROL... Maximal load estimations of P-V and Hex model (Wrong) Idea: Given density of BS s λ BS find maximal density of mobiles λ M, such that power allocation is feasible with probability. Res. Given density of BS s λ BS, for any λ M > in both P-V and Hex model, the spectral radius of A is equal with probability, and thus power allocation in not feasible! Conclusion: A reduction of mobiles (admission control) is necessary for any λ M >. Calculate blocking probabilities. POWER CONTROL / Maximal load estimations... Explicit mean formulas M mean number of users per cell, R mean distance between BS s; λ BS = /(πr 2 ), L(r) = (Kr) α path-loss, α path-loss exponent, κ (downlink) orthogonality factor, H bit rate (SINR threshold) P max power limit downlink: M P cch /P max H(κ + f + L(R)g( M, α)/p max ) f H = /(α 2), fp V = 2/(α 2), ḡ H ()..., ḡ P V () =... uplink: M H( + f + L(R)h( M, α)/p max ) h H ()..., h P V () = POWER CONTROL / Maximal load estimations... POWER CONTROL M 3 25 Feasibility probabilities for DPAP R ( P H j total path loss of user i i own-bs path loss of user i < X j i N j M } {{ } DPAP satisfied ). Hexagonal model Poisson-Voronoi model Mean maximal load for downlink with power constraints; number of users in cell as a function of the distance between BS s. It says says how often an non-constrained Poisson configuration of users in a given cell cannot be entirely accepted by the admission scheme DPAP. 79 8

21 POWER CONTROL / Feasibility probabilities... Pr. of DPAP λ M Simulated DPAP failure probability for P-V model (more flat curve) and Hex model (more steep) curve. The straight line corresponds to the explicit mean capacity of the P-V model. λ BS =.8 BS/km 2 C =.797 γ = κ =.2 α = 3 POWER CONTROL... Blocking rates under DPAP spatial dymamic modeling Fix one BS, say Y =. Denote its cell, considered as a subset of R 2, by C. Spatial Birth-and-Death (SBD) process of call arrivals to C : for a given subset A C, call inter-arrival times to A are independent exponential random variables with mean /λ(a), where λ( ) is some given intensity measure of arrivals to C unit of time, call holding times are independent exponential random variables with mean τ. Call acceptance/rejection: given some configuration of calls in progress {X m inc }, accept a new call at x if f(x) + m f(x m) <, where f( ) is the call weight function defined on C, and reject otherwise POWER CONTROL / Blocking rates... Define blocking rate associated with a given location in the cell is the fraction of users arriving according to the SBD process at this location that are rejected. Res. The stationary distribution (time-limit) of the (non-constrained) SBD process of call arrivals is the distribution of a (spatial) Poisson process Π with density λ( ). The stationary distribution of calls accepted is the distribution of a Poisson process truncated to the state space { m f(x m) < }. Blocking rate b x at x C is given by the spatial Erlang formula b x = Π{ f(x) m f(x m) < } Π{ m f(x m) < } Rem. Note that m f(x m) is a compound Poisson r.v., whose distribution can be effectively approximated by Gaussian distribution.. 83 POWER CONTROL / Blocking rates... Blocking rate Numerical results (assumptions correspond to UMTS) approximation I approximation II normalized distance Approximations of the blocking probability as functions of the distance to BS for the mean number M = 27 of users per cell. 84

22 POWER CONTROL / Blocking rates / Numerical results... POWER CONTROL / Blocking rates / Numerical results... Blocking rate average cell edge infeasibility Blocking rate average, R= R=3 R=5 cell edge, R= R=3 R=5 infeasibility, R= R=3 R= M M Blocking probability at the cell edge, average blocking probability, and feasibility probability as functions of the mean number of users in hexagonal cell. 85 Blocking probability at the cell edge, average blocking probability, and feasibility probability as functions of the mean number of users in hexagonal cell; comparison for various cell radii R. 86 POWER CONTROL... References Baccelli & Blaszczyszyn & Tournois (23) Downlink admission/congestion control and maximal load in CDMA networks, IEEE INFOCOM Baccelli & Blaszczyszyn & Karray (24) Up and Downlink Admission/Congestion Control and Maximal Load in Large Homogeneous CDMA Networks, Mobile Networks 9(6), Baccelli & Blaszczyszyn & Karray (25) Blocking Rates in Large CDMA Networks via a Spatial Erlang Formula, IEEE INFOCOM. 87