AN EXTERNALITY BASED DECENTRALIZED OPTIMAL POWER ALLOCATION SCHEME FOR WIRELESS MESH NETWORKS Shruti Sharma and Demos Teneketzis Department of Electrical Engineering and Computer Science University of Michigan, Ann Arbor June 15, 2007 University of California, San Diego, CA, USA
Outline of the optimization of the power Formulation as an externality A decentralized optimal power
: A new perspective Future wireless communication High demand high competition Quality of Service (QoS) satisfaction Interference control More flexibility at users ends Decentralized power control In mesh networks centrally operated control adds on infrastructure and latency Decentralized power control draws interest Decentralized resource - well studied in Mathematical Economics Shruti Sharma, Demos Teneketzis 3/36 EECS, University of Michigan, Ann Arbor
: A new perspective Future wireless communication High demand high competition Quality of Service (QoS) satisfaction Interference control More flexibility at users ends Decentralized power control In mesh networks centrally operated control adds on infrastructure and latency Decentralized power control draws interest Decentralized resource - well studied in Mathematical Economics Mathematical Economics - to study power Shruti Sharma, Demos Teneketzis 3/36 EECS, University of Michigan, Ann Arbor
Literature survey Classification of power Configuration of the network Cellular uplink/downlink network Mesh/Ad-hoc network Interference control Interference temperature constraint (ITC) Application Voice vs. data network Fixed rate vs. elastic data rate network Cellular uplink Cellular downlink Mesh network Shruti Sharma, Demos Teneketzis 4/36 EECS, University of Michigan, Ann Arbor
Contribution of our work Main Contribution Formulation of power for wireless mesh networks as a resource with externalities Characteristics of Philosophically similar to Laffont and St. Pierre s of Economies with externalities Allows us to appropriately modify an of Lions & Temam for static decentralized optimization so as to obtain optimal power Shruti Sharma, Demos Teneketzis 5/36 EECS, University of Michigan, Ann Arbor
The model R 1 h 31 h 11 h 21 T 1 p 1 P 1 = [0, P max 1 ] R 3 R 2 T 2 p 2 P 2 = [0, P max 2 ] T 3 p 3 P 3 = [0, P max 3 ] M transmitter receiver pairs (Users), M := {1, 2,..., M} Transmissions of a user create interference to other users Interference depends on the transmission powers Performance determined by utilities: U i (p) = U i (p 1, p 2,..., p M ), i M Shruti Sharma, Demos Teneketzis 6/36 EECS, University of Michigan, Ann Arbor
The model (cont ) Interference Temperature Constraint (ITC) R 1 h 31 h 11 h 21 R 2 h T 101 1 h MC1 301 p 1 P 1 = [0, P max 1 ] h 201 R 3 T 2 p 2 P 2 = [0, P max 2 ] T 3 p 3 P 3 = [0, P max 3 ] Interference Temperature (IT): Net radio frequency (RF) power measured at a receiving antenna per unit bandwidth ITC: A measure to keep the RF noise floor below a safe threshold MX p i h i01 P 1 i=1 Multiple ITCs: K measurement centers (MCs)/ Users 0 K := {0 1, 0 2,..., 0 K } Shruti Sharma, Demos Teneketzis 7/36 EECS, University of Michigan, Ann Arbor
