Node-based Distributed Optimal Control of Wireless Networks

Transcription

1 Node-based Distributed Optimal Control of Wireless Networks CISS March 2006 Edmund M. Yeh Department of Electrical Engineering Yale University Joint work with Yufang Xi

2 Main Results Unified framework for jointly optimal power control, routing, and congestion control in interference-limited wireless networks. Approach based on flow models. Optimal (minimum delay) power control and multi-path routing in wireless networks with link costs. Node-based algorithms for jointly optimal power control and multi-path routing. Fast convergence, distributed computation, adaptability to changing topology and traffic demands. Natural incorporation of congestion control. Extends optimal routing framework of Gallager (77) to wireless setting.

3 Previous work Optimal multi-path routing in wired networks: Gallager (77), Bertsekas et al. (84), Tsitsiklis and Berksekas (86). Congestion control and pricing in wired networks: Kelly et al. (98), Low and Lapsley (00), Srikant (04). Congestion control and multi-path routing in wired network: Kar et al. (01), Lin and Shroff (03), Wang, Palaniswami and Low (03). Power control in wireless networks: Foschini and Miljanic (95), Hanly and Tse (99), Bambos et al, (00), Julian et al. (01), Cruz and Santhanam (02, 03). Cross-layer approaches: Johansson et al. (02), Chiang (04), Huang, Berry, Honig (05), Karnik, Mazumbar, Rosenberg (06).

4 Network Model Multi-hop wireless network. Model: directed, connected graph G = (N, E). W = set of sessions (origin-destination O(w)-D(w) pairs). Slowly varying traffic: flow model. r w = end-to-end flow rate of session w W. f ij (w) = rate of session w s flow on (i, j) E. F ij = w W f ij(w) = aggregate rate of flow on (i, j).

5 Capacity Constraints C ij = transmission capacity of (i, j) E. - variable due to power control, interference, fading. Assume relatively static channel conditions. Assume C ij = C(SINR ij ), where SINR ij (P ) = G ij P ij θ i G ij n j P in + m i G mj n P mn + N j. θ i [0, 1] = self-noise factor. C( ) increasing, concave, twice continuously differentiable. e.g. CDMA with single-user decoding: C ij = log(1 + KSINR ij ).

6 Link Costs Cost on link (i, j) = D ij (F ij, C ij ). Queueing delay: D ij = message arrival rate on (i, j) expected message delay on (i, j). D ij increasing and convex in F ij, decreasing and convex in C ij, twice continuously differentiable. e.g. Kleinrock s M/M/1 approximation: D ij (F ij, C ij ) = 1 C ij F ij. Network cost D = (i,j) E D ij(f ij, C ij ).

7 Jointly Optimal Power control and Routing Minimize (i,j) E D ij(f ij, C ij ) subject to f ij (w) 0 (i, j) E, w W j O i f ij (w) = k I i f ki (w) w W, i O(w), D(w) j O i f ij (w) = r w w W, i = O(w) f ij (w) = 0 w W, i = D(w), j O i F ij = w W f ij(w) (i, j) E C ij = C(SINR ij (P )) (i, j) E P ij 0 (i, j) E j O i P ij P i i N O i = node i s next hop neighbors. I i = node i s immediate upstream neighbors. Each node i knows only O i and I i, not global topology.

8 Change of Variables and Concavity (i,j) E D ij(f ij, C ij ) concave in F but not concave in P. Define log-powers S ij = ln P ij (Johansson et al. 03, Chiang 04). If C( ) satisfies xc (x) C (x) 1, then C ij (S) concave in S = (S ij ) (Huang et al. 05). High-SINR CDMA: C ij = log(1 + KSINR ij ) log(ksinr ij ).

9 Jointly Optimal Power Control and Routing If D ij (F ij, C ij ) jointly convex in (F ij, C ij ) (e.g. 1 C ij F ij ), then (i,j) E D ij(f ij, C ij (S)) = D(F, S) jointly convex in (F, S). Jointly Optimal Power Control and Routing (JOPR) Problem: Minimize D(F, S) subject to flow conservation constraints C ij = C(SINR ij (S)) (i, j) E j O i e S ij P i i N

10 Paradigm Shift: Node-based Wireless Routing Power control: done at each node. Multi-path routing: path-based source routing (most previous literature): node-by-node routing Wireless: frequent topology changes and node activity hard for sources to get detailed, current path information. Consider node-by-node routing. Each node decides on total transmission power, power allocation, and traffic allocation on outgoing links.

