Interference Calculus

Transcription

1 Interference Calculus Part I: Axiomatic Characterization of Interference in Wireless Networks Part II: Algorithms for Resource Allocation Martin Schubert joint work with Holger Boche e for Telecommunications, Berlin, Germany Fraunhofer German-Sino Lab for Mobile Communications (MCI) Heinrich Hertz Chair for Mobile Communications Technical University of Berlin MCI Fraunhofer German Sino Lab Mobile Communications

2 Outline 1 Introduction and Motivation Interference in Wireless Systems: The Beamforming Example Joint Beamforming and Power Allocation SIR Balancing and Utility Optimization 2 Representation and Classification of Interference Functions General Interference Functions Concave Interference Functions Convex Interference Functions Log-Convex Interference Functions 3 SIR-Constrained Power Minimization 4 Utility Optimization Strategies 5 Cooperative Game Theory 6 Conclusions

4 Improving the Spectral Efficiency of Wireless Systems Motivation from information theory: the system performance is generally maximized by tolerating interference (in a controlled way) the system can no longer be regarded as a collection of point-to-point links techniques that were originally designed for wireline networks do not necessarily perform well in a wireless context some interference-related issues: resource allocation interference mitigation adaptivity cooperation and interference coordination

6 Example: Interference Mitigation by Beamforming user 1 user K Array x (t) 1 x (t) 2 x (t) 3 x (t) M downconv. + ADC channel estimation w (1) 1 w (2) 1 w (3) 1 w beamforming received array signal: x = K h k s k + n k=1 output of the kth beamformer: y k = w H k x = wh k } {{ h ks k + } l k desired signal (M) 1 user 1 w H k h ls l } {{ } interference y 1 (t) y K (t) user K + w H k n }{{} noise

7 Signal-to-Interference Ratio (SIR) Signal-to-Interference(-plus-Noise) Ratio: E [ w H k h ks k 2] E [ w H k (x h ks k ) 2] = w H k p k w H k h k 2 ( σ 2 n I + l k p lh l h H l useful power interference+noise power ) wk where p k = E[ s k 2 ] is the transmit power of user k this is maximized by w k = ( σni 2 + l k p lh l h H ) 1 l hk SIR max k (p) = p k h H ( k σ 2 n I + p l h l h H ) 1hk l l k

8 Max-SINR Beamforming ( Optimum Combining ) signal-to-interference-(plus-noise) ratios (SIR) depend on the Tx power allocation SIR 5 p = [p 1, p 2,..., p } {{ K, σn 2 } Tx powers }{{} noise ] T SIR 4 residual interference: I k (p) = h H k 1 ( σ 2 n I + l k p lh l h H l ) 1hk SIR 1 SIR 2 SIR 3 the function I k (p) has a nice analytical structure (concave, non-negative,monotonic,... ), which has facilitated many interesting results and algorithms in the past is there a more general underlying concept?

9 Spatial Matched Filter the matched filter w k = h k / h k 2 is a single user receiver assuming quasi-static channels, we have a constant link gain matrix V = [v 1,..., v K ] T 0 the interference of the kth user is I k (p) = p T v k this linear interference function has a longstanding tradition in power control theory [Aein 73]

10 General Interference Functions Definition We say that I : R K + R + is an interference function if it fulfills the axioms: A1 (non-negativeness) I(p) 0 A2 (scale invariance) I(αp) = αi(p) α R + A3 (monotonicity) I(p) I(p ) if p p this framework generalizes the framework of standard interference functions [Yates 95] the beamforming example is a special case of this framework

12 The Downlink Power Minimization Problem joint optimization of beamformers u 1,..., u K and powers p = [p 1,..., p K ], for given channels h 1,..., h K problem formulation: achieve SIR targets γ = [γ 1,..., γ K ] with minimum total power: min p>0,u 1,...,u K s.t. K i=1 p i p k u H k h k 2 l k p l u H l h k 2 + σn 2 γ k, u k = 1 k = 1,..., K

13 Different Approaches to Downlink Beamforming [Rashid-Farrokhi/Tassiulas/Liu,1998] fixed-point iteration general approach [Bengtsson/Ottersten,1999] semidefinite programming, interior-point algorithms exploits the special structure of the beamforming problem convexity is commonly considered as the dividing line between easy and difficult problems. special properties of interference functions (axioms A1-A3) also enable easy solutions

14 Reformulation Based on Interference Functions exploiting uplink/downlink duality, the problem can be rewritten in terms of uplink interference functions min p>0 K p l s.t. p k γ k I k (p), k = 1, 2,..., K, l=1 where I k (p) = h H k 1 ( σ 2 n I + l k p lh l h H l ) 1hk. the special structure of I k (p) can be exploited (a globally convergent algorithm will be discussed in part II)