Assumptions Information available to the users R 1 h 31 h 11 h 21 R 2 h T 101 1 h MC1 301 p 1 P 1 = [0, P max 1 ] h 201 R 3 T 2 p 2 P 2 = [0, P max 2 ] T 3 p 3 P 3 = [0, P max 3 ] User i M P i = [0, P max i ] Utility U i User 0 k, k K (MC k ) Channel gains h j0k, j M Common knowledge P = [0, P max ] i M P i # of active users M M remains constant Shruti Sharma, Demos Teneketzis 8/36 EECS, University of Michigan, Ann Arbor
Assumptions (cont ) Assumption on utility functions For all i M, U i (p) from R M into R is a non-negative, strictly concave, continuous function of p. U i (p : p i = 0) = 0, i M The utilities of the MCs are zero i.e. U 0k (p) = 0, k K. The utilities remain fixed throughout the power period. Shruti Sharma, Demos Teneketzis 9/36 EECS, University of Michigan, Ann Arbor
The optimization Objective Allocate transmission powers to the users to maximize the social welfare function for the wireless mesh network considered. Problem (A) subject to: p S := {p max p X U i (p) = max p i M 0 K U(p) (1) MX p i h i0k P k, k K, p i P i i M} (2) i=1 For each i M the utility function U i ( ) is known only to user i. (3) P i is known only to user i. (4) A set P = [0, P max ] i M P i is common knowledge. (5) For each k K, MC k knows the channel gains h j0k, j M. (6) i M, U i (p) : R M R is a non-negative, strictly concave, continuous function of p and U i (p : p i = 0) = 0 (7) U 0k (p) = 0, k K (8) Shruti Sharma, Demos Teneketzis 10/36 EECS, University of Michigan, Ann Arbor
Corresponding centralized Centralized Problem (A C ) subject to: max p p S := {p X U i (p) = max p i M 0 K U(p) (9) MX p i h i0k P k, k K, p i P i i M} (10) i=1 i M, U i (p) : R M R is a non-negative, strictly concave, continuous function of p and U i (p : p i = 0) = 0 (11) U 0k (p) = 0, k K (12) Note: Problem (A C ) has a unique optimum. Shruti Sharma, Demos Teneketzis 11/36 EECS, University of Michigan, Ann Arbor
Corresponding centralized Centralized Problem (A C ) subject to: max p p S := {p X U i (p) = max p i M 0 K U(p) (9) MX p i h i0k P k, k K, p i P i i M} (10) i=1 i M, U i (p) : R M R is a non-negative, strictly concave, continuous function of p and U i (p : p i = 0) = 0 (11) U 0k (p) = 0, k K (12) Note: Problem (A C ) has a unique optimum. Objective To solve Problem (A) and obtain, if possible, the optimal solution of Problem (A C ) Satisfy the constraints imposed by the decentralized system Shruti Sharma, Demos Teneketzis 11/36 EECS, University of Michigan, Ann Arbor
of the optimization F1) Problem (A) is a decentralized optimization where the information is distributed among the users according to Assumptions (3) (6). Users need to communicate and exchange information with one another; Make decisions based on information available to them; Eventually reach an optimal. Assumption Design an mechanism that achieves the above goal. Users will follows the rules of the mechanism for communication and decision making. (Realization Theory point of view.) E π A µ h Mechanism: (M, µ, h) h(µ(e)) π(e) e E M Shruti Sharma, Demos Teneketzis 12/36 EECS, University of Michigan, Ann Arbor
of the optimization (cont ) F2) The utility a user receives from power depends not only on the power allocated to it but also on the powers allocated to all other users. Shruti Sharma, Demos Teneketzis 13/36 EECS, University of Michigan, Ann Arbor
