Metropolis Markov chains for wireless random-access networks - PowerPoint PPT Presentation

Stochastic representation for the transition time T s 1 → s 2 N k � � d T ( i ) 0 → ( k , 1) + T ( i ) � � ˆ + ˆ T s 1 → s 2 = T s 1 → 0 + T 0 → s 2 . ( k , 1) → 0 k � =2 i =1 � � L 2 d where N k = Geo L 2 + L k 14

Stochastic representation for the transition time T s 1 → s 2 N k � � d T ( i ) 0 → ( k , 1) + T ( i ) � � ˆ + ˆ T s 1 → s 2 = T s 1 → 0 + T 0 → s 2 . ( k , 1) → 0 k � =2 i =1 � � L 2 d where N k = Geo L 2 + L k 14

Stochastic representation for the transition time T s 1 → s 2 N k � � T ( i ) 0 → ( k , 1) + T ( i ) � � ˆ + ˆ T s 1 → s 2 ≈ T s 1 → 0 + T 0 → s 2 . ( k , 1) → 0 k � =2 i =1 � � L 2 d where N k = Geo L 2 + L k 14

Stochastic representation for the transition time T s 1 → s 2 N k � � T ( i ) 0 → ( k , 1) + T ( i ) � � ˆ + ˆ T s 1 → s 2 ≈ T s 1 → 0 + T 0 → s 2 . ( k , 1) → 0 k � =2 i =1 � � L 2 d where N k = Geo L 2 + L k 14

Stochastic representation for the transition time T s 1 → s 2 N k � � T ( i ) 0 → ( k , 1) + T ( i ) � � ˆ + ˆ T s 1 → s 2 ≈ T s 1 → 0 + T 0 → s 2 . ( k , 1) → 0 k � =2 i =1 � � L 2 d where N k = Geo L 2 + L k Define L ∗ := max k � =2 L k and K ∗ = { k � = 2 : L k = L ∗ } 14

Stochastic representation for the transition time T s 1 → s 2 N k � � T ( i ) 0 → ( k , 1) + T ( i ) � � ˆ + ˆ T s 1 → s 2 ≈ T s 1 → 0 + T 0 → s 2 . ( k , 1) → 0 k ∈ K ∗ i =1 � L 2 � d where N k = Geo L 2 + L k k � =2 L k and K ∗ = { k � = 2 : L k = L ∗ } Define L ∗ := max 14

Stochastic representation for the transition time T s 1 → s 2 M � � � T ( i ) ˆ 0 → ( k , 1) + T ( i ) + ˆ T s 1 → s 2 ≈ T s 1 → 0 + T 0 → s 2 . ( k , 1) → 0 i =1 � � L 2 d where N k = Geo . L 2 + L k Define L ∗ := max k � =2 L k and K ∗ = { k � = 2 : L k = L ∗ } � | K ∗ | L ∗ � M d = Geo . | K ∗ | L ∗ + L 2 14

Asymptotics for the transition time T s 1 → s 2 Theorem � 1 { s 1 ∈ K ∗ } + | K ∗ | � ν L ∗ − 1 , E T s 1 → s 2 ( ν ) ∼ as ν → ∞ , L ∗ L 2 and 15

Asymptotics for the transition time T s 1 → s 2 Theorem � 1 { s 1 ∈ K ∗ } + | K ∗ | � ν L ∗ − 1 , E T s 1 → s 2 ( ν ) ∼ as ν → ∞ , L ∗ L 2 and M T s 1 → s 2 ( ν ) 1 d � − → Y i , as ν → ∞ , E T s 1 → s 2 ( ν ) E M i =1 where M d | K ∗ | L ∗ = Geo ( p ∗ ) + 1 { s 1 ∈ K ∗ } , p ∗ = | K ∗ | L ∗ + L 2 , and Y i are i.i.d. Exp(1). 15

Asymptotics for the transition time T s 1 → s 2 Theorem � 1 { s 1 ∈ K ∗ } + | K ∗ | � ν L ∗ − 1 , E T s 1 → s 2 ( ν ) ∼ as ν → ∞ , L ∗ L 2 and M T s 1 → s 2 ( ν ) 1 d � → − Y i , as ν → ∞ , E T s 1 → s 2 ( ν ) E M i =1 where M d | K ∗ | L ∗ = Geo ( p ∗ ) + 1 { s 1 ∈ K ∗ } , p ∗ = | K ∗ | L ∗ + L 2 , and Y i are i.i.d. Exp(1). In particular, T s 1 → s 2 ( ν ) d s 1 ∈ K ∗ ⇒ − → Exp (1) , as ν → ∞ . = E T s 1 → s 2 ( ν ) 15

Mixing time 16

Mixing time Relabel the components such that L 1 ≥ L 2 ≥ · · · ≥ L K . 16

Mixing time Relabel the components such that L 1 ≥ L 2 ≥ · · · ≥ L K . Theorem t mix ( ε, ν ) = Θ( ν L 2 − 1 ) , ν → ∞ . 16