16 The Linear Interference Model transmit power vector p = [p 1,..., p K ] T ( power allocation ) non-negative irreducible link gain matrix transmitter k link gain V kk receiver k V = [v 1,..., v K ] T 0 interference of the kth user: gains V kl, l k I k (p) = p T v k

17 Irreducibility V is reducible the directed graph is not fully connected 0 V 12 V 13 V 14 V = 0 0 V 23 V V V V 12 V 24 V 13 V is irreducible the directed graph is fully connected 0 V 12 V 13 V 14 V = 0 0 V 23 V V 34 V V 14 V 34 3 V V 41 V 12 V 24 V 13 V 34 3 V 23

18 SIR Balancing and Perron-Frobenius Theory SIR feasible set: S = {γ : ρ(γv) 1} where Γ = diag{γ} and ρ(γv) = inf max [ΓVp] k p>0 k p k illustrating example: eigenvalues of the non-neg. matrix ΓV in the complex plane Perron root ρ(γv ) (Collatz/Wielandt) if V is irreducible, then the SIR balancing problem is solved by the unique principal eigenvector associated with the spectral radius ΓVp = ρ(γv)p

19 The SIR Feasible Region of the Linear Model observation: the function ρ(γ) := ρ(γv) = ρ(diag{γ}v) is an interference function SIR requirements γ are jointly feasible if ρ(γ) 1 the SIR region is defined as S = {γ R K + : ρ(γ) 1} γ 2 feasible ρ(γ) 1 infeasible ρ(γ) > 1 later, it will be shown that every SIR region is a sub-level set of an interference function interference calculus is not restricted to power control problems, another application is the analysis of performance tradeoff regions γ 1

20 SIR Balancing with Adaptive Beamforming [Gerlach/Paulraj 96] have studied the problem of maximizing the minimum SIR (also referred to as SIR balancing) ( p ) k max min p>0,u 1,...,u K 1 k K l k p l u H l h k 2 s.t. u H k h k 2 = 1 [Montalbano/Slock 98]: uplink/downlink duality leads to the problem of Perron root optimization ( [ γ1 0 ] ) ρ opt (γ) = min ρ... V(u) u={u 1,...,u K } 0 γ K where ρ( ) is the spectral radius ( Perron root ) and V(u) is a beamformer-dependent coupling matrix

21 SIR Balancing Reformulation the Perron root minimization problem can be rewritten as γ k I k (p) ρ opt (γ) = inf max p>0 1 k K p k where I k (p) = min p T v k (u k ) u k the optimum ρ opt (γ) is a single criterion for the joint quality of all K users a globally convergent algorithm can be derived by exploiting that ρ opt (γ) is a concave interference function (in part II)

22 Utility Optimization maximize the weighted sum utility: I U (w) = max u U K w k u k k=1 I U (w) is a convex interference function u 2 max u Pk w ku k utility region U u 1

23 Discussion The examples show that there exist many different types of interference functions, which are useful for different applications, e.g., physical layer modeling resource allocation fairness Is there a unifying framework for interference functions? Next, we discuss the structure of different classes of interference functions and applications

26 General Interference Functions Definition We say that I : R K + R + is an interference function if it fulfills the axioms: A1 (non-negativeness) I(p) 0 A2 (scale invariance) I(αp) = αi(p) α R + A3 (monotonicity) I(p) I(p ) if p p this framework generalizes the framework of standard interference functions [Yates 95] the beamforming example is a special case of this framework

27 Example: Robust Nullsteering interference can be reduced by nullsteering beamforming: assume that the interference direction is only known up to an uncertainty c from a region C array gain worst-case interference interference direction h incertainty region C beampattern u the beamformer u minimizes the worst-case interference power: ( I(p) = min max p l u H h l (c) 2) u =1 c C this is also an interference function (A1 A3 fulfilled) l angle of arrival

28 The Impact of Noise in order to include noise, we consider the extended power [ p ] allocation p = where the last component p p K+1 K+1 stands for noise power an additional strict monotonicity property is required: p K+1 > p K+1, with p p I(p) > I(p ) If p K+1 > 0 is constant, and, then I( p) is standard as defined in [Yates 95]

29 Fixed-Point Iteration For standard interference functions it was shown [Yates 95] If target SIR γ = [γ 1,..., γ K ] are feasible, i.e., C(γ) 1, under a sum-power constraint, then for an arbitrary initialization p (0) 0, the iteration p (n+1) k = γ k I k (p (n) ), k = 1, 2,..., K converges to the optimum of the power minimization problem K inf p k p>0 k=1 s.t. p k I k (p) γ k, k,

