Upper bound 1 [Gupta, Kumar]: tighter upper bound of λ ( n ) = O ( √ n log n ) (discussed in Part II) Theorem (Yi et al., 2003) Expected per-connection throughput is O ( 1 √ n ) . Proof sketch Let L denote the average distance between the source and destination of a connection Each connection has rate of λ ⇒ transport capacity of n λ L per second. Consider the b th bit, where 1 ≤ b ≤ λ nT . Suppose it moves from its source to its destination in a sequence of h ( b ) hops, where the h th hop covers a distance of r h b units. We have: h ( b ) λ nT X X r h b = λ nTL b =1 h =1 Anil Vullikanti (Virginia Tech) Capacity of Wireless Networks 19 / 100
Proof of upper bound (continued) Let indicator Γ( h , b , s ) be 1 if the h th h ( b ) λ nT Γ( h , b , s ) ≤ Wn hop of bit b occurs during slot s . We X X 2 have b =1 h =1 Summing over all slots over the T - λ nT h ( b ) ≤ WTn H . X = second period: 2 b =1 Because of Tx-model of interference, disks of radius (1 + ∆) times the lengths of hops centered at the trans- mitters are disjoint. λ nT h ( b ) b ) 2 ≤ W X X Γ( h , b , s ) π (1+∆) 2 ( r h b =1 h =1 Anil Vullikanti (Virginia Tech) Capacity of Wireless Networks 20 / 100
Proof of upper bound (continued) λ nT h ( b ) X X π (1 + ∆) 2 ( r h b ) 2 ≤ WT b =1 h =1 λ nT h ( b ) 1 WT X X H ( r h b ) 2 ⇒ ≤ π (1 + ∆) 2 H b =1 h =1 2 0 1 h ( b ) h ( b ) λ nT λ nT 1 1 X X X X H ( r h H ( r h b ) 2 ( convexity) b ) ≤ @ A b =1 h =1 b =1 h =1 h ( b ) s λ nT 1 WT X X H ( r h ⇒ b ) ≤ π (1 + ∆) 2 · H b =1 h =1 Anil Vullikanti (Virginia Tech) Capacity of Wireless Networks 21 / 100
Proof of upper bound (continued) s WTH λ nTL ≤ π (1 + ∆) 2 (1 + ∆) W √ n bit-meters / second 1 1 ⇒ λ nL ≤ √ 2 π O ( 1 ⇒ λ = √ n ) Tighter upper bound using cuts and flows (discussed later) Anil Vullikanti (Virginia Tech) Capacity of Wireless Networks 22 / 100
Lower bound Theorem (Kulkarni et al., 2004) 1 Expected per-connection throughput is Ω( √ n log n ) . Proof strategy: reduction to permutation routing. Anil Vullikanti (Virginia Tech) Capacity of Wireless Networks 23 / 100
Step 1: partition into grid 1 Grid formed by horizontal and vertical lines uniformly spaced s n apart: 1 n squarelets s 2 of area s 2 n . 2 Crowding factor : maximum number of nodes in any squarelet Anil Vullikanti (Virginia Tech) Capacity of Wireless Networks 24 / 100
Reduction to permutation routing 1 ℓ × ℓ lattice of processors 2 Each processor can communicate with its adjacent vertical and horizontal neighbors in a single slot simultaneously (with one packet being a unit of communication with any neighbor during a slot). 3 Each processor is the source and destination of exactly k packets. 4 The k × k permutation routing problem: routing all the k ℓ 2 packets to their destinations. Lemma (Kauffman et al., 1994, Kunde, 1993) k × k permutation routing in a ℓ × ℓ mesh can be performed deterministically in k ℓ 2 + o ( k ℓ ) steps with maximum queue size at each processor equal to k. Anil Vullikanti (Virginia Tech) Capacity of Wireless Networks 25 / 100
Step II: Reduction to permutation routing 1 Map nodes in each specific squarelet onto a particular 1 processor ( ℓ = s n ). 2 Each node has m packets and set k = mc n . Map to permutation routing on lattice. 3 Equivalence class for each squarelet s : squarelets whose vertical and horizontal separation from s is an integral multiple of K squarelets: K depends on ∆. 1 Transmissions only within squarelet, or to neighboring 2 squarelets ⇒ for any transmission on e = ( u , v ), √ d ( u , v ) ≤ 5 s n . Minimum distance between two transmitters in the 3 same equivalence class is ( K − 2) s n . By interference condition: 4 √ √ ( K − 2) s n > 2(1 + ∆) 5 s n , or K > 4 + 2 5∆. √ Thus, we could set K = 5 + ⌈ 2 5∆ ⌉ . Number of equivalence classes = K 2 (a fixed 5 constant dependent only on ∆). Anil Vullikanti (Virginia Tech) Capacity of Wireless Networks 26 / 100
Step II: Reduction to permutation routing (contd.) 1 Construct schedule for packets on mesh. Each processor in the mesh can transmit and receive up to four packets in the same slot. 2 Serialize transmissions of the processors not in the same equivalence class: Expands the total number of steps in the mesh routing algorithm by a factor of K 2 (# 1 of equivalence classes). Serialize the transmissions of a single processor: increases the total number of steps in 2 the mesh routing by a further factor of 4. 2 = Θ( K 2 mc n 3 m packets from all nodes reach in time 4 K 2 k ℓ ) s n Lemma Assuming each squarelet has at least one node, the per-connection throughput for a network with squarelet size s n and crowding factor c n is Ω( s n c n ) . Anil Vullikanti (Virginia Tech) Capacity of Wireless Networks 27 / 100
