✬ ✩ WIRELESS QUEUING NETW ORKS UNDER CHURN AND MOTION F. Ba elli UT Austin & Inria P a ris Simons Center for Communication, Information & Network Mathematics UT Austin Wiopt 2017, Paris ✫ ✪
✬ ✩ 1 Stru ture of the Le ture Background and Motivation Methodology Part 1 Birth and Death of Wireless Queues Joint work with A. Sankararaman and S. Foss Part 2 Motion of Wireless Queues under Multihop Routing Joint work with S. Rybko, S. Shlosman and S. Vladimirov ✫ ✪ Wireless Queueing Networks under Churn and Motion
✬ ✩ 2 Motivations in Wireless Net w o rks Lack of understanding and analysis of Space-time interactions – Static spatial setting well understood: Stochastic Geometry [FB, Blaszczyszyn 01] – Churn partly taken into account in flow-based queuing [Bonald, Proutiere 06], [Shakkottai, De Veciana 07] Contents of this lecture: First models with such dynamics in stochastic geometry ✫ ✪ Wireless Queueing Networks under Churn and Motion
✬ ✩ 3 Metho dology Everything Should Be Made as Simple as Possible, But Not Simpler A. E. ✫ ✪ Wireless Queueing Networks under Churn and Motion
✬ ✩ 4 Setting Infrastructureless Wireless Network: Ad-hoc Networks, D2D Networks, IoT Markov Models: Poisson, Exponential Mathematical tools: Stochastic Geometry, Fluid, Mean-Field ✫ ✪ Wireless Queueing Networks under Churn and Motion
✬ ✩ 5 I: Churn: Birth and Death Problem Statement Summary of Results Proof Overview ✫ ✪ Wireless Queueing Networks under Churn and Motion
✬ ✩ 6 ✫ ✪ Wireless Queueing Networks under Churn and Motion
✬ ✩ 7 Sto hasti Net w o rk Mo del S = [ − Q, Q ] × [ − Q, Q ] : torus where the wireless links live Links: (Tx-Rx pairs) Links: arrive as a PPP on I R × S with intensity λ : Prob. of a point arriving in space dx and time dt: λ dxdt Each Tx has an i.i.d. exponential file size of mean L bits to transmit to its Rx A point exits after the Tx finishes transmitting its file Φ t : set of locations of links present at time t: Φ t = { x 1 , . . . , x N t } , x i ∈ S ✫ ✪ Wireless Queueing Networks under Churn and Motion
✬ ✩ 8 Interferen e and Servi e Rate Interference seen at point x due to configuration Φ � I ( x , Φ ) = l ( || x − x i || ) x i ∈ Φ � = x – Distance on the torus R + → I R + : path loss function – l ( · ) : I The speed of file transfer by link at x in configuration Φ is � 1 � R ( x , Φ ) = B log 2 1 + N + I ( x , Φ ) B , N Positive constants ✫ ✪ Wireless Queueing Networks under Churn and Motion
✬ ✩ 9 B& D Master Equation A point born at x p and time b p with file-size L p dies at time t � d p = inf t > b p : R ( x p , Φ u ) du ≥ L p u = b p Spatial Birth-Death Process – Arrivals from the Poisson Rain – Departures happen at file transfer completion ✫ ✪ Wireless Queueing Networks under Churn and Motion
✬ ✩ 10 Prop erties of the Dynami s The statistical assumptions imply that Φ t is a Markov Pro- cess on the set of simple counting measures on S Euclidean extension of the flow-level models of [Bonald, Proutiere 06], [Shakkottai, De Veciana 07] ✫ ✪ Wireless Queueing Networks under Churn and Motion
✬ ✩ 11 Questions Existence and uniqueness of the stationary regimes of Φ t Characterization of the stationary regime(s) if existence ✫ ✪ Wireless Queueing Networks under Churn and Motion
✬ ✩ 12 Main Stabilit y Results � a := l ( || x || ) dx x ∈ S Theorem B – If λ > ln( 2 ) La , then Φ t admits no stationary regime. B – If λ < ln( 2 ) La , and r → l ( r ) bounded and monotone, then Φ t admits a unique stationary regime Sufficient condition by fluid limit Corollary For the path-loss model l ( r ) = r − α , α ≥ 2, for all λ > 0, and all mean file sizes, the process Φ t admits no stationary-regime ✫ ✪ Wireless Queueing Networks under Churn and Motion
✬ ✩ 13 Main Qualitative Result Φ stationary point-process on S with Palm distribution P 0 Clustering Φ is clustered if for all bounded, positive, non-increasing functions f ( · ) : R + → R + , the shot noise; � F ( x , Φ ) := f ( || y − x || ) y ∈ Φ \{ x } satisfies E 0 [ F ( 0 , Φ )] ≥ E [ F ( 0 , Φ )] ✫ ✪ Wireless Queueing Networks under Churn and Motion