30 Properties of the Fixed-Point Iteration The fixed-point iteration has the following properties: component-wise monotonicity optimum achieved iff = γ k I k (p (n) ), k p (n+1) k optimizer lim n p (n) is unique x 10 4 convergence to the optimal power levels C(γ) =

31 The Weighted Max-Min Optimum C(γ) example: K interference functions I 1,..., I K and weighting factors γ = [γ 1,..., γ K ] (e.g. SIR requirements). The optimum of the weighted SIR balancing problem is SIR feasible region ( C(γ) = inf p>0 max 1 k K S = {γ : C(γ) 1} γ k I k (p) p k ) level set of the interference function C(γ)

32 Comparison of Min-Max and Max-Min Balancing alternative approach to SIR balancing: ( c(γ) = sup p>0 min 1 k K γ k I k (p) p k ) ( = sup p>0 in general, c(γ) C(γ). fairness gap min 1 k K γ ) k γ k (p) consider a fixed coupling matrix V R K K +. Under special conditions (V irreducible), both max-min and min-max fairness equal the spectral radius ρ: C(γ) = ρ(diag{γ}v) = inf max γ k [Vp] k p>0 k p k = c(γ)

33 Discussion C(γ) and c(γ) are constructed by underlying interference functions I 1,..., I K certain important operations are closed within the framework of interference functions for all general interference functions (only the basic properties A1 A3 fulfilled) algorithmic solutions exist (e.g. fixed point iteration) but more efficient solutions can be designed by exploiting the structure of the interference functions

34 Representation of General Interference Functions Theorem Let I be an arbitrary interference function, then I(p) = min max p k ˆp L(I) k ˆp k = max min ˆp L(I) k I(p) can always be represented as the optimum of a weighted max-min (or min-max) optimization problem The weights ˆp are elements of convex/concave level sets p k ˆp k L(I) = {ˆp > 0 : I(ˆp) 1} L(I) = {ˆp > 0 : I(ˆp) 1}

35 Interference Functions and Utility/Cost Regions ˆp 2 the set L(I) is closed bounded and monotonic decreasing L(I) ˆp ˆp ˆp, ˆp L(I) = ˆp L(I) ˆp 1 the set L(I) is closed and monotonic increasing ˆp 2 L(I) ˆp ˆp ˆp, ˆp L(I) = ˆp L(I) ˆp 1 every interference function can be interpreted as a utility/cost resource allocation problem

37 Concave Interference Functions Definition We say that I : R K + R + is a concave interference function if it fulfills the axioms: A1 (non-negativeness) I(p) 0 A2 (scale invariance) I(αp) = αi(p) α R + A3 (monotonicity) I(p) I(p ) if p p C1 (concavity) I(p) is concave on R K +

38 Examples for Concave Interference Functions beamforming: I k (p) = h H k 1 ( σ 2 n I + l k p lh l h H l ) 1hk generalization: receive strategy z k I k (p, σ 2 n) = min z k Z k ( p T v(z k ) } {{ } Interference + σnn 2 ) k (z k ), k = 1, 2,..., K } {{ } Noise

39 Cost/Loss Minimization minimize the weighted sum cost: u 2 utility region U I U (w) = max u U K w k u k k=1 I U (w) is a concave interference function w max u Pk w ku k u 1

40 Representation of Concave Interference Functions Theorem Let I(p) be an arbitrary concave interference function, then I(p) = min w N 0 (I) k=1 K w k p k, for all p > 0. where N 0 (I) = {w R K + : I (w) = 0} ( and I K ) (w) = inf p>0 l=1 w lp l I(p) is the conjugate of I.

41 Interpretation of Concave Interference Functions I(p) = the set N 0 (I) is closed, convex, and monotonic increasing, i.e., w N 0 (I) implies w w belongs to N 0 (I) min w N 0 (I) k=1 K w k p k any concave interference function can be interpreted p as the solution of a loss/cost minimization problem w 1 w 2 region N 0 (I)

43 Convex Interference Functions Definition We say that I : R K + R + is a convex interference function if it fulfills the axioms: A1 (non-negativeness) I(p) 0 A2 (scale invariance) I(αp) = αi(p) α R + A3 (monotonicity) I(p) I(p ) if p p C2 (convexity) I(p) is convex on R K +

44 Example: Robustness Another example is the worst-case model I k (p) = max c k C k p T v(c k ), k, where the parameter c k models an uncertainty (e.g. caused by channel estimation errors or system imperfections). the optimization is over a compact uncertainty region C k I k (p) is a convex interference function

45 Utility Maximization maximize the weighted sum utility: I U (w) = max u U K w k u k k=1 I U (w) is a convex interference function u 2 max u Pk w ku k utility region U u 1

46 Representation of Convex Interference Functions Theorem Let I(p) be an arbitrary convex interference function, then I(p) = max w W 0 (I) k=1 K w k p k, for all p > 0. where W 0 (I) = {w R K + : Ī (w) = 0} ( K ) and Ī (w) = sup p>0 l=1 w lp l I(p) is the conjugate of I.