Step II: Reduction to permutation routing (contd.) q 3 log n 1 Set s n = n 2 With high probability, no squarelet is empty (union bound) 3 c n ≤ 3 e log n (Chernoff bound). Anil Vullikanti (Virginia Tech) Capacity of Wireless Networks 28 / 100
Extensions: directional antennas β α Transmission beamwidth: α Reception beamwidth: β Lemma (Yi et al., 2007) The expected per-connection throughput in random networks with directed antennas with transmission and reception beamwidth α and β , respectively is: cW 8 (1+∆) 2 √ n log n , Omni Tx, Omni Rv > > 2 π cW > (1+∆) 2 √ n log n , Dir Tx, Omni Rv > < α λ ( n ) = 2 π cW (1+∆) 2 √ n log n , Omni Tx, Dir Rv β > > > 4 π 2 cW > (1+∆) 2 √ n log n , Dir Tx, Dir Rv : αβ Anil Vullikanti (Virginia Tech) Capacity of Wireless Networks 29 / 100
Extensions: delays and mobility End-to-end delay D ( n ): average delay between packet arrival at source and delivery at destination v ( n ): speed of a node T ( n ): expected per-node throughput Delay-throughput tradeoffs How does T ( n ) vary with D ( n ) and v ( n )? Anil Vullikanti (Virginia Tech) Capacity of Wireless Networks 30 / 100
Delay-throughput tradeoffs in mobile networks Theorem (El Gamal et al., 2004) In a mobile network with average delay D ( n ) and per-connection throughput T ( n ) , we have D ( n ) = Θ( nT ( n )) for T ( n ) = O (1 / √ n log n ) D ( n ) = O ( √ n / v ( n )) when T ( n ) = Θ(1) Several unrealistic assumptions, e.g., arbitrarily large packets and buffering Anil Vullikanti (Virginia Tech) Capacity of Wireless Networks 31 / 100
Extensions: hybrid networks n nodes distributed randomly, each choosing a random destination m hybrid base stations distributed randomly hybrid nodes are all connected by high capacity wired links Theorem (Liu et al., 2003) In a hybrid network with n nodes and m base stations, the per-connection throughput λ ( m , n ) satisfies: 8 q q 1 n Θ( n log n W ) if m = O ( log n ) < λ ( m , n ) = q Θ( mW n n ) if m = ω ( log n ) : Anil Vullikanti (Virginia Tech) Capacity of Wireless Networks 32 / 100
Part II: approximating the capacity of arbitrary networks Anil Vullikanti (Virginia Tech) Capacity of Wireless Networks 33 / 100
Related work: algorithms for computing capacity Small sample of results... Formulation of rate region using LPs and conflict graphs: [Hajek, Sasaki, 1988], [Jain et al., 2003], [Kodialam and Nandagopal, 2003],... Constant factor approximation of the capacity under primary interference [Kodialam and Nandagopal, 2003] Constant factor approximation of the capacity for uniform power levels in disk graph models: [Lin, Schroff, 2005], [Kumar et al, 2005], [Kar, Sarkar, Chaporkar, 2005] Local multi-commodity flow algorithms [Awerbuch-Leighton, 1993] Stability based on Max-weight matching policy [Tassiulas-Ephrimedes, 1993] Convex programming methods for capacity [Low et al.] Anil Vullikanti (Virginia Tech) Capacity of Wireless Networks 34 / 100
Feasible Schedules and link rates (recap) Assumption: synchronous time slots of uniform length τ Schedule S specifies the time slots when packets move on links: X ( e , t ) = 1 if packet moves on edge e in time slot t S is feasible if: ∀ t , X ( e , t ) = X ( e ′ , t ) = 1 ⇒ e , e ′ do not interfere Link utilization vector, ¯ x , corresponding to S is defined as P t ≤ T X ( e , t ) ∀ e : x ( e ) = lim T T →∞ Flow rate vector, ¯ f , corresponding to S is defined as ∀ e : f ( e ) = x ( e ) · cap ( e ) , where cap ( e ) is the capacity of edge e . Definition A rate vector ¯ f is feasible if it has a corresponding feasible/stable schedule S that achieves rate ¯ f and is able to schedule all the packets in bounded time. Anil Vullikanti (Virginia Tech) Capacity of Wireless Networks 35 / 100
Example The flow vector � f with f 1 = 2 / 8, f 2 = 1 / 8 corresponds to periodic schedule S , and is feasible Anil Vullikanti (Virginia Tech) Capacity of Wireless Networks 36 / 100
Example f 1 = f 2 = 1 / 5 for this schedule Goal : Given a network, and source-destination pairs, find a feasible flow vector � f with high total throughput Anil Vullikanti (Virginia Tech) Capacity of Wireless Networks 37 / 100
General strategy Define suitable interference set ˆ N ( e ) for each link e Construct LP P ( λ ) with flow constraints, and congestion constraints of the form X x ( e ′ ) ≤ λ, x ( e ) + e ′ ∈ ˆ N ( e ) for each e Prove that P ( c 1 ) gives necessary conditions – any feasible solution � f ,� x satisfies the constraints of P ( c 1 ) Prove that P ( c 2 ) gives sufficient conditions – corresponding to any feasible solution � x of P ( c 2 ), we can construct a schedule S that corresponds to � f ,� f ,� x Anil Vullikanti (Virginia Tech) Capacity of Wireless Networks 38 / 100