of the optimization (cont ) F2) The utility a user receives from power depends not only on the power allocated to it but also on the powers allocated to all other users. Market mechanisms cannot achieve optimal solution of the. [S. Reichelstein, Information and incentives in economic organizations, Ph.D. dissertation, Northwestern University, Evanston, IL, June 1984.] Shruti Sharma, Demos Teneketzis 13/36 EECS, University of Michigan, Ann Arbor
Formulation as an externality The ingredients Shruti Sharma, Demos Teneketzis 14/36 EECS, University of Michigan, Ann Arbor
Formulation as an externality The ingredients (cont ) Technically possible power profiles S i := {p p i P i, p i P M 1 }, i M Shruti Sharma, Demos Teneketzis 15/36 EECS, University of Michigan, Ann Arbor
Formulation as an externality The ingredients (cont ) Technically possible power profiles S i := {p p i P i, p i P M 1 }, i M Shruti Sharma, Demos Teneketzis 16/36 EECS, University of Michigan, Ann Arbor
Formulation as an externality The ingredients (cont ) Technically possible power profiles S i := {p p i P i, p i P M 1 }, i M k semi-feasible power profiles S 0k := {p P M i=1 p i h i0k P k, p i P i}, k K Shruti Sharma, Demos Teneketzis 17/36 EECS, University of Michigan, Ann Arbor
Formulation as an externality The ingredients (cont ) Technically possible power profiles S i := {p p i P i, p i P M 1 }, i M k semi-feasible power profiles S 0k Feasible power profiles := {p P M i=1 p i h i0k P k, p i P i}, k K S = T i M 0 K S i = {p P M i=1 p i h i0k P k, k K, p i P i i M} Shruti Sharma, Demos Teneketzis 17/36 EECS, University of Michigan, Ann Arbor
The externality for decentralized power (A modification of Lions-Temam (1971) ) 0) Initialization: Users (including users 0 K ) agree upon a common power profile p (0) {p p i P i} (13) Shruti Sharma, Demos Teneketzis 18/36 EECS, University of Michigan, Ann Arbor
The externality (cont ) 0) Initialization (cont ): A sequence of modification parameters {τ (n) } n=1 is chosen that satisfies, The counter n is set to 0. lim N σ(n) = 0 < τ (n+1) τ (n), n 1 (14) lim n = 0 (15) NX lim τ (n) = (16) N n=1 Example τ (n) = 1 n or τ (n) = 1 n for real n satisfies (14) (16). Shruti Sharma, Demos Teneketzis 19/36 EECS, University of Michigan, Ann Arbor
The externality (cont ) 1) n th iteration: (Individual optimization) User i, i = 1, 2,..., M, solves ˆp (n+1) i = argmax p Si U i (p) 1 τ (n+1) p p(n) 2 (17) MC i (user 0 i ), i = {0 1, 0 2,..., 0 K }, solves ˆp (n+1) i = argmax p S0k 1 τ (n+1) p p(n) 2 (18) Individual optimals ˆp (n+1) i i are broadcast to all the users. Shruti Sharma, Demos Teneketzis 20/36 EECS, University of Michigan, Ann Arbor
The externality (cont ) 2) Calculation of user and time averages Upon receiving ˆp (n+1) i i, users compute for (n + 1)th iteration p (n+1) 1 = M + K X i M 0 K ˆp (n+1) i (19) Shruti Sharma, Demos Teneketzis 21/36 EECS, University of Michigan, Ann Arbor
The externality (cont ) p (n+1) is used as a reference point in the (n + 1)th iteration. Shruti Sharma, Demos Teneketzis 22/36 EECS, University of Michigan, Ann Arbor