47 Interpretation of Convex Interference Functions I(p) = max w W 0 (I) k=1 K w k p k the set W 0 (I) is closed, convex, and monotonic decreasing, i.e., w W 0 (I) implies w w belongs to W 0 (I) any convex interference function can be interpreted as the solution of a utility maximization problem w 2 W 0 (I) p w 1

49 Log-Convex Interference Functions Definition We say that I : R K + R + is a log-convex interference function if it fulfills the axioms: A1 (non-negativeness) I(p) 0 A2 (scale invariance) I(αp) = αi(p) α R + A3 (monotonicity) I(p) I(p ) if p p C3 (log-convexity) I k (e s ) is log-convex on R K

50 Log-Convexity Let f (s) := I(exp{s}). The function f : R K R + is said to be log-convex on R K if log f is convex, i.e., log f ( (1 λ)ŝ+λš ) (1 λ) log f (ŝ)+λ log f (š), λ (0, 1), ŝ, š R K taking exp on both sides, this is equivalent to [e.g. Boyd/Vandenbergh] f ( (1 λ)ŝ + λš ) f (ŝ) 1 λ f (š) λ

51 Example Let I 1,..., I K be log-convex interference functions, then the SIR-balancing optimum ( C(γ) = inf p>0 max 1 k K is a log-convex interference function γ k I k (p) p k )

52 Basic Properties the properties of log-convex interference functions are preserved under certain operations example: let I 1 and I 2 be log-convex interference functions, and I (p) = α 1 I 1 (p) + α 2 I 2 (p), α 1, α 2 R + I (p) = ( I 1 (p) ) α ( I1 (p) ) 1 α, α [0, 1] I (p) and I (p) are log-convex interference functions (note: this is not valid for log-concave interference functions) log-convex interference functions have a rich analytical and algebraic structure.

53 Representation of Log-Convex Interference Functions Theorem Every log-convex interference function I(p), with p > 0, can be represented as I(p) = ( max f I (w) w L(I) K ) (p l ) w l. l=1 where I(p) f I (w) = inf p>0 K l=1 (p, w R K +, l) w l L(I) = { w R K + : f I (w) > 0 } k w k = 1

54 Connection between Convex and Log-Convex Functions every convex function I(p) can be expressed as I(p) = max w W 0 log k w ke s k is convex = log max w W0 k w ke s k is convex = I(e s ) is log-convex k w k p k if I(p) is convex then I(e s ) is log-convex (but the converse is not true)

55 Categories of Interference Functions convex interference functions concave interference functions log-convex interference functions general interference functions

57 Fixed-Point Iteration consider the power minimization problem K inf p k p>0 k=1 s.t. p k I k (p) γ k, k, with feasible) target SIR γ = [γ 1,..., γ K ] and interference functions I 1,..., I K. it was shown [Yates 95] that the global minimum is achieved by the fixed-point iteration p (n+1) k = γ k I k (p (n) ), k = 1, 2,..., K

58 Concave Interference Functions and Receive Strategies let I 1,..., I K be arbitrary concave interference functions from the representation result, it is clear that I k (p) has a matrix-based structure I k (p) = min z k Z k ( p T v(z k ) } {{ } Interference ) + n k (z k ), k = 1, 2,..., K } {{ } Noise the parameter z k can be interpreted as a receive strategy for K users, we have an interference coupling matrix V(z) = [v 1 (z 1 ),..., v K (z K )] T

59 Example: MMSE Beamforming The interference coupling V can depend on adaptive receive beamforming vectors u 1,..., u K, with u k = 1. In this case, the normalized coupling matrix V(u) is defined as u H k R l u k [V(u)] kl = u H k R l k, ku k where R l = E[h l h H l ] 0 k = l. Under this model, we have an interference function ( [V(u) I k (p, σn) 2 ] = min p u k =1 k + σ2 ) n u H k R ku k The function (-1) fulfills A1 A3 and is concave. For every p, u 1,..., u K are the respective MMSE beamformers (-1)

60 Example: Zeroforcing beamforming Under a different normalization u H k R ku k = 1 we have I k (p, σn) 2 [ ] = min V(u) p u H k R k + σ2 n u k 2. (0) ku k =1 Assuming that K is less or equal to the number of antennas, we can introduce the constraint u H k h l = 0, l k. I k (p, σ 2 n) = min u H k R ku k =1 u H k h l =0,l k u k 2 σ 2 n. (1) This is solved by the well-known least squares zeroforcer. The function (1) is a concave interference function (though a trivial one since I k (p, σ 2 n) does no longer depend on p).