Summary: techniques used 1 Linearization of joint physical and MAC constraints: upper bounds on the rate region expressed by weaker linear constraints 2 Scheduling based on inductive ordering: packets on edge e scheduled after those on edges in N ≥ ( e ) - lower bounds on the optimum Anil Vullikanti (Virginia Tech) Capacity of Wireless Networks 39 / 100
Characterizing the Rate Region e2 For any edge e : e1 P x ( e ) = lim T →∞ t ≤ T X ( e , t ) / T e3 Capacity Constraint: One Primary Interference: For any node, at most packet per edge one incident edge is used at a time ⇒ X ( e i , t ) ≤ 1 ⇒ ∀ t : X ( e 1 , t ) + X ( e 2 , t ) + X ( e 3 , t ) ≤ 1 ⇒ x ( e i ) ≤ 1 ⇒ x ( e 1 ) + x ( e 2 ) + x ( e 3 ) ≤ 1 Anil Vullikanti (Virginia Tech) Capacity of Wireless Networks 40 / 100
Throughput capacity under Primary Interference Objective: max P i f i Subject to: Lemma (Kodialam and X Nandagopal, 2003) ∀ i , f i = f ( e ) e =( s i , v ) Any solution to the program P (2 / 3) can X ( P ( λ )) − f ( e ) be scheduled feasibly. e =( v , s i ) ∀ e , x ( e ) = f ( e ) / cap ( e ) Theorem (Kodialam and X ∀ v , x ( e ) ≤ λ (C) Nandagopal, 2003) e ∈ N ( v ) The optimum solution to the program ∀ e , f ( e ) ≥ 0 P (2 / 3) gives a 2 / 3 -approximation to the total throughput capacity, under Observation Any feasible link utilization primary interference constraints. vector ¯ x is a feasible solution to P (1). Anil Vullikanti (Virginia Tech) Capacity of Wireless Networks 41 / 100
Effects of Secondary Interference Linearization e6 e3 X ( e , t ) + P 6 i =1 X ( e i , t ) ≤ 6 ⇒ x ( e ) + P 6 i =1 x ( e i ) ≤ 6 e1 e e5 Lemma Any feasible utilization vector ¯ x satisfies the e2 e4 congestion constraints: e ′ ∈ N ( e ) x ( e ′ ) ≤ λ . ∀ e = ( u , v ) , x ( e ) + P X ( e , t ) = 1 ⇒ X ( e i , t ) = 0 , ∀ e i N ( e ) = { e ′ = ( u ′ , v ′ ) : u ′ ∈ N ( u ) ∪ N ( v ) } . X ( e , t ) = 0 ⇒ all edges e i can simultaneously transmit ⇒ non-linear constraints Anil Vullikanti (Virginia Tech) Capacity of Wireless Networks 42 / 100
Formulation P uniform ( λ ) : uniform disks Objective: max P i f i Subject to: X X ∀ i , f i = f ( e ) − f ( e ) e =( s i , v ) e =( v , s i ) ∀ e , x ( e ) = f ( e ) / cap ( e ) X x ( e ′ ) ∀ e , x ( e ) + ≤ λ (Congestion Constraints) e ′ ∈ N ( e ) ∀ e , f ( e ) ≥ 0 Lemma The constraints of program P uniform ( λ ) are necessary for some constant λ : every feasible utilization vector ¯ x is a feasible solution to the program P uniform ( λ ) . Anil Vullikanti (Virginia Tech) Capacity of Wireless Networks 43 / 100
Capacity under Uniform Power Levels Lemma The optimum solution to the program P uniform (1) can be scheduled feasibly. The solution ¯ x to P uniform (1) can be scheduled using a periodic greedy schedule. Theorem Program P uniform (1) gives an O (1) -approximation to the total throughput capacity of a wireless network with uniform power levels. Anil Vullikanti (Virginia Tech) Capacity of Wireless Networks 44 / 100
Non-uniform power levels: problem with P uniform (1) X ( e , t ) + P i X ( e i , t ) could be large e ′ ∈ N ( e ) x ( e ′ ) ≤ 1 could be highly ⇒ x ( e ) + P suboptimal Large number of edges e i can transmit simultaneously Anil Vullikanti (Virginia Tech) Capacity of Wireless Networks 45 / 100
Non-uniform power levels Idea: Inductive ordering - ignore “small” edges in the constraint For e = ( u , v ), define r ( e ) = max { r ( u ) , r ( v ) } N ≥ ( e ) = { e ′ ∈ N ( e ) : r ( e ′ ) ≥ r ( e ) } Lemma e ′ ∈ N ≥ ( e ) X ( e ′ , t ) ≤ λ , for a constant λ . ∀ e , t, X ( e , t ) + P Anil Vullikanti (Virginia Tech) Capacity of Wireless Networks 46 / 100
Non-uniform power levels: Formulation P non − uniform ( λ ) Objective: max P i f i Subject to: X X ∀ i , f i = f ( e ) − f ( e ) e =( s i , v ) e =( v , s i ) ∀ e , x ( e ) = f ( e ) / cap ( e ) X x ( e ′ ) ∀ e , x ( e ) + ≤ λ (Congestion Constraints) e ′ ∈ N ≥ ( e ) ∀ e , f ( e ) ≥ 0 Anil Vullikanti (Virginia Tech) Capacity of Wireless Networks 47 / 100
Non-uniform power levels: Formulation P non − uniform ( λ ) Lemma e ′ ∈ N ≥ ( e ) x ( e ′ ) ≤ λ are There is a constant λ such that the constraints ∀ e , x ( e ) + P necessary: every feasible vector ¯ x is a feasible solution to program P non − uniform ( λ ) . Lemma e ′ ∈ N ≥ ( e ) x ( e ′ ) ≤ 1 are sufficient: the solution to The constraints ∀ e , x ( e ) + P P non − uniform (1) can be scheduled feasibly. Theorem The program P non − uniform (1) gives an O (1) -approximation to the total throughput capacity under non-uniform power levels. Anil Vullikanti (Virginia Tech) Capacity of Wireless Networks 48 / 100
Necessary condition Lemma 1 e ∈ E ′ ⇒ | E ′ ∩ N ≥ ( e ) | = 1 For any edge e and any D-2 matching E ′ , | E ′ ∩ N ≥ ( e ) | ≤ λ 2 Let e �∈ E ′ Suppose e 1 = ( u 1 , v 1 ) , e 2 = ( u 2 , v 2 ) ∈ E ′ ∩ N ≥ ( e ) u 1 , v 1 �∈ D ( u 2 ) ∪ D ( v 2 ) D ( u ) ∪ D ( v ) can be partitioned into disjoint regions of area π r ( e ) 2 /λ 3 Let n ( e ) =# packets sent on e in time T 4 ∀ e , n ( e ) + P e ′ ∈ N ≥ ( e ) n ( e ′ ) ≤ λ T 5 Set x ( e ) = n ( e ) / T Anil Vullikanti (Virginia Tech) Capacity of Wireless Networks 49 / 100
