Connectivity in large mobile ad-hoc networks

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1 Weiertraß-Intitut für Angewandte Analyi und Stochatik Connectivity in large mobile ad-hoc network WOLFGANG KÖNIG (WIAS und U Berlin) joint work with HANNA DÖRING (Onabrück) and GABRIEL FARAUD (Pari) Mohrentraße Berlin el

2 A probabilitic model for large mobile ad-hoc network large bounded domain D in R d (telecommunication area) many particle X (1),..., X (N) (the uer) move randomly and independently relay principle: meage are tranmitted via a equence of hop from uer to uer; each hop ditance i R Our main quetion today: Connectivity I there a hop-path from a given uer to any other uer? How many pair of uer have a connection at a given time, over a given time lag? (in term of percentage) What i the amount of time during a given time lag over which a large percentage of the uer are connected with each other? What i the large-time aymptotic of the probability that a non-zero percentage ha a bad ervice (i.e., low connectivity)? We will attack uch quetion for N in term of a law of large number. Connectivity in large mobile ad-hoc network Edinburgh, January 2016 Seite 2 (10)

3 Connection time hermodynamic limit: D = D N R d with volume N. Equivalently: D independent of N, but the tranmiion radiu R i replaced by R N = N 1/d. Communication zone at time : Connectivity at time : x Connection time: D (N) = D N i=1 B ( X (i), N 1/d) N y x and y lie in the ame component of D (N). τ (N) := d 1l{X (1) N X (2) }, 0 We will invetigate the aymptotic of τ (N) in the limit N in probability w.r.t. P( X (1), X (2) ) for almot all (X (1), X (2) ). Connectivity in large mobile ad-hoc network Edinburgh, January 2016 Seite 3 (10)

4 Movement cheme and continuum percolation Aumption on the movement cheme he location of X (1) ha a continuou denity f : D [0, ), and P(X (1) = x X (1) = y) = 0 for <. Sufficient: the exitence of a jointly continuou denity for (X (1), X (1) ). Notion from continuum percolation θ(λ) percolation probability for a Poion point proce in R 2 with intenity λ. λ c = inf{λ > 0: θ(λ) > 0} the critical threhold. Write x > y : there exit a path from x to y within {f > λ c}. (Analogouly with and and < intead of >.) For {, >}, define τ ( ) (X (1), X (2) ) = 0 d 1l{X (1) X (2) }θ ( ) ( f(x (1) ) ) θ ( ) ( f(x (2) ) ), with θ (>) (λ) = θ(λ ) and θ ( ) (λ) = θ(λ+) the left- and right-continuou verion of θ. Connectivity in large mobile ad-hoc network Edinburgh, January 2016 Seite 4 (10)

5 Main reult: aymptotic of the connection time heorem: Bound on the connection time For almot every path X (1), X (2), in probability with repect to P 1,2 := P( X (1), X (2) ), τ (>) (X (1), X (2) ) lim inf τ (N) N lim up τ (N) τ ( ) (X (1), X (2) ). (1) N Comment: Global (determinitic) effect: he two walker are only connected at time if {X (1) X (2) }, i.e., if they belong to the ame component of the region where the denity of uer i high enough. Local (tochatic) effect: hey have a connection only if they locally belong to the infinitely large cluter, which ha probability θ(f (X (1) ))θ(f (X (2) )). Doe the limit exit? Under many abtract condition, all the four expreion in (1) are equal to each other almot urely, but it i cumberome to formulate and prove reaonably general and explicit one. (Difficulty here: geometry of the et {f = λ c}) Connectivity in large mobile ad-hoc network Edinburgh, January 2016 Seite 5 (10)

6 On the proof: convergence of the expectation (I) Auxiliary event: { G (i) N,,δ = X (i) N [ X (i) + ( δ/2, δ/2) d]}. hen connection of X (1) and X (2) i roughly equal to G (1) G (2), if they are in the ame component of {f > λ c} or {f < λ c}: Lemma 1: Approximation with G (i) For P-almot all X (1) and X (2), for almot any [0, ] and on the event {f (X (1) ) λ c} {f (X (2) ) λ c} {X (1) X (2) }, ) ( 0 = lim lim X (1) N X (2) G (1) N,,δ G(2) (1) N,,δ {X [( δ 0 N P1,2 )] > X (2) }. Connectivity in large mobile ad-hoc network Edinburgh, January 2016 Seite 6 (10)