The externality (cont ) 2) (cont ) User i, i M 0 K, also computes the weighted averages ŵ (N+1) i = = where σ (N+1) = N+1 1 X σ (N+1) n=1 1 σ (N+1) (n) τ (n)ˆp i, i M 0 K σ (N) ŵ (N) i + τ (N+1) (N+1)ˆp i, (20) N+1 X τ (n) = σ (N) + τ (N+1) (21) n=1 The average calculated in (20) is stored in users memories. The counter n is increased to n + 1 and the process repeats from Step 1). For (n + 1)th iteration, τ (n+2) τ (n+1) is chosen from the predefined sequence in Step 0). Shruti Sharma, Demos Teneketzis 23/36 EECS, University of Michigan, Ann Arbor
Convergence to optimal solution Theorem The externality results in a power which is the unique global optimum of the centralized (A C ). The optimal is obtained as the limit of the sequences {ŵ (N) i } N=1, i M 0 K, all of which converge to the same limit. Shruti Sharma, Demos Teneketzis 24/36 EECS, University of Michigan, Ann Arbor
Investigated the decentralized power for a wireless mesh network with multiple ITCs. Formulated the power from an externality perspective. Proposed a decentralized for solving it. The proposed obtains a globally optimal power. It provides a guaranteed convergence to the optimum solution. It satisfies the informational constraints of the. Shruti Sharma, Demos Teneketzis 25/36 EECS, University of Michigan, Ann Arbor
Modifying the present to make it applicable for more general class of utility functions. Extending the analysis for time-varying channels. Designing mechanisms that implement optimal centralized power s in Nash Equilibria. Shruti Sharma, Demos Teneketzis 26/36 EECS, University of Michigan, Ann Arbor
THANK YOU! Shruti Sharma, Demos Teneketzis 27/36 EECS, University of Michigan, Ann Arbor
Questions? Reference (1) S. Sharma and D. Teneketzis, An externality based decentralized optimal power scheme for wireless mesh networks, Control Group Report CGR-07-02, Department of EECS, University of Michigan, Ann Arbor. (Submitted for IEEE journal publication.) Shruti Sharma, Demos Teneketzis 28/36 EECS, University of Michigan, Ann Arbor
Proof of Theorem Proof of Theorem For simplicity, we prove the Theorem only for a single ITC. We call the single MC in this case as user 0 and associate with it, a semi-feasible set S 0 MX S 0 := {p p i h i0 P, p i P i} (22) Since ˆp (n+1) i i=1 is the optimal solution of, ˆp (n+1) i = argmax p Si U i (p) 1 τ (n+1) p p(n) 2, i M {0} (23) it follows that, p S i, i M {0}, τ (n+1) U i (ˆp (n+1) i ) ˆp (n+1) i p 2 + p (n) p 2 ˆp (n+1) i p (n) 2 τ (n+1) U i (p) (24) Adding (24) over all i gives, p S = i M {0} S i, τ (n+1) M X U i (ˆp (n+1) i ) MX ˆp (n+1) i p 2 + (M + 1) p (n) p 2 i=0 i=0 MX MX ˆp (n+1) i p (n) 2 τ (n+1) U i (p) = τ (n+1) U(p) (25) i=0 i=0 Shruti Sharma, Demos Teneketzis 29/36 EECS, University of Michigan, Ann Arbor
Proof of Theorem (cont ) Proof of Theorem Since 2 is a convex function, p (n+1) p 2 1 M + 1 MX i=0 ˆp (n+1) i p 2 (26) Replacing the second term in (25) using (26), adding over n = 0, 1,..., N 1, and dividing by M + 1 we obtain, p S, N 1 1 X τ (n+1) M + 1 n=0 1 M + 1 MX N 1 X i=0 n=0 M X i=0 By concavity of U i (p) in p, 1 M + 1 MX N 1 X i=0 n=0 U i (ˆp (n+1) i ) p (N) p 2 ˆp (n+1) i p (n) 2 σ(n) M + 1 U(p) p(0) p 2 (27) τ (n+1) U i (ˆp (n+1) i ) σ(n) M + 1 MX i=0 Since S i, i M {0}, is a convex set and ˆp (n+1) i ŵ (N) i = 1 τ (k)ˆp (k) i S i, i M {0}. U i (ŵ (N) i ), i M {0} (28) S i n, it follows that P N σ (N) k=1 Shruti Sharma, Demos Teneketzis 30/36 EECS, University of Michigan, Ann Arbor