Sufficient condition Lemma e ′ ∈ N ≥ ( e ) x ( e ′ ) ≤ 1 are sufficient: the solution to The constraints ∀ e , x ( e ) + P P non − uniform (1) can be scheduled feasibly. Objective: Need to show existence of stable schedule that can send all packets Different approaches: 1 Periodic scheduling: stable, not necessarily polynomial time, in general 2 Randomized scheme: stable, centralized 3 Random access scheduling: completely local Lose a factor of 1 e for synchronous random access 1 Lose a factor of O ( 1 γ ), where γ is the ratio of the maximum transmission duration to 2 the minimum transmission duration 4 Distributed collision free scheduling: based on access hash functions Anil Vullikanti (Virginia Tech) Capacity of Wireless Networks 50 / 100
Periodic Scheduling Step I: Choosing time slots 1 Choose W s.t. S ( e ) = Wx ( e ) integral for each e 2 Order edges so that r ( e 1 ) ≥ . . . ≥ r ( e m ) 3 (Inductive Scheduling) Choose time slots S ( e ) for edges in this order: • For edge e i choose any Wx ( e i ) slots from the set { 1 , . . . , W } \ ( ∪ j ≤ i − 1 , e j ∈ N ≥ ( e i ) S ( e j )) Step II: Periodic scheduling For each packet, move one edge in W steps Anil Vullikanti (Virginia Tech) Capacity of Wireless Networks 51 / 100
Example W = 7 Need Wx ( e ) = 1 slot for all links other than (3 , 5); Wx (3 , 5) = 2 Assign slots: S (1 , 2) = { 1 } , S (2 , 3) = { 2 } , S (3 , 4) = { 3 } , S (3 , 5) = { 4 , 5 } ,... Anil Vullikanti (Virginia Tech) Capacity of Wireless Networks 52 / 100
When does greedy fail? Consider the utilization vector: W = 8. Assign slots { 1 , . . . , 8 } Consider an ordering with link (3 , 5) in the end Suppose greedy assigns: S (1 , 2) = { 1 , 2 } , S (2 , 3) = { 3 , 4 } , S (3 , 4) = { 5 } , S (5 , 6) = { 1 , 2 } , S (6 , 7) = { 3 , 4 } Not enough free slots for (3 , 5) Anil Vullikanti (Virginia Tech) Capacity of Wireless Networks 53 / 100
Periodic scheduling: proof Lemma For each edge e i , |{ 1 , . . . , W } \ ( ∪ j ≤ i − 1 , e j ∈ N ≥ ( e i ) S ( e j )) | ≥ Wx ( e i ) Proof. If not, X Wx ( e i ) + Wx ( e j ) > W j ≤ i − 1 , e j ∈ N ≥ ( e i ) which violates the congestion constraint in P non − uniform (1). ⇒ S ( e ) = W · x ( e ) slots can be allocated for each edge e Anil Vullikanti (Virginia Tech) Capacity of Wireless Networks 54 / 100
Sufficient condition: proof (continued) Schedule is valid since N () is symmetric: Suppose e ∈ N ( e ′ ) , e ′ ∈ N ( e ), r ( e ′ ) ≥ r ( e ) ⇒ e ′ ∈ N ≥ ( e ) Suppose e ′ is scheduled at time t . Then, t ∈ S ( e ′ ). Since e ′ ∈ N ≥ ( e ), slot t is not assigned to edge e Schedule is stable (constant bit rate): in a frame of length W , number of packets required to flow through e is x ( e ) W , and exactly this many slots are assigned for this edge. Lyapunov technique for proving stability for stochastic arrivals Anil Vullikanti (Virginia Tech) Capacity of Wireless Networks 55 / 100
Adding additional constraints: fairness Objective: max P i f i Subject to: X X ∀ i , f i = f ( e ) − f ( e ) e =( s i , v ) e =( v , s i ) ∀ e , x ( e ) = f ( e ) / cap ( e ) X x ( e ′ ) ∀ v , x ( e ) + ≤ 1 e ′ ∈ N ≥ ( e ) ∀ e , f ( e ) ≥ 0 ∀ i , j , f i ≤ f j /γ Fairness constraints Fairness: γ = 1 ⇒ completely fair γ = 0 ⇒ throughput maximization Anil Vullikanti (Virginia Tech) Capacity of Wireless Networks 56 / 100
Adding additional constraints: fairness Same approximation ratio holds Can quantify the relationship between fairness and capacity Maximum throughput vs Number of flows Throuput vs. Fairness 5 3 4.5 2.8 Maximum throughput 2.6 4 2.4 3.5 Throughput 2.2 3 2 2.5 1.8 k=2 2 k=5 1.6 k=7 1.5 1.4 k=9 k=11 1 single run 1.2 averaged 0.5 1 0 5 10 15 20 25 30 35 40 0.001 0.01 0.1 1 Number of flows Fairness index Anil Vullikanti (Virginia Tech) Capacity of Wireless Networks 57 / 100
Extensions: SINR model SINR model: If pairs ( v 1 , v ′ 1 ), ( v 2 , v ′ 2 ), . . . communicate P 1 d ( v 1 , v ′ 1 ) α ≥ β P i N + P d ( v i , v ′ 1 ) α i > 1 ∀ e : N ( e ) = E ∀ e = ( u , v ) : N ≥ ( e ) = { e ′ = ( u ′ , v ′ ) : ℓ ( e ′ ) ≥ max { ℓ ( e ) , a · d ( u , u ′ ) } Assumptions: Power levels for all links are fixed, For each edge e , cap ( e ) is fixed under an additive white Gaussian noise assumption Anil Vullikanti (Virginia Tech) Capacity of Wireless Networks 58 / 100
Extensions: SINR model ∆ = max e { ℓ ( e ) } / min e ′ { ℓ ( e ′ ) } Lemma The program P non − uniform ( λ ) gives necessary conditions for a constant λ , while the program P non − uniform (1 / log ∆) gives sufficient conditions. Anil Vullikanti (Virginia Tech) Capacity of Wireless Networks 59 / 100