7 On the proof: convergence of the expectation (I) Auxiliary event: { G (i) N,,δ = X (i) N [ X (i) + ( δ/2, δ/2) d]}. hen connection of X (1) and X (2) i roughly equal to G (1) G (2), if they are in the ame component of {f > λ c} or {f < λ c}: Lemma 1: Approximation with G (i) For P-almot all X (1) and X (2), for almot any [0, ] and on the event {f (X (1) ) λ c} {f (X (2) ) λ c} {X (1) X (2) }, ) ( 0 = lim lim X (1) N X (2) G (1) N,,δ G(2) (1) N,,δ {X [( δ 0 N P1,2 )] > X (2) }. For ufficiently mall δ, the δ-boxe around x = X (1) and y = X (2) are dijoint. Ue reult of [PENROSE (1995)]: With probability tending to one, the only way to realie G (1) i that X (1) lie in the unique giant cluter of it δ-box. Analogouly for X (2) and G (2). Contruct a equence of mutually overlapping δ-boxe from x to y inide {f > λ c} and argue imilarly for thi et. Connectivity in large mobile ad-hoc network Edinburgh, January 2016 Seite 6 (10)

8 On the proof: convergence of the expectation (II) Lemma 2: Convergence of the probability of G (i) Under the ame aumption, θ(f (X (1) ) )θ(f (X (2) ) ) lim inf δ 0 lim up δ 0 ( ) lim inf N P1,2 G (1) N,,δ G(2) N,,δ ( lim up P 1,2 G (1) N,,δ G(2) N,,δ N θ(f (X (1) )+)θ(f (X (2) )+). ) Aymptotic independence of G (1) and G (2) in the limit N, followed by δ 0. Connectivity in large mobile ad-hoc network Edinburgh, January 2016 Seite 7 (10)

9 On the proof: convergence of the expectation (II) Lemma 2: Convergence of the probability of G (i) Under the ame aumption, θ(f (X (1) ) )θ(f (X (2) ) ) lim inf δ 0 lim up δ 0 ( ) lim inf N P1,2 G (1) N,,δ G(2) N,,δ ( lim up P 1,2 G (1) N,,δ G(2) N,,δ N θ(f (X (1) )+)θ(f (X (2) )+). ) Aymptotic independence of G (1) and G (2) in the limit N, followed by δ 0. Sofar, we have proved that the expectation of τ (N) and τ (N,δ, ) = d 1l (1) G 1l (2) 0 N,,δ G 1l{ X (1) X (2) 3δ} 1l{X (1) X (2) } N,,δ are aymptotically equal in the limit N, followed by δ 0 (in term of an upper bound with = and a lower bound for =>). Connectivity in large mobile ad-hoc network Edinburgh, January 2016 Seite 7 (10)

10 On the proof: Vanihing variance We ue the econd-moment method and till have to prove: Propoition: τ (N,δ, ) i aymptotically determinitic Almot urely, under P 1,2 := P( X (1), X (2) ), the variance of τ (N,δ, ) N, followed by δ 0. vanihe in the limit Write out the variance, uing integral 0 d and 0 d. Left to how: independence of G (1) N,,δ G(2) N,,δ and G(1) N,,δ G(2) N,,δ. Connectivity in large mobile ad-hoc network Edinburgh, January 2016 Seite 8 (10)

11 On the proof: Vanihing variance We ue the econd-moment method and till have to prove: Propoition: τ (N,δ, ) i aymptotically determinitic Almot urely, under P 1,2 := P( X (1), X (2) ), the variance of τ (N,δ, ) N, followed by δ 0. vanihe in the limit Write out the variance, uing integral 0 d and 0 d. Left to how: independence of G (1) N,,δ G(2) N,,δ and G(1) N,,δ G(2) N,,δ. Let C (,N) x,δ be the bigget component of the N 1/d -ball around X (1),..., X (N) in X (1) + ( δ, δ). Abbreviate x = X (1), x = X (1), y = X (2) and ỹ = X (2). Idea: the dependence of C (,N) x,δ C (,N) y,δ of the walker (only O(δ 2d N)). and C (,N) x,δ C (,N) ỹ,δ come from only very few Main argument: our aumption that P(X (1) = x S = y) = 0 implie that the ma of thoe walker that are at time cloe to y and at time cloe to x i mall. Connectivity in large mobile ad-hoc network Edinburgh, January 2016 Seite 8 (10)