Mixing time Relabel the components such that L 1 ≥ L 2 ≥ · · · ≥ L K . Theorem t mix ( ε, ν ) = Θ( ν L 2 − 1 ) , ν → ∞ . Idea of the proof: 16

Mixing time Relabel the components such that L 1 ≥ L 2 ≥ · · · ≥ L K . Theorem t mix ( ε, ν ) = Θ( ν L 2 − 1 ) , ν → ∞ . Idea of the proof: • Upper bound by coupling argument 16

Mixing time Relabel the components such that L 1 ≥ L 2 ≥ · · · ≥ L K . Theorem t mix ( ε, ν ) = Θ( ν L 2 − 1 ) , ν → ∞ . Idea of the proof: • Upper bound by coupling argument • Lower bound by giving an upper bound for the conductance (bottleneck ratio) S ⊂ Ω: π ( S ) ≤ 1 / 2 Q ( S , S c ) /π ( S ) Φ = min 16

Heterogeneous complete K -partite graph [ZBvL14] Heterogeneity: users in component k have a back-off rate f k ( ν ). 17

Heterogeneous complete K -partite graph [ZBvL14] Heterogeneity: users in component k have a back-off rate f k ( ν ). Asymptotics for the transition time T s 1 → s 2 (part I) E T s 1 → s 2 ( ν ) ∼ 1 1 f s 1 ( ν ) L s 1 − 1 + � f k ( ν ) L k , as ν → ∞ . L s 2 f s 2 ( ν ) L s 1 k ∈ K ∗ 17

Heterogeneous complete K -partite graph [ZBvL14] Heterogeneity: users in component k have a back-off rate f k ( ν ). Asymptotics for the transition time T s 1 → s 2 (part I) E T s 1 → s 2 ( ν ) ∼ 1 1 f s 1 ( ν ) L s 1 − 1 + � f k ( ν ) L k , as ν → ∞ . L s 2 f s 2 ( ν ) L s 1 k ∈ K ∗ f k ( ν ) Lk j � = s 2 f j ( ν ) Lj and K ∗ := { k � = 2 : γ k > 0 } . where γ k := lim ν →∞ � 17

Heterogeneous complete K -partite graph [ZBvL14] Heterogeneity: users in component k have a back-off rate f k ( ν ). Asymptotics for the transition time T s 1 → s 2 (part II) T s 1 → s 2 ( ν ) d − → Z = α Y + (1 − α ) W , as ν → ∞ , E T s 1 → s 2 ( ν ) � − 1 � γ k s where Y d � � = Exp (1) and L W ( s ) = 1 + + s γ k . 1 + γ k s /β k k ∈A k ∈S 18

Heterogeneous complete K -partite graph [ZBvL14] Heterogeneity: users in component k have a back-off rate f k ( ν ). Asymptotics for the transition time T s 1 → s 2 (part II) T s 1 → s 2 ( ν ) d − → Z = α Y + (1 − α ) W , as ν → ∞ , E T s 1 → s 2 ( ν ) � − 1 � γ k s where Y d � � = Exp (1) and L W ( s ) = 1 + + s γ k . 1 + γ k s /β k k ∈A k ∈S L k f k ( ν ) where β k := lim ν →∞ L s 2 f s 2 ( ν ) and K ∗ is partitioned into three subsets • k ∈ N if β k = 0, • k ∈ A if β k ∈ R + , • k ∈ S if β k = ∞ . 18

Heterogeneous complete K -partite graph [ZBvL14] Possible asymptotic distribution for Z α A S Limiting distribution Z ∅ ∅ δ 0 (trivial r.v. identical to 0) � G i =1 H i ( β 1 β A , . . . , β m β A , β 1 γ 1 , . . . , β m n.e. ∅ γ m ) � G i =1 Exp i ( λ ) if β k /γ k = λ ∀ k ∈ A 0 ∅ n.e. Exp (1 /γ S ) n.e. n.e. W ∅ ∅ Exp (1 /α ) � � Exp (1 /α ) + � G β A , . . . , β m β 1 β 1 β m i =1 H i β A , (1 − α ) γ 1 , . . . , (1 − α ) γ m � � Exp (1 /α ) + � G n.e. ∅ λ/ (1 − α ) if β k /γ k = λ ∀ k ∈ A i =1 Exp i � k =1 β k ) − 1 � α − 1 (1 + � m Exp if β k /γ k = (1 − α ) /α ∀ k ∈ A (0 , 1) if β k /γ k = (1 − α ) /α = � m Exp (1) i =1 β k ∀ k ∈ A ∅ Exp (1 /α ) + Exp (1 / (1 − α ) γ S ) n.e. Erlang (2 , 1 /α ) if α = γ S / (1 + γ S ) n.e. n.e. Exp (1 /α ) + (1 − α ) W 1 - - Exp (1) 19