Extensions: power constraints Setting: S has to determine which edges e to use at time t , and what power level to use Capacity of link e at power level p p cap ( e , p ) = W log 2 (1 + d ( u , v ) α N 0 W ) Anil Vullikanti (Virginia Tech) Capacity of Wireless Networks 60 / 100
Extensions: power constraints Setting: S has to determine which edges e to use at time t , and what power level to use Capacity of link e at power level p p cap ( e , p ) = W log 2 (1 + d ( u , v ) α N 0 W ) J = set of possible choices of power levels; need not be finite Define T ( J ) = { ( e , p ) ∈ E × J } Define N ( e , p ) = { ( e ′ = ( u ′ , v ′ ) , p ′ ) : e ′ ∈ V 2 , p ′ ∈ J , d ( u , u ′ ) ≤ (1 + ∆)( range ( p ) + range ( p ′ )) } Define N ≥ ( e , p ) = { ( e ′ = ( u ′ , v ′ ) , p ′ ) ∈ N ( e , p ) : p ′ ≥ p } Anil Vullikanti (Virginia Tech) Capacity of Wireless Networks 60 / 100
Program P pctm ( λ ) X max f i s.t.: i X X ∀ i , f i = f ( e , p ) − f ( e , p ) ( e =( s i , v ) , p ) ∈T ( e =( v , s i ) , p ) ∈T ∀ ( e , p ) ∈ T , x ( e , p ) = f ( e , p ) / cap ( e , p ) X ∀ ( e , p ) ∈ T , x ( e , p ) + x ( e , p ) ≤ λ ( e ′ , p ′ ) ∈ N ≥ ( e , p ) X X ∀ i , ∀ u � = s i , t i f ( e , p ) = f ( e , p ) e ∈ N out ( u ) e ∈ N in ( u ) X x ( e , p ) · p ≤ B ( e , p ) ∈T B = total bound on power usage Anil Vullikanti (Virginia Tech) Capacity of Wireless Networks 61 / 100
Joint power and throughput capacity optimization: special case Lemma Any feasible rate vector and power assignment must satisfy the constraints of P ( c ) for a constant c. Further, any solution to P (1) is feasible. Anil Vullikanti (Virginia Tech) Capacity of Wireless Networks 62 / 100
Joint power and throughput capacity optimization: special case Lemma Any feasible rate vector and power assignment must satisfy the constraints of P ( c ) for a constant c. Further, any solution to P (1) is feasible. Assumption: | J | ≤ poly ( n ) ⇒ |P pctm | is polynomial sized. Anil Vullikanti (Virginia Tech) Capacity of Wireless Networks 62 / 100
Joint power and throughput capacity optimization: general case Let p max = max { p ∈ J } and p min = min { p ′ ∈ J } Assumption: p max / p min ≤ poly ( n ) J ′ = { p min , (1 + ǫ ) p min , . . . , p max } Lemma The program P pctm (1) defined using set J ′ (instead of set J) gives a constant factor approximation to throughput capacity under a given bound on total power consumption. Anil Vullikanti (Virginia Tech) Capacity of Wireless Networks 63 / 100
Extension: capacity with random access scheduling T id : idle slot length T xmit ( ℓ ): length of transmission on link ℓ N pri ( ℓ ): links within primary interference of ℓ N sec ( ℓ ) = N ( ℓ ) \ N pri ( ℓ ) Node v attempts to transmit on link e = ( v , w ) only if no neighbor of v is Probability of accessing the link ℓ : τ ( ℓ ) = 1 − e − x ( ℓ ) currently transmitting If channel free, v transmits on e with probability τ ( e ) Anil Vullikanti (Virginia Tech) Capacity of Wireless Networks 64 / 100
Synchronous Random Access Networks Lemma x be a feasible solution to the program P (1) . Then, 1 Let ¯ e ¯ x can be achieved by synchronous random access scheduling. Proof: Choose τ ( ℓ ) = 1 − e − x ( ℓ ) /λ , for each ℓ . Probability of collision free transmission on edge ℓ : Π ℓ ′ ∈ I ( ℓ ) (1 − τ ( ℓ ′ )) η ( ℓ ) = ℓ ′∈ I ( ℓ ) − x ( ℓ ′ ) P = e e x ( ℓ ) − 1 ≥ Anil Vullikanti (Virginia Tech) Capacity of Wireless Networks 65 / 100
Synchronous Random Access Networks Successful flow through ℓ = cap ( ℓ ) · τ ( ℓ ) · η ( ℓ ) cap ( ℓ ) · (1 − e − x ( ℓ ) ) · e x ( ℓ ) − 1 ≥ cap ( ℓ ) · ( e x ( ℓ ) − 1 − e − 1 ) = „ 1 + x ( ℓ ) « − 1 ≥ cap ( ℓ ) · e e f ( ℓ ) = e e ¯ 1 ⇒ f is stable Anil Vullikanti (Virginia Tech) Capacity of Wireless Networks 66 / 100
Random Access Scheduling in an Asynchronous Network T id : idle slot length T xmit ( ℓ ): transmission duration on ℓ max ℓ T xmit ( ℓ ) γ = min ℓ ′ T xmit ( ℓ ′ ) ∆: max #simultaneous transmissions possible in N ( ℓ ) (interference degree) Theorem Let � x be a feasible solution to P (1) . The random access protocol with channel access probability − x ( ℓ ) Tid ∆( ℓ ) · Txmit ( ℓ )(1+ γ ) , τ ( ℓ ) = 1 − e achieves a link utilization of � 1 h ≥ e ( γ +1)∆ � x. Anil Vullikanti (Virginia Tech) Capacity of Wireless Networks 67 / 100
Asynchronous Random Access: effect of packet sizing policies Random access is more competitive when the packet sizes on links are non-uniform, and are proportional to the link capacity Flow 1 = 6Mbps, Flow 2 = 24Mbps 14 500B 750B 12 1000B Flow 2’s rate (Mbps) 1250B 10 1500B 1750B 8 2000B 6 4 2 0 0 0.5 1 1.5 2 2.5 Flow 1’s rate (Mbps) ℓ 1 and ℓ 2 : hidden interfering links c ( ℓ 1 )= 6Mbps, c ( ℓ 2 )=24Mbps packet size on ℓ 1 : 500 Bytes packet size on ℓ 2 varied from 500 Bytes to 2000 Bytes Anil Vullikanti (Virginia Tech) Capacity of Wireless Networks 68 / 100
