Networks: space-time stochastic models for networks

Transcription

1 Mobile and Delay Tolerant Networks: space-time stochastic models for networks Philippe Jacquet Alcatel-Lucent Bell Labs France

2 Plan of the talk Short history of wireless networking Information in space-time embeding The limit of wireless networking Space scaling capacity paradox Time scaling capacity paradox Delay Tolerant Networks

3 wireless performance from Traffic density 1900: Marconi to Wifi 10 bit/s/ km 2, 1000 watt 2008: 10,000,000 bit/s/ha, 0.01 watt A factor ,000,000, 000,000

4 La progression de Marconi à Wifi

5 Comparison: road traffic Road traffic increase : < 10 5 =100,000?

6 One of the futures of wireless mobile networking Internet of things Galaxies of mobile or static sensors Everywhere with short ranges Increasing number

7 Networks: information in space-time Basically: a network is A set of physical objects: routers Which relay information From arbitrary sources To arbitrary destination. time destination Concept of information path (simple or multiple) source space

8 The limits of mobile Transport capacity networking Point to point Shannon law Bit per second per Hz (no spatial reuse) I p2p = 1 log2 log 1+ S N 0 S z N 0

9 Random spatial reuse Emitters are distributed as Poisson process in the plan density ν z j Signals sum α S = z z j α N 0 = z z i i j Removing spatial reuse: I p2p = α 4π sin 4π α S N 0 2 /α 1 log2 log 1+ S N 0 z - PJ, «Éléments de théorie analytique de l'information, modélisation et évaluation de performances, Baccelli, Blaszczyszyn «Stochastic geometry and wireless networks», 2009

10 Random spatial reuse

11 Optimal spatial reuse Random is not far from optimal - PJ, Malik, «Optimizing Local Capacity of Wireless Ad Hoc Networks», 2011

12 - PJ, «Shannon capacity in poisson wireless network model», 2009 The limits of mobile networking multi-point capacity I m2p = α D log2 z i S = z z i α z i D = 4 3

13 The space capacity paradox Increasing density increases capacity Overall Capacity (including relaying) C b = n I p2p Transport Capacity C n = O C b = O(I n p2p n) n - Gupta, Kumar, «The capacity of wireless networks», 2000 n

14 Introducing time in mobile networking A mobility model: independent random walks Average speed s z j (t) z j (0) L L

15 Time capacity paradox Increasing delay increases capacity Flying postman of Grossglauser-Tse Total transport capacity C n = n 2 I p2p D(t) Average delay: D n = O L s n S(0) Much larger than «hot potatoes» routing - Grossglauser, Tse, «Mobility increases the capacity of ad-hoc wireless networks» 2001

16 Angle routing Capacity C n = O n logn I p2p delay D n = O L s PJ, Malik, Mans, Silva, «On the Throughput-Delay Trade-off in Georouting Networks», Infocom 2012

17 Algorithm Direct routing (hot potatoes) Delayed routing (postman routing) Georouting (angle routing) On urban area Total transport capacity Delivery delay 100 Mbps immediate 50 Gbps One month 10 Gbps One hour I p2p =100 kbps L =1 hour s n =1,000,000

18 Time capacity paradox Mobility can create connectivity in disconnected networks S End-to-end path D S S X X path path disruption! X D D node link Delay Tolerant Networks

19 Time capacity paradox

20 Time capacity paradox Mobility creates capacity capacity capacity Information propagation time T( z ) Permanently disconnected time Permanently connected time

21 Model of network Radio propagation Nodes communicates within a unit disk Transmission/reception are instantaneous c =

22 Information propagation speed Density ν Turn rate τ z speed s z

23 Epidemic broadcast Gossips propagate from node to node Node forwards information to all neighbor Repeats information to new neighbor Broadcast delay T(n) : An arbitrary source Time to reach all mobile nodes

24 Dynamic Properties At any time t Neighbor probability Contact rate P( x i (t) x j (t) 1)= πν n P(x fν i met x j during [ t,t + dt[) = dt n f = 2 (s) Exponential Time to next contact

25 Erdos-Renyi DTN model Conjecture on DTN (Towsley, Chlamtac, Nain, etc) contact times are iid exponential and have frequency f. The contact tree grows exponentially T(n) = O(log n) Pellegrini, Miorandi, Carreras, Chlamtac, «A graph-based model for disconnected ad hoc networks,» 2007 Zhang, Neglia, Kurose, Towsley, «Performance modeling of epidemic routing», 2007

26 Erdos-Renyi epidemic contact theory Time to reach k+1 nodes from k E( k T +1 k (n))= n (n k)kfν Logarithmic estimate of epidemic delay k E( T(n) )= E T +1 k (n) k<n ( ) E( T(n) ) = 2logn fν + O 1 = O logn n s

27 How to kill a conjecture Simulation first Epidemic flooding n ( ) O(logn) Is Ω n not

28 Why it fails? Discrete plane Based on network size N nodes Contact times are I.I.d Neighbor states are I.I.d Asymptotics on generating functions Continuous plane Based on time Speed s and density ν I.I.d Random walk model Exponential time to contact Average case analysis

29 Why ER model fails! A Contact times and neighbor status are not I.I.d Triangular inequality B P( AC <1 AB <1& BC <1)> P(AC <1) Ω(1) O( ν n ) B Devroye 1988

30 What is missing to ER model Space considerations! Nodes are confined to move in a finite dimension space Collocation are not I.I.d Source We define source T(x n ) x 0 We define as time to reach x n

31 Space-time path cumulation Accumulate all possible paths Restricted to 2D embedding Weighted with probabilities z j (t) S(0)

32 Path density We define q(z,t) = E(# path ending on (z,t)) q(z,t) Warning: is not a probability distribution p(z,t) = P( path ending on (z,t)) q(z,t) time space

33 Information propagation speed p n (z,t) = P T(x n ) < t x n (0) x 0 (0) = z ( ) n n n 0 Quantity c is a propagation speed upper bound if lim z,n p n z, z = 0 c c x 0 z x n The information propagates like a wave. n It takes Ω to cover n nodes (for the c same density)

34 Information propagation speed Upper bound of information propagation speed via Bessel functions Any quantity c such that lim z P(T( z ) < z z ) = 0 c θ Is the smallest ratio in the kernel of ρ D(ρ,θ) = (τ + θ) 2 ρ 2 v 2 τ 2πνsI 0(ρ) I k () are modified Bessel functions 1 πν 2 ρ I 1 (ρ)

35 Information propagation speed time theory speed s =1 turn rate τ = 0.1 node density ν = 0.25 space

36 Wave propagation PJ, Mans, Rodolakis, «Information Propagation Speed in Mobile and Delay Tolerant Networks», 2010