✬ ✩ 14 Main Qualitative Result ( continued ) Theorem The steady-state point process, when it exists, is clustered Follows from Palm calculus + the FKG inequality Interpretation of the result The steady-state interference measured at a uniformly ran- domly chosen point of is larger in mean than that at an uniformly random location of space. Key Observation – Dynamics Shapes Geometry – Geometry Shapes Dynamics ✫ ✪ Wireless Queueing Networks under Churn and Motion
✬ ✩ 15 15 10 5 0 0 5 10 15 A sample of Φ when λ = 0 . 99 and l ( r ) = ( r + 1 ) − 4 . ✫ ✪ Wireless Queueing Networks under Churn and Motion
✬ ✩ 16 First Quantitative Results Mean-field approximations for the intensity of the steady- state process 1. Poisson heuristic β f - derived by neglecting clustering and assuming Poisson 2. Second-order heuristic β s based on a second-order cavity approximation of the dynamics ✫ ✪ Wireless Queueing Networks under Churn and Motion
✬ ✩ 17 P oisson Heuristi Exact Rate Conservation Law: � � �� 1 λ L = β E 0 log 2 1 + . Φ N + I ( 0 ) Poisson Heur.: Largest solution to the fixed point equation: ∞ e − Nz ( 1 − e − z ) β f � x ∈ S ( 1 − e − zl ( || x || ) ) dx dz � e − β f λ L = ln( 2 ) z z = 0 Ignores the Palm effect and uses that if X , Y are non-negative and independent, ∞ e − az � � X �� � z ( 1 − E [ e − zX ]) E [ e − zY ] dz . ln 1 + = E Y + a z = 0 ✫ ✪ Wireless Queueing Networks under Churn and Motion
✬ ✩ 18 Se ond Order Heuristi The intensity β s is given by λ L β s = � � 1 B log 2 1 + N + I s where I s is the smallest solution of the fixed-point equation l ( || x || ) � I s = λ L � dx � 1 B log 2 1 + N + I s + l ( || x || ) x ∈ S ✫ ✪ Wireless Queueing Networks under Churn and Motion
✬ ✩ 19 Second Order Heuristic ( continued ) Rationale based on ρ 2 ( x , y ) : second moment measure of Φ Rate Conservation for ρ 2 : when considering I s as a constant � � ρ 2 ( x , y ) 1 1 LB log 2 1 + = λβ s N + I s + l ( || x − y || ) From the definition of second moment measure, � l ( || x || ) ρ 2 ( 0 , x ) I s = dx β s x ∈ S which gives the fixed point equation for I s The formula for β s follows from Rate Conservation for ρ 1 = β s ✫ ✪ Wireless Queueing Networks under Churn and Motion
✬ ✩ 20 30 Simulations Second−Order Heuristic 25 Poisson Heuristic 20 β 15 10 5 0 0.4 0.5 0.6 0.7 0.8 0.9 1 λ / λ c 95% confidence interval when l ( r ) = ( r + 1 ) − 4 ✫ ✪ Wireless Queueing Networks under Churn and Motion
✬ ✩ 21 Tightness Results & Extensions The Poisson heuristic is tight in heavy and light traffic Recent Extensions obtained with S. Foss: – Exact expression for the intensity β of Φ in the Low SINR regime when replacing the death rate by B S ln( 2 ) L N + I ( x , Φ ) – Scalability result: extension to dynamics on R 2 using Coupling from the Past techniques. Future: introduction of scheduling or multi-user IT ✫ ✪ Wireless Queueing Networks under Churn and Motion
✬ ✩ 22 Summa ry A new basic representation of space time interactions in wireless networks A generative model for clustering as assumed in 3GPP sim- ulation standards A new dynamic notion of capacity involving both queuing and IT First analytical results in the Low SINR case and good heuristics in general ✫ ✪ Wireless Queueing Networks under Churn and Motion
✬ ✩ 23 Stru ture of the Le ture Background and Motivation Methodology Part 1 Birth and Death of Wireless Queues Joint work with A. Sankararaman and S. Foss Part 2 Motion of Wireless Queues under Multihop Routing Joint work with S. Rybko, S. Shlosman and S. Vladimirov ✫ ✪ Wireless Queueing Networks under Churn and Motion
✬ ✩ 24 I I: Motion and Multi-Hop Routing Problem Statement Summary of Results ✫ ✪ Wireless Queueing Networks under Churn and Motion
✬ ✩ 25 Problem Statement Setting: Grossglauser & Tse 02 scaling law problem – Multihop relaying – Opportunistic geographic routing – Motion of nodes New SG+QT view of the problem ✫ ✪ Wireless Queueing Networks under Churn and Motion
✬ ✩ 26 Example of Geographi Routing Nearest Neighbor Geographic Routing The next hop on the route from S to D is X S the nearest among the nodes which are closer from D than X . D On a Poisson P.P., a.s. – No ties – Converges in finite number of steps ✫ ✪ Wireless Queueing Networks under Churn and Motion
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