Limits on the competitive ratio of asynchronous random access scheduling ℓ 1 ℓ 0 � f = � 1 / 2 , . . . , 1 / 2 � is feasible for greedy ℓ 2 scheduling ℓ ∆ Lemma λ� f is feasible for random access scheduling only if λ ≤ c log ∆ γ ∀ i ≥ 1, ℓ i ∈ hidden ( ℓ 0 ) ∆ γ ∀ i ≥ 1, ℓ 0 ∈ hidden ( ℓ i ) Assume T xmit ( ℓ i ) = T xmit = a 1 T id and T xmit ( ℓ 0 ) = γ T xmit Anil Vullikanti (Virginia Tech) Capacity of Wireless Networks 69 / 100
Characterizing the capacity region for random access protocols New formulation to approximate the throughput capacity of an asynchronous random access network within an O (∆)-factor: Theorem (Necessary Conditions) � x is feasible for asynchronous random access protocol only if: x ( ℓ ′ ) · (1 + T xmit ( ℓ ) − T id X x ( ℓ ′ ) + X ∀ ℓ : x ( ℓ ) + ) ≤ ∆ T xmit ( ℓ ′ ) ℓ ′ ∈ exposed ( ℓ ) ℓ ′ ∈ hidden ( ℓ ) Theorem (Sufficient Conditions) � x is feasible for asynchronous random access protocol if: x ( ℓ ′ ) · (1 + T xmit ( ℓ ) − T id ) ≤ 1 X x ( ℓ ′ ) + X ∀ ℓ : x ( ℓ ) + T xmit ( ℓ ′ ) e ℓ ′ ∈ exposed ( ℓ ) ℓ ′ ∈ hidden ( ℓ ) Anil Vullikanti (Virginia Tech) Capacity of Wireless Networks 70 / 100
Extension: multi-channel multi-radio networks Graph G = ( V , E ) Induced Radio Network G = ( V , L ): V is the set ∪ u Radios ( u ) and L = For each node u ∈ V , Radios ( u ): set ∪ e =( u , v ) ∈ E Radios ( u ) × Radios ( v ) of wireless interfaces associated with it. For link ℓ = ( ρ, ρ ′ ), Set Ψ of channels available parent ( ℓ ) = ( u , v ) if ρ ∈ Radios ( u ) Schedule + channel assignment: at and ρ ′ ∈ Radios ( v ) each time t , choose links e = ( u , v ) Consider set which will transmit, which radio T = { ( ℓ, ψ ) : ℓ ∈ L , ψ ∈ Ψ } interfaces to use at u , v and which channel to use Anil Vullikanti (Virginia Tech) Capacity of Wireless Networks 71 / 100
Necessary conditions for scheduling For link ℓ = ( ρ, ρ ′ ) in induced radio network G = ( V , L ): Pri ( ℓ ) = { ℓ ′ sharing a radio with ℓ } Pri ≻ ( ℓ ) = { ℓ ′ ∈ Pri ( ℓ ) : parent ( ℓ ′ ) ≻ parent ( ℓ ) } Sec ( ℓ ) = { ℓ ′ : parent ( ℓ ′ ) ∈ Pri ( parent ( ℓ )) } ∪ { ℓ ′ : parent ( ℓ ′ ) ∈ Sec ( parent ( ℓ )) } Sec ≻ ( ℓ ) = { ℓ ′ ∈ Succ ( ℓ ) : parent ( ℓ ′ ) ≻ parent ( ℓ ) } Theorem Flow constraints with the following congestion constraints are necessary for any feasible flow+utilization vector: X X X x ( ℓ, ψ ) + x ( ℓ, ρ ) + x ( f , χ ) ρ ∈ Ψ \{ ψ } χ ∈ Ψ f ∈ Pri ≻ ( ℓ ) X + x ( g , ψ ) ≤ λ + 2 g ∈ Sec ≻ ( ℓ ) Anil Vullikanti (Virginia Tech) Capacity of Wireless Networks 72 / 100
Sufficient conditions Theorem The rate vector satisfying the following conditions can be scheduled feasibly: X X X ∀ ( ℓ, ψ ) , x ( ℓ, ψ ) + x ( ℓ, ρ ) + x ( f , χ ) ρ ∈ Ψ \{ ψ } χ ∈ Ψ f ∈ Pri ( ℓ ) x ( g , ψ ) ≤ 1 X + e − ǫ g ∈ Sec ( ℓ ) Anil Vullikanti (Virginia Tech) Capacity of Wireless Networks 73 / 100
Random Access Hash Functions Need access-hash function H ( ℓ, ψ, t ) such that: 1 with probability 1 − e − e · x ( ℓ,ψ ) H ( ℓ, ψ, t ) = with probability e − e · x ( ℓ,ψ ) 0 Key Property : Value of H ( ., ., ) fixed no matter who invokes it with the same arguments Also known as random oracles in Cryptography SHA-1 works well in practice Anil Vullikanti (Virginia Tech) Capacity of Wireless Networks 74 / 100
Algorithm PLDS Executed by each radio ρ : 1 ∀ ℓ incident on ρ : compute H ( ℓ, ψ, t ), for each ψ , t . 2 Randomly pick a pair ( ℓ, ψ ) s.t. H ( ℓ, ψ, t ) = 1 • if no such pair exists, sleep during time t 3 If selected link ℓ ∈ L out ( ρ ), then schedule an outgoing transmission across ℓ on channel ψ at time t 4 if selected link ℓ ∈ L in ( ρ ), then tune to channel ψ and await an incoming transmission across ℓ on channel ψ at time t Anil Vullikanti (Virginia Tech) Capacity of Wireless Networks 75 / 100
Extensions: Bounding Delays Goal : choose flow vector � f so that: P i f i is maximized For each session i such that f i > 0, average delay for each packet is at most D Our Result Careful choice of paths plus random access scheduling to get joint bounds on throughput and delays. Anil Vullikanti (Virginia Tech) Capacity of Wireless Networks 76 / 100
Choosing routes and flows Choose flow � f that maximizes P i f i subject to: X ∀ i , f ( p ) cost ( p ) ≤ Df i p ∈ P ( i ) X ∀ ( e , i ) , x ( e , i ) = f ( p ) / cap ( e ) p ∈ P ( i ): e ∈ p X X X x ( e ′ , i ) ∀ e , x ( e , i ) + ≤ 1 i e ′ ∈ N ( e ) i (Filter) Drop flows on paths longer than 2 D for each i (Round) Choose a subset S of sessions and a path p i for each i ∈ S by iterative rounding (Choose flows) Choose flow f ( p i ) = K log log D / log D Anil Vullikanti (Virginia Tech) Capacity of Wireless Networks 77 / 100