11 Control Variables Routing: φ ik (w) = f ik(w) t i (w), k O i t i (w) = total incoming rate of w traffic at node i. φ ik 0, k O i φ ik = 1. Allows nodes to adjust variables independently (Gallager 77). Power allocation: η ik = P ik P i, k O i. η ik 0, k O i η ik = 1. Power control: γ i = S i S i, γ i 1, where S i = ln P i (P i > 1). Control overhead for power allocation and power control different. Update all three sets of variables based on exchange of information.

12 Jointly Optimal Power Control and Routing Minimize (i,j) E D ij(f ij, C ij ) subject to φ ij (w) 0 (i, j) E, w W j O i φ ij (w) = 1 w W, i D(w) η ij 0 j O i η ij = 1 γ i 1 capacity constraints flow conservation constraints (i, j) E i N i N

13 Marginal Cost Indicators Routing: Power Allocation: δφ ik (w) D ik F ik + D r k (w) δη ik D ik C ik C ik G ik IN ik (1 + SINR ik ) Power Control: δγ i P i (m,n) E D mn C mn C mng mn G in P mn IN 2 mn + δη ij η ij j O i IN mn = θ m G mn (P m P mn ) + l m G lnp l + N n.

14 Optimality Conditions Theorem 1 If D ik (F ik, C ik ) is jointly convex in (F ik, C ik ) for all (i, k), then necessary conditions for optimality are: for all w W and i N such that t i (w) > 0, δφ ik (w) = λ i (w), if φ ik (w) > 0 δφ ik (w) λ i (w), if φ ik (w) = 0 (1) and δη ik = ν i, k O i δγ i = 0, if γ i < 1 (2) δγ i 0, if γ i = 1 These conditions are sufficient when (1) holds for all w W and i N whether t i (w) > 0 or not.

15 Control Algorithms Scaled gradient projection with marginal cost exchange. Scale descent direction using second derivatives of objective (constrained Newton Method) for fast convergence. Large dimensionality and need for distributed computation: scaling matrices to upper bound Hessians. Guaranteed convergence from all initial conditions. Algorithms adaptive to changing topology and traffic conditions. Generalize class of algorithms by Bertsekas, Gafni and Gallager (84) for joint power control and routing.

16 Routing Algorithm Scaled gradient projection (Bertsekas et al. 84): φ k+1 i (w) = RT (φ k i ) = [ φ k i (w) (M k i (w)) 1 δφ k i (w) Diagonal, positive definite scaling matrix M k i (w). ] + M k i (w) Projection on on feasible set relative to norm induced by M k i (w): [ φ i ] + M k i = arg min φ i F k φ i φ i φ i, M k i (φ i φ i ). F k φ i = {φ i 0 : φ ij = 0, j B k i and j O i φ ij = 1} In general, iteration involves quadratic program. Special case: constrained steepest descent (Gallager 77).

17 Routing Algorithm Requires δφ ik (w) D ik F ik + D r k (w). D r k (w) = marginal delay due to increment of w input at k. D r k (w) = 0, if k = D(w) D r i (w) = φ ik (w) k O i [ Dik F ik (C ik, F ik ) + ] D r k (w), i D(w) Each node i provides marginal cost D r i (w) to upstream neighbors. B i (w) = set of blocked nodes for loop-free routing graph. RT ( ) generates loop-free routing graph given loop-free input (Gallager 77).

18 Scaling Matrix for Routing Choose M k i to upper bound H k,λ φ i. H k,λ φ i = Hessian matrix of D with respect to φ i, evaluated at λφ k i + (1 λ)φ k+1 i for some λ [0, 1]. Scaling matrix: M k i ( ) = tk i 2 diag A k ij(d 0 ) + AN k i h k j A k (D 0 ) j AN k i, A k ij (D0 ) max Fij :D ij (C k ij,f ij) D 0 A k (D 0 ) max (m,n) E A k mn(d 0 ). 2 D ij F 2 ij, AN k i O i \B k i, h k j = maximum number of hops (or any upper bound on maximum) on path from j to D(w).