12 Drawback of the model For mathematical reaon, we made a number of unrealitic aumption: Number of hop for tranmitting a given meage i unbounded. All travel of any uer are global. here are no fixed additional relay intalled. No interference nor capacity problem are conidered, jut the poibility of tranferring a ingle meage. Connectivity in large mobile ad-hoc network Edinburgh, January 2016 Seite 9 (10)

13 Improved model More realitic movement cheme: Contained in our aumption: the random waypoint model with arbitrary dependence on time and arbitrary denity of the tarting ite. Improved verion: Pick i.i.d. home in D and an N -dependent way point meaure centred at the home with action radiu N α and infinite upport after caling. Connectivity in large mobile ad-hoc network Edinburgh, January 2016 Seite 10 (10)

14 Improved model More realitic movement cheme: Contained in our aumption: the random waypoint model with arbitrary dependence on time and arbitrary denity of the tarting ite. Improved verion: Pick i.i.d. home in D and an N -dependent way point meaure centred at the home with action radiu N α and infinite upport after caling. More realitic relay ytem: Bounded-hop percolation: Inert bae tation on a grid D S N Z d with ome S N 0, forbid equence of more than k hop for a meage, and look only at connection with ome of the bae tation. (Some firt reult on ergodicity of the tatic model in [HIRSCH (2015)].) Connectivity in large mobile ad-hoc network Edinburgh, January 2016 Seite 10 (10)

15 Improved model More realitic movement cheme: Contained in our aumption: the random waypoint model with arbitrary dependence on time and arbitrary denity of the tarting ite. Improved verion: Pick i.i.d. home in D and an N -dependent way point meaure centred at the home with action radiu N α and infinite upport after caling. More realitic relay ytem: Bounded-hop percolation: Inert bae tation on a grid D S N Z d with ome S N 0, forbid equence of more than k hop for a meage, and look only at connection with ome of the bae tation. (Some firt reult on ergodicity of the tatic model in [HIRSCH (2015)].) Include interference: Invetigate how many ignal can be uccefully received if many ignal are floating around. Criterion: Signal-to-interference ratio (SIR) and variant. Connectivity in large mobile ad-hoc network Edinburgh, January 2016 Seite 10 (10)

16 Improved model More realitic movement cheme: Contained in our aumption: the random waypoint model with arbitrary dependence on time and arbitrary denity of the tarting ite. Improved verion: Pick i.i.d. home in D and an N -dependent way point meaure centred at the home with action radiu N α and infinite upport after caling. More realitic relay ytem: Bounded-hop percolation: Inert bae tation on a grid D S N Z d with ome S N 0, forbid equence of more than k hop for a meage, and look only at connection with ome of the bae tation. (Some firt reult on ergodicity of the tatic model in [HIRSCH (2015)].) Include interference: Invetigate how many ignal can be uccefully received if many ignal are floating around. Criterion: Signal-to-interference ratio (SIR) and variant. Conider capacity: Handle each trajectory of a meage like a tochatic proce and inert upper bound for the number of meage per relay and time unit. Connectivity in large mobile ad-hoc network Edinburgh, January 2016 Seite 10 (10)

c n b n 0. c k 0 x b n < 1 b k b n = 0. } of integers between 0 and b 1 such that x = b k. b k c k c k

c n b n 0. c k 0 x b n < 1 b k b n = 0. } of integers between 0 and b 1 such that x = b k. b k c k c k 1. Exitence Let x (0, 1). Define c k inductively. Suppoe c 1,..., c k 1 are already defined. We let c k be the leat integer uch that x k An eay proof by induction give that and for all k. Therefore c n