Joint Delay-Throughput Tradeoffs Theorem The flow vector � f along with random access scheduling ensures that P i f i = Ω( OPT · log log D / log D ) , and at least (1 − 1 / n ) -fraction of the packets for each session i are delivered within a delay of O ( D · (log D / log log D ) · log n ) . Adaptive channel switching delays can be incorporated into the framework in terms of cost ( p ) to quantify the throughput gains of adaptive channel switching Similar tradeoffs for adaptive power switching Anil Vullikanti (Virginia Tech) Capacity of Wireless Networks 78 / 100
Summary: General strategy Define suitable interference set ˆ N ( e ) for each link e Construct LP P ( λ ) with flow constraints, and congestion constraints of the form X x ( e ′ ) ≤ λ, x ( e ) + e ′ ∈ ˆ N ( e ) for each e Prove that P ( c 1 ) gives necessary conditions – any feasible solution � f ,� x satisfies the constraints of P ( c 1 ) Prove that P ( c 2 ) gives sufficient conditions – corresponding to any feasible solution � x of P ( c 2 ), we can construct a schedule S that corresponds to � f ,� f ,� x Anil Vullikanti (Virginia Tech) Capacity of Wireless Networks 79 / 100
Summary Two techniques for cross-layer formulation of the end-to-end capacity of wireless networks Linearization of interference constraints Inductive ordering to deal with non-uniform power levels Framework extends to a number of models, constraints and objective functions Anil Vullikanti (Virginia Tech) Capacity of Wireless Networks 80 / 100
Part III: Dynamic control for network stability Anil Vullikanti (Virginia Tech) Capacity of Wireless Networks 81 / 100
Outline for Part III Background: arrival processes, queuing Backpressure algorithm and its analysis Approximate version of backpressure algorithm Random access approach Summary of related research Anil Vullikanti (Virginia Tech) Capacity of Wireless Networks 82 / 100
Background “Arrivals at all sources are well-behaved” Anil Vullikanti (Virginia Tech) Capacity of Wireless Networks 83 / 100
Background “Arrivals at all sources are well-behaved” 1 Let A i ( t ) be the exogenous arrival process for connection i with rate λ i 2 An arrival process A i ( t ) is admissible with rate λ i if The time averaged expected arrival rate satisfies: 1 t − 1 1 X E [ A i ( τ )] = λ lim t →∞ t τ =0 Let H ( t ) represent the history until time t There exists A max such that 2 E [( A i ( t )) 2 | H ( t )] ≤ A 2 max for t . For any δ > 0, there exists an interval size T , possibly dependent on δ , such that for 3 any initial time t 0 : " T − 1 # 1 X A i ( t 0 + k ) | H ( t 0 ) E ≤ λ + δ T k =0 Other models: adversarial arrivals Anil Vullikanti (Virginia Tech) Capacity of Wireless Networks 83 / 100
Background (continued) Each node v maintains queues for each link ( v , w ) and each connection i Assume unbounded buffer sizes – no packet drops because of buffer overflows Let U i v ( t ) denote the queue at node v for connection i at time t ; let U ( t ) = � U i v ( t ) � µ i ( u , v ) ( t ) ≤ c ( u , v ): data rate allocated to commodity i during slot t across the link ( u , v ) by the network controller. Anil Vullikanti (Virginia Tech) Capacity of Wireless Networks 84 / 100
Capacity region revisited I ⊂ E is a conflict free subset if for every e , e ′ ∈ I , e and e ′ are conflict-free. Let I denote the set of all possible conflict-free subsets I ⊂ E Let µ ( I ) denote the vector of transmission rates for each e ∈ I . Let Γ . = Conv ( { � µ ( I ) | I ∈ I} ) denote the convex hull of all transmission-rate matrices Let inflow i ( w , v ) ∈ E µ i v ,µ ( t ) = P ( w , v ) ( t ) denote the flow of commodity i into node v for policy µ at time t Let outflow i ( v , w ) ∈ E µ i v ,µ ( t ) = P ( v , w ) ( t ) denote the flow of commodity i out of node v for policy µ at time t Let netflow i v ,µ ( t ) = outflow i v ,µ ( t ) − inflow i v ,µ ( t ) denote the total flow of commodity i out of node v for policy µ at time t Anil Vullikanti (Virginia Tech) Capacity of Wireless Networks 85 / 100
Example Assume primary interference: edges Traffic matrix corresponding to with common end-point conflict µ = 2 3 I 1 + 1 3 I 2 Two connections ( s 1 , t 1 ) and ( s 2 , t 2 ) inflow 1 2 ,µ ( t ) = µ 1 (1 , 2) = 2 / 3 Γ = { α I 1 + β I 2 : α + β ≤ 1 } Anil Vullikanti (Virginia Tech) Capacity of Wireless Networks 86 / 100
Capacity region Theorem (Grigoriadis et al., 2006) The connection rate vector � λ i � is within the network-layer capacity region Λ if and only if there exists a randomized network control algorithm that makes valid µ i ( u , v ) ( t ) decisions, and yields: ∀ i , E [ netflow i s i ,µ ( t )] = λ i ∈ { s i , t i } , E [ netflow i ∀ i , ∀ w / w ,µ ( t )] = 0 Anil Vullikanti (Virginia Tech) Capacity of Wireless Networks 87 / 100
