Spatial Birth and Death Peer-to-Peer Point Processes F. Baccelli - PowerPoint PPT Presentation

✬ ✩ Spatial Birth and Death Peer-to-Peer Point Processes F. Baccelli INRIA–ENS Joint work with F. Mathieu and I. Norros Stochastic Network Conference 2012, MIT ✫ ✪

✬ ✩ 1 STRUCTURE OF THE TALK 1. P2P Motivations 5. Simulation 2. Stochastic Model 6. Design & Scaling 3. Dimensional Analysis 7. Limitations 4. Stochastic Analysis 8. Extensions Spatial Birth and Death Peer-to-Peer Point Processes ✫ ✪ F. B.

✬ ✩ 2 PEER-TO-PEER CONTENT DISTRIBUTION Content Distribution Common Features – Filesharing – Lot of stress on the network – Streaming – P2P solutions: ∗ OnDemand large family of algorithms and implementations to cope with ∗ Live churn, load, latency... Spatial Birth and Death Peer-to-Peer Point Processes ✫ ✪ F. B.

✬ ✩ 3 P2P STOCHASTIC MODELING State of the Art: Queuing Theory [Yang and De Veciana 04], [Qiu and Srikant 04] Assumptions Three main types of nodes – Access-limited – Servers: provide, don’t scale up (physical/software) – Leechers: need, provide – No network limitation – Seeders: provide, scale – Poisson arrivals This presentation: New models with network rate limitations Spatial Birth and Death Peer-to-Peer Point Processes ✫ ✪ F. B.

✬ ✩ 4 SPATIAL BIRTH AND DEATH STOCHASTIC MODEL R 2 Peers live in a finite or infinite subset D of the Euclidean plane I Natural extensions to R d ; – General metric spaces (semantic spaces) eg I – Torus (approximation of the whole plane) Dynamics: arrivals – Poisson rain: new peers arrive according to a Poisson process with time space intensity λdxdt on D × I R Spatial Birth and Death Peer-to-Peer Point Processes ✫ ✪ F. B.

✬ ✩ 5 SPATIAL BIRTH AND DEATH STOCHASTIC MODEL ( continued ) Dynamics: service – Service requirement: each peer p is born with an individual service requirement F p > 0 i.i.d. exponential with mean F . – Bit rate function: two peers at locations x and y serve each other at rate f ( || x − y || ) , where f is the bit rate function (BRF) – Service rate: the service rate of a peer at x in configuration φ is � f ( || x − y || ) . µ ( x, φ ) = y ∈ φ \{ x } – Service completion: for a system with state history { φ t } t , a peer p born at point x p at time t p leaves at time t � τ p = inf { t > t p : µ ( x p , φ s ) ds ≥ F p } . t p Spatial Birth and Death Peer-to-Peer Point Processes ✫ ✪ F. B.

✬ ✩ 6 SPATIAL BIRTH AND DEATH PROCESS N ( D ) : the space of counting measures in ( D, D ) The state φ t at time t is a Markov process living in the space N ( D ) : – a peer has birth intensity λ at x – a peer located at x has death intensity µ ( x, φ t ) /F Class of spatial birth-and-death process with a death rate defined as a shot-noise of the configuration. Spatial Birth and Death Peer-to-Peer Point Processes ✫ ✪ F. B.

✬ ✩ 7 SPATIAL BIRTH AND DEATH PROCESS ( continued ) Lemma If D is compact and f is bounded from below by a positive constant on some non-degenerate interval, then the Markov process { φ t } t is ergodic for any birth rate λ > 0 . Proof by stochastic domination: M/M/ ∞ queue that is modified so that a lone customer cannot leave. Existence/uniqueness of stationary regimes in the infinite Euclidean plane currently under investigation using – Extensions of the Garcia & Kurtz martingale method; – Coupling methods. Non reversible Markov process non Gibbsian point process Spatial Birth and Death Peer-to-Peer Point Processes ✫ ✪ F. B.

✬ ✩ 8 EXAMPLES OF BRF: TCP R 2 and TCP model: D is the Euclidean plane I f ( r ) = C r 1 r ≤ R . Justification: – peers use TCP Reno – on the path between two peers, if the packet loss probability is p and the round trip time is RTT , then the rate obtained on this path is η RTT √ p with η = ∼ 1 . 309 square root formula – the RTT is proportional to distance r – only peers at distance less than R are retained. Spatial Birth and Death Peer-to-Peer Point Processes ✫ ✪ F. B.

✬ ✩ 9 EXAMPLES OF BRF: TCP ( continued ) Variants – Affine RTT model: RTT = ar + b , where a accounts for propagation delays in the Internet path and b for the mean access latency: C f ( r ) = r + q 1 r ≤ R – Additional overhead cost: c bits per second: � C � + f ( r ) = r + q − c 1 r ≤ R – Upload (or Download) rate limitations: � C � � + � f ( r ) = min U, r + q − c 1 r ≤ R with U the individual rate limitation Spatial Birth and Death Peer-to-Peer Point Processes ✫ ✪ F. B.

✬ ✩ 10 EXAMPLES OF BRF: UDP UDP assumptions: R 2 – D is the Euclidean plane I – only peers within distance R are retained – peers use UDP with prescribed rate C regardless of distance f ( r ) = C 1 r ≤ R . Spatial Birth and Death Peer-to-Peer Point Processes ✫ ✪ F. B.