Grid graphs [ZBvLN13] 20

Grid graphs [ZBvLN13] “even chessboard” E 20

Grid graphs [ZBvLN13] “odd chessboard” O 20

Boundary conditions 2 K × 2 L open grid networks 21

Boundary conditions 2 K × 2 L toric grid networks 21

Boundary conditions 2 K × 2 L cylindrical grid networks 21

Transition and mixing times results Let G be a grid graph 22

Transition and mixing times results Let G be a grid graph Theorem (Mean transition time) E T E→O ( ν ) � ν Γ( G ) − 1 , as ν → ∞ . 22

Transition and mixing times results Let G be a grid graph Theorem (Mean transition time) E T E→O ( ν ) � ν Γ( G ) − 1 , as ν → ∞ . log E T E→O ( ν ) lim inf ≥ Γ( G ) − 1 log ν ν →∞ 22

Transition and mixing times results Let G be a grid graph Theorem (Mean transition time) E T E→O ( ν ) � ν Γ( G ) − 1 , as ν → ∞ . log E T E→O ( ν ) lim inf ≥ Γ( G ) − 1 log ν ν →∞ Theorem (Mixing time) t mix ( ε, ν ) ≥ C ε ν Γ( G ) − 1 , as ν → ∞ . 22

Analysis of the state space Ω( G ) Key ingredient = analysis of the “bottlenecks” of the state space Ω 23

Analysis of the state space Ω( G ) Key ingredient = analysis of the “bottlenecks” of the state space Ω Intuition During the transition from E to O the process has to visit “mixed configurations” with fewer active nodes 23

Analysis of the state space Ω( G ) Key ingredient = analysis of the “bottlenecks” of the state space Ω Intuition During the transition from E to O the process has to visit “mixed configurations” with fewer active nodes 23

Analysis of the state space Ω( G ) Key ingredient = analysis of the “bottlenecks” of the state space Ω Intuition During the transition from E to O the process has to visit “mixed configurations” with fewer active nodes Bottleneck = subset of low-probability configurations in the middle to go from E to O 23

Where is the bottleneck? 24

Where is the bottleneck? State space Ω for the complete bipartite grid 24

Where is the bottleneck? 25

Where is the bottleneck? State space Ω for the 4 × 4 toric grid 25

Bottleneck analysis In general, the bottleneck is a large and intricate subset of Ω( G ), which consists of “mixed configurations” 26

Bottleneck analysis In general, the bottleneck is a large and intricate subset of Ω( G ), which consists of “mixed configurations” N ∆( x ) := N � 2 − x i i =1 26

Bottleneck analysis In general, the bottleneck is a large and intricate subset of Ω( G ), which consists of “mixed configurations” N ∆( x ) := N � 2 − “inefficiency” of x ∈ Ω( G ) x i i =1 26

Bottleneck analysis In general, the bottleneck is a large and intricate subset of Ω( G ), which consists of “mixed configurations” N ∆( x ) := N � 2 − “inefficiency” of x ∈ Ω( G ) x i i =1 To understand how the transition E → O occurs, the key quantity is efficiency gap Γ( G ) := ω : E→O max min x ∈ ω ∆( x ) 26

Metropolis Markov chains Setting ν = e β , the stationary distribution can be rewritten e − β H ( x ) π β ( x ) = y ∈ Ω e − β H ( y ) , x ∈ Ω , � where H : Ω → R is defined as H ( x ) := − � N i =1 x i for every x ∈ Ω. 27

Metropolis Markov chains Setting ν = e β , the stationary distribution can be rewritten e − β H ( x ) π β ( x ) = y ∈ Ω e − β H ( y ) , x ∈ Ω , � where H : Ω → R is defined as H ( x ) := − � N i =1 x i for every x ∈ Ω. After uniformization, the resulting Markov chain has Metropolis transition probabilities � q ( x , y ) e − β [ H ( y ) − H ( x )] + , if x � = y , P β ( x , y ) = 1 − � z � = y P β ( x , z ) , if x = y . 27

Efficiency gap for grid networks We can rewrite ∆( x ) = H ( x ) − min y ∈ Ω H ( y ) ≥ 0 28

Efficiency gap for grid networks We can rewrite ∆( x ) = H ( x ) − min y ∈ Ω H ( y ) ≥ 0 and Γ( G ) = ω : E→O max min x ∈ ω ∆( x ) , is the minimum energy barrier to overcome to go from E to O . 28

Efficiency gap for grid networks We can rewrite ∆( x ) = H ( x ) − min y ∈ Ω H ( y ) ≥ 0 and Γ( G ) = ω : E→O max min x ∈ ω ∆( x ) , is the minimum energy barrier to overcome to go from E to O . Theorem  min { 2 K , 2 L } + 1 if G is a 2 K × 2 L toric grid,   Γ( G ) = min { K , L } + 1 if G is a 2 K × 2 L open grid,  min { 2 K , L } + 1 if G is a 2 K × 2 L cylindrical grid.  28

Proof idea 1 29

Proof idea 1 Lemma (Inefficiency on a band) ⇐ ⇒ x | B = E | B or x | B = O | B ∆ B ( x ) = 0 29