Backpressure algorithm At each time t For each link ( v , w ): let i = i ∗ be the commodity with maximum differential backlog ∆ U i v − U i w For each link ( v , w ), define weight ( v , w ) to be the maximum differential backlog Choose independent set I with maximum weight wt ( I ) = P e ∈ I wt ( e ) Schedule all links in I simultaneously, and send as much as possible Anil Vullikanti (Virginia Tech) Capacity of Wireless Networks 88 / 100
Example ∆ U 1 (1 , 2) = 5, ∆ U 2 (1 , 2) = − 35 ⇒ i ∗ (1 , 2) = 1 , W ∗ (1 , 2) = 5 ∆ U 1 (2 , 3) = 15, ∆ U 2 (2 , 3) = 5 ⇒ i ∗ (2 , 3) = 1 , W ∗ (2 , 3) = 15 Assume primary interference: edges ∆ U 1 (3 , 4) = 0, ∆ U 2 (3 , 4) = 30 with common end-point conflict ⇒ i ∗ (3 , 4) = 2 , W ∗ (3 , 4) = 30 Two connections ( s 1 , t 1 ) and ( s 2 , t 2 ) wt ( I 1 ) = 5 + 30 = 35, Γ = { α I 1 + β I 2 : α + β ≤ 1 } wt ( I 2 ) = 15 Anil Vullikanti (Virginia Tech) Capacity of Wireless Networks 89 / 100
Backpressure algorithm At each time t For each link ( v , w ): let i ∗ ( v , w ) ( t ) denote the connection which maximizes the differential backlog i ∗ i ∗ ( v , w ) ( t ) ( v , w ) ( t ) W ∗ ( v , w ) ( t ) = U ( t ) − U ( t ). v w Choose conflict-free link set I ∗ ∈ I which maximizes ( u , v ) ∈ I ∗ W ∗ P ( u , v ) ( t ) · c ( u , v ) The network controlled chooses links e = ( u , v ) ∈ I ∗ and connection i ∗ ( u , v ) ( t ) if W ∗ ( u , v ) ( t ) > 0 (if there is not enough backlogged data, i.e., i ∗ ( u , v ) ( t ) U ( t ) < c ( u , v ) use dummy bits) ( u , v ) Anil Vullikanti (Virginia Tech) Capacity of Wireless Networks 90 / 100
Analysis µ i Consider any valid resource allocation policy that assigns a rate of ˜ ( u , v ) ( t ) to commodity i across link ( u , v ) at time t . Let µ i ( u , v ) ( t ) denote the corresponding values for the dynamic backpressure algorithm. By construction: X X µ i ( u , v ) ( t )[ U i u ( t ) − U i X X µ i ( u , v ) ( t ) W ∗ ˜ v ( t )] ≤ ˜ ( u , v ) ( t ) ( u , v ) i ( u , v ) i X W ∗ ≤ ( u , v ) ( t ) · µ ( u , v ) ( u , v ) Anil Vullikanti (Virginia Tech) Capacity of Wireless Networks 91 / 100
Analysis (continued) Rearranging the terms: “ P v of queue-size at v · netflow( v ) = P e flow( e ) · backlog( e )” Anil Vullikanti (Virginia Tech) Capacity of Wireless Networks 92 / 100
Analysis (continued) Rearranging the terms: “ P v of queue-size at v · netflow( v ) = P e flow( e ) · backlog( e )” X X U i X µ i X µ i v ( t ) · [ ( v , w ) ( t ) − ( u , v ) ( t )] i v w u X X µ i ( u , v ) ( t )[ U i u ( t ) − U i = v ( t )] ( u , v ) i Lemma (Property) µ i ( u , v ) ( t ) denotes any resource allocation policy, and µ i If ˜ ( u , v ) ( t ) denotes the resource allocation for the Backpressure scheme, we have: X X X X U i µ i µ i v ( t )[ ˜ ( v , w ) ( t ) − ˜ ( u , v ) ( t )] v i w u "X # X X U i µ i X µ i ≤ v ( t ) ( v , w ) ( t ) − ( u , v ) ( t ) v i w u Anil Vullikanti (Virginia Tech) Capacity of Wireless Networks 92 / 100
Lyapunov functions Define: X X ( U i v ( t )) 2 L ( U ( t )) = i v Theorem (Grigoriadis et al., 2006) If there exist constants B > 0 and ǫ > 0 such that for all slots t: X X U i E [ L ( U ( t + 1)) − L ( U ( t )) | U ( t )] ≤ B − ǫ v ( t ) (1) v i then, the network is strongly stable. Anil Vullikanti (Virginia Tech) Capacity of Wireless Networks 93 / 100
Analysis of backpressure Theorem Let � λ denote the vector of arrival rates; if there exists an ǫ > 0 such that � λ + � ǫ ∈ Λ (where � ǫ is the vector such that ǫ i = 0 if λ i = 0 , and ǫ i = ǫ otherwise), then the dynamic backpressure algorithm stably services the arrivals. Anil Vullikanti (Virginia Tech) Capacity of Wireless Networks 94 / 100
Analysis of backpressure If V , U , µ, A ≥ 0 and V ≤ max { U − µ, 0 } + A , then, V 2 ≤ U 2 + µ 2 + A 2 − 2 U ( µ − A ) Since U i v ( t + 1) ≤ max { U i e =( v , w ) µ i i A i ( t ) + P e =( u , v ) µ i v ( t ) − P e ( t ) , 0 } + P e ( t ), we have: ´ 2 + ´ 2 − v ( t + 1) 2 ≤ U i v ( t ) 2 + U i `P w µ i ` A i u µ i ( v , w ) ( t ) v ( t ) + P ( u , v ) ( t ) 2 U i w µ i ( v , w ) ( t ) − A i u µ i `P v ( t ) − P ´ v ( t ) · ( u , v ) ( t ) Anil Vullikanti (Virginia Tech) Capacity of Wireless Networks 95 / 100
Analysis (continued) j z 2 j z j ) 2 , if z j ≥ 0, Summing over all indices ( v , i ) and since P j ≤ ( P X X U i L ( U ( t + 1)) − L ( U ( t )) ≤ 2 BN − 2 v ( t ) · v i X ! µ i ( v , w ) ( t ) − A i X µ i v ( t ) − ( u , v ) ( t ) , w u where B . v [(max w µ ( v , w )) 2 + (max i A i + max u µ ( u , v )) 2 ]. 1 2 N · P = X U i s i ( t ) · E [ A i ⇒ E [ L ( U ( t + 1)) − L ( U ( t )) | U ( t )] ≤ 2 BN + 2 · s i ( t ) | U ( t )] − i 0 1 X X @X X U i µ i A | U ( t )] 2 E [ v ( t ) · ( v , w ) ( t ) − µ ( u , v ) ( t ) v w i ( u , v ) Anil Vullikanti (Virginia Tech) Capacity of Wireless Networks 96 / 100
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