✬ ✩ 11 EXAMPLES OF BRF: WIRELESS SNR SNR model: the rate between a transmitter and its receiver at dis- tance r is � � f ( r ) = 1 1 + C 2 log 1 r<R r α with – α > 2 the path loss exponent – C the signal to noise power ratio at distance 1 – R the transmission range Requirement: all point-to-point channels are mutually orthogonal Spatial Birth and Death Peer-to-Peer Point Processes ✫ ✪ F. B.

✬ ✩ 12 DEFAULT MODEL Default option model throughout the talk: – D is the Euclidean plane or a large torus – TCP Bit Rate Function: f ( r ) = C r 1 r<R + comments on the other Bit Rate Functions Spatial Birth and Death Peer-to-Peer Point Processes ✫ ✪ F. B.

✬ ✩ 13 DIMENSIONAL ANALYSIS 4 basic parameters: – R in meters (m), – F in bits, – λ in m − 2 per second (s) – C in bit · m · s − 1 . π -Theorem In the TCP case, all system properties only depend on the parameter ρ = λFR 3 . C Extension for more general f Spatial Birth and Death Peer-to-Peer Point Processes ✫ ✪ F. B.

✬ ✩ 14 DIMENSIONAL ANALYSIS ( continued ) Sketch of proof – choose R as a new distance unit, then ∗ the arrival intensity becomes l = λR 2 ∗ the download constant becomes c = C/R – now define F as an information unit, then ∗ the download speed constant becomes c = C/ ( RF ) – take a time unit such that the download speed constant is 1 , then ∗ all parameters are equal to 1 ∗ the arrival rate becomes l = λFR 3 C Spatial Birth and Death Peer-to-Peer Point Processes ✫ ✪ F. B.

✬ ✩ 15 DIMENSIONAL ANALYSIS ( continued ) Terminology: Three cases – ρ ≫ 1 is called fluid – ρ ≪ 1 is called hard core – ρ inbetween is called intermediate Spatial Birth and Death Peer-to-Peer Point Processes ✫ ✪ F. B.

✬ ✩ 16 NOTATION In the steady state regime of the P2P dynamics: – β o the density of the peer point process – µ o the mean rate of a typical peer – W o the mean latency of a typical peer – N o the mean number of peers in a ball of radius R around a typical peer Spatial Birth and Death Peer-to-Peer Point Processes ✫ ✪ F. B.

✬ ✩ 17 f -REPULSION Theorem 1 For all BRF f , in the stationary regime, � � E [ f ( || x i || )] ≥ E 0 [ f ( || x i || )] , x i ∈ φ x i ∈ φ \ 0 where I P 0 is the Palm probability w.r.t. Φ . Spatial Birth and Death Peer-to-Peer Point Processes ✫ ✪ F. B.

✬ ✩ 18 SKETCH OF PROOF - TORUS Φ t : state of the SBD at time t . A t : total rate � A t = A t ( X ) , X ∈ Φ t with, for all X ∈ Φ t : � f ( || X − Y || )) A t ( X ) = Y ∈ Φ t ,Y � = X Spatial Birth and Death Peer-to-Peer Point Processes ✫ ✪ F. B.

✬ ✩ 19 SKETCH OF PROOF - TORUS ( continued ) Miyazawa rate conservation principle applied to A t : – E ↑ : (time) Palm probability of the SBD at birth epochs – E ↓ at death epochs. r ↑ E + ( I ) = r ↓ E ↓ ( |D| ) with – I = A 0+ − A 0 the total rate increase, r ↑ the inc. intensity – D = A 0+ − A 0 the total rate decrease, r ↓ the dec. intensity Spatial Birth and Death Peer-to-Peer Point Processes ✫ ✪ F. B.

✬ ✩ 20 SKETCH OF PROOF - TORUS ( continued ) Since r ↑ = r ↓ , E ↑ ( I ) = E ↓ ( D ) . From PASTA E ↑ ( I ) = 2 E ( n 0 ) a | D | . with n 0 the total population and � f ( || x || ) m ( d x ) . a = T with T the torus of area | D | . Spatial Birth and Death Peer-to-Peer Point Processes ✫ ✪ F. B.

✬ ✩ 21 SKETCH OF PROOF - TORUS ( continued ) The (total) death point process admits a stochastic intensity w.r.t. the filtration F t = σ (Φ s , s ≤ t ) equal to A t . From Papangelou’s theorem d P ↓ A 0 d P | F 0 − = E ( A 0 ) . Since the decrease (in state Φ 0 − ) is of magnitude A 0 ( X ) (w.r.t. Φ 0 − ) with probability A 0 ( X ) (w.r.t. Φ 0 − ), A 0 � � ( A 0 ( X )) 2 � E   A 0 ( X )  A 0 X ∈ Φ 0 � E ↓ ( D ) = 2 E  = 2 A 0 ( X ) � � E ( A 0 ) A 0 X ∈ Φ 0 � A 0 ( X ) E X ∈ Φ 0 � ( A 0 (0)) 2 � = 2 E 0 E 0 ( A 0 (0)) Spatial Birth and Death Peer-to-Peer Point Processes ✫ ✪ F. B.

✬ ✩ 22 SKETCH OF PROOF - TORUS ( continued ) Miyazawa rate conservation principle for total rate: � ( A 0 (0)) 2 � | D | = E 0 E ( n 0 ) a E 0 ( A 0 (0)) . Using the fact that ≥ E 0 ( A 0 (0)) 2 , ( A 0 (0)) 2 � � E 0 we get E ( n 0 ) a | D | ≥ E 0 ( A 0 (0)) . Spatial Birth and Death Peer-to-Peer Point Processes ✫ ✪ F. B.