19 Power Allocation Algorithm Power allocation update: η k+1 i = P A(η k i ) = [ η k i (Q k i ) 1 δη k i ] + Q k i. δη ik = D ik C ik G ik C ik IN ik (1 + SINR ik ) requires only local information. Q k i diagonal, positive definite, chosen to upper bound Hk,λ η i.

20 Power Control Algorithm Network-wide power control update: γ k+1 = P C(γ k ) = [ γ k (V k ) 1 δγ k] + V k. V k diagonal for distributed implementation: γ k+1 i = P C(γ k i ) = min{1, γ k i (v k i ) 1 δγ k i }. V k positive definite chosen to upper bound H k,λ = n i [ G in m I n δγ i P i γ. D mn C mn C mn SINR mn IN mn ] + n I i δη in η in. Power messages from faraway nodes less important.

21 Power Control Message Exchange Protocol Each node j adds measures on all its incoming links and forms one power control message: MSG j = m I j D mj C mj C mj SINR mj IN mj. Node j broadcasts MSG j to network, collected and processed by all other nodes i. D C SINR C l ' lj lj lj lj MSG j IN lj D C j D C ' kj kj kj kj C SINR j C SINR ' mj mj mj IN IN m I mj mj kj k D C SINR C ' ij ij ij ij IN ij i

22 Convergence Theorem Theorem 2 Let a valid loop-free initial configuration with finite total network cost be given. Under RT, P A, P C algorithms, φ k i (w) φ i (w), η k i η i, γk i γ i as k, where ({φ i (w)}, {η i }, {γ i }) is the optimal solution. Each iteration of any one of RT, P A, and P C (with all other variables fixed) strictly reduces network cost unless corresp. optimality conditions satisfied. Does not require particular order of iteration.

23 Adaptation to Changing Topology and Traffic Node-based framework: nodes don t need knowledge of traffic matrix {r w, w W}. Nodes only need to identify session of traffic stream, and measure t i (w). All control done through marginal cost updates. Nodes don t need knowledge of global topology. Topology changes also reflected through marginal costs. Control variables as fractions: always feasible in spite of changes. Algorithms can chase shifting optimum.

24 Congestion Control Seamlessly incorporated into framework. Utility of session w W : U w (r w ) U w ( ) increasing, concave r w = admitted end-to-end flow rate Assume each session has threshold of utility satiation: U w (r w ) = U w (r w ) = U w, for all r w r w. Overflow (blocked) rate F wb = r w r w 0. E w (F wb ) = U w U w (r w ) = utility loss from rejecting F wb. E w ( ) increasing, convex.

25 Congestion Control Maximize total source utility - total network cost U w (r w ) D ij (F ij, C ij ) w W (i,j) E = U w E w (F wb ) w W w W (i,j) E D ij (F ij, C ij ) Minimize (i,j) E D ij(f ij, C ij ) + w W E w(f wb ) Same as pure routing problem + overflow link with cost function E w (F wb ) directly connecting O(w) and D(w) (Bertsekas and Gallager 92). Jointly Optimal Power Control, Routing, and Congestion Control (JOPRC). Algorithms directly extend.

26 Network with Overflow Link f O ( w ) j j r w O(w) f O ( w ) k k D(w) F wb Figure 1: Virtual Network with Overflow Link

27 Summary Unified framework for jointly optimal power control, routing, and congestion control in multi-hop ad hoc wireless networks. Based on flow model with link costs: for minimizing delay. Node-based scaled gradient projection algorithms for power control and routing. Fast convergence, distributed computation, adaptability to changing topology and traffic demands. Natural extension to congestion control.

28 Extensions and Applications Results generalized to quasi-convex cost functions and convex capacity regions: Pareto optimality (see Xi and Yeh, Optimal Power Control, Routing, and Congestion Control in Wireless Networks. Submitted to IEEE Transactions on Information Theory). Send/receive constraints can be introduced through frequency division (small number of bands). Algorithms used for distributed implementation of the Maximum Differential Backlog Policy (Tassiulas and Ephremides 92) for stochastic wireless networks (Xi and Yeh, WiOpt 06). Algorithms used to find optimal coding subgraph for network coding for wireless networks (Xi and Yeh, Allerton 05, NetCod 06).