Proof of Theorem (cont ) Proof of Theorem Substituting (28) in (27) and multiplying by (M + 1)/σ (N) we get, p S, MX U i (ŵ (N) i ) M + 1 MX N 1 σ (N) p (N) p 2 1 X σ (N) ˆp (n+1) i p (n) 2 i=0 i=0 n=0 U(p) M + 1 σ (N) p (0) p 2 (29) S i, i M {0}, and S are compact, numerators of the 2nd terms on both LHS and RHS of (29) are bounded. The 2nd terms go to zero as N, 1/σ (N) 0 as N. Numerator of the 3rd term on the LHS of (29) grows with N. S i, i M {0}, is compact, ŵ (N) i S i, and U i ( ) is a continuous function on S i, C 0, C 1 : N, ŵ (N) i C 0, i M {0} U i (ŵ (N) i ) C 1, i M {0} (30) the 1st terms on both the LHS and the RHS of (29) are bounded, and so the third term on the LHS. Shruti Sharma, Demos Teneketzis 31/36 EECS, University of Michigan, Ann Arbor
Proof of Theorem (cont ) Proof of Theorem C 2 : N N 1 1 X σ (N) ˆp (n+1) i p (n) 2 C 2 (31) n=0 S i is compact, (30) a subsequence {ŵ (N ) i } (N) N of ŵ =1 i, such that ŵ (N ) i Next we show that, ŵ i S i, i M {0}. 1 The subsequence {w (N 1) } N =1 of w (N) converges to a limit w. 2 ŵ i = ŵ j = w i, j M {0}. Lemma lim ŵ (N N i ) w (N 1) 2 = ŵ i w 2 = 0, i Shruti Sharma, Demos Teneketzis 32/36 EECS, University of Michigan, Ann Arbor
Proof of Theorem (cont ) Proof of Lemma Proof of Theorem We must show that, i, ɛ > 0, N 0 : N N 0, ŵ (N ) i w (N 1) 2 ɛ, (32) τ (n) 0, we can find an n 0 such that, τ (n0) ɛ (33) 2C 2 Knowing n 0 we can compute the finite quantity, n 0 1 X A 0i = ˆp (n+1) i p (n) 2 (34) n=0 σ (N) as N, N 0 i large enough such that, σ (N) is increasing in N, σ (N 0 i ) 2τ (1) A 0i ɛ (35) N N 0 = max N 0 i i, σ (N ) σ (N 0 ) σ (N 0 ) i i (36) Shruti Sharma, Demos Teneketzis 33/36 EECS, University of Michigan, Ann Arbor
Proof of Theorem (cont ) Proof of Lemma (cont ) Proof of Theorem τ (n+1) τ (n) n, 1 σ (N ) N X 1 n=0 τ (n+1) ˆp (n+1) i p (n) 2 τ (1) n 0 1 X ˆp (n+1) σ (N ) i p (n) 2 + τ (n N 0) X 1 ˆp (n+1) σ (N ) i p (n) 2 (37) n=0 n=n 0 Substituting (31) and (34) in (37) and using (36) we get, ŵ (N ) i w (N 1) 2 1 σ (N 0 ) τ (1) A 0i + τ (n 0) C 2 The second inequality in (38) follows from (33) and (35). ɛ 2 + ɛ 2 = ɛ (38) This proves the Lemma. Shruti Sharma, Demos Teneketzis 34/36 EECS, University of Michigan, Ann Arbor
Proof of Theorem (cont ) Proof of Theorem Subsequences ŵ (N ) i, i M {0}, converge to the same limit ŵi = w. ŵi S i, i M {0}, w S = T i M {0 1,0 2,...,0 K } S i. w is a feasible solution for Problem (A). Next we show that, w is an optimal solution of Problem (A). Since 2 is a convex function, for i M {0}, ŵ (N ) i w (N 1) 2 1 N X 1 τ (n+1) ˆp (n+1) σ (N ) i p (n) 2 n=0 1 N X 1 ˆp (n+1) σ (N ) i p (n) 2 for τ (n+1) 1 (39) n=0 Substituting (39) in (29) and taking the limit as N, Shruti Sharma, Demos Teneketzis 35/36 EECS, University of Michigan, Ann Arbor
Proof of Theorem (cont ) Proof of Theorem or lim { X M ŵ (N ) N i w (N 1) 2 + i=0 MX i=0 U i (ŵ (N ) i )} U(p) MX U i (w ) U(p) (40) i=0 The inequality in (40) follows from (38). This proves the optimality of w and the optimality of the externality for decentralized power. Shruti Sharma, Demos Teneketzis 36/36 EECS, University of Michigan, Ann Arbor