61 Example: Base Station Assignment Consider the problem of combined beamforming and base station assignment [(?;?;?;?)]. The kth user is received by a base station with index b k B k. I k (p, σ 2 n) = min b k B k ( min u k : u k =1 u H k ( l k p lr (b k) l + σni 2 ) u k u H k R(b k) k u k This is a concave interference function which fulfills A1 A3. ).

62 Proposed Matrix-Based Iteration By exploiting the special structure of concave interference functions, a new iteration is obtained: Alternating optimization of receive strategies z (n) and power allocation p (n) ] 1 z (n) k = arg min zk Z k [V(z)p (n) + n(z), k {1, 2,..., K} 2 p (n+1) = (I ΓV(z (n) )) 1 Γn(z (n) ) k

63 Convergence Behavior 3.5 power (5 users) improved algorithm that exploits concavity fixed point iteration iterations

64 Required Iteration Steps vs. System Load 1000 the convergence behavior of the proposed iteration is almost independent of the system load iterations fixed-point iteration proposed algorithm spectral radius min z Z ρ(ψ(z))

65 Direct Step-by-Step Comparison fixed-point: p (n) matrix-based iteration: p (n) p (0) joint initialization p (1) p (2) p (3) f (1) (p (1) ) p (1) f (2) (p (1) ) f (1) (p (2) ) p (2) the proposed iteration has the following advantages step-wise better than the fixed point iteration, p (n) p (n) achieves the SI(N)R targets Γ in each step componentwise monotonicity both iterations converge to the global optimum of the power minimization problem p (3)

66 Problem Reformulation approach: introduce auxilliary function d(p) = p ΓI(p) the global optimum of the power minimization problem is completely characterized by d(p) = 0 (fixed-point) d k (p) tangential hyperplane g (n) k (p) global optimum ˆp p (n+1) p (n) p framework can be extended to non-smooth functions d(p)

67 Super-Linear Convergence Theorem Let p (0) be an arbitrary feasible initialization, then the new algorithm has super-linear convergence p (n+1) p lim n p (n) p = 0 quadratic convergence is achieved for the typical case of semi-smooth interference functions (C 2n 1 compared to C n 2 ) the fixed-point iteration achieves only linear convergence in general

69 Resource Allocation can be regarded as the search for an optimal operating point in the quality-of-service (QoS) feasible region total power minimum QoS link 2 max min k QoS k (max-min fairness) max P k α kqos k (weighted sum optimization) Q 2 QoS feasible region Q 1 max P k QoS k (best overall efficiency) QoS link 1 Here, QoS stands for some performance measure which still needs to be specified

70 Proportional Fairness [Kelly 98]: The proportionally fair equilibrium û is the one, at which the difference to any other utility vector u U measured in the aggregated proportional change k (u k û k )/û k is non-positive. This operating point can be found by solving utility user max min k u k (max-min fairness) 15,2 max k log uk (prop. fairness) 16,4 max k uk (utility efficiency) 16,5 max u U K log u k k= utility user 1

71 QoS Model for Wireless Systems signal-to-interference ratio SIR(p) = p k I k (p) the QoS is a strictly monotonic function of the SIR QoS(p) = φ ( SIR(p) ) examples: φ(x) = x SIR φ(x) = log(x) SIR in db φ(x) = 1/(1 + x) Min. Mean Squared Error (MMSE) φ(x) = x α BER slope, diversity order α φ(x) = log(1 + x) capacity for Gaussian signals...

72 Some Examplary Resource Allocation Problems max-min fairness ( min k sup p>0 p k I k (p) ) sum-power minimization K inf p k p>0 k=1 s.t. p k I k (p) γ k, k, total power minimum QoS link 2 max-min fairness Q 2 QoS feasible region Q 1 proportional fairness [Kelly 97] inf K p>0 k=1 prop. fairness log I k(p) p k. efficient algorithmic solutions exist for certain types of interference functions I k (p) QoS link 1

73 Feasible QoS Region for Log-Convex Interference Functions the log-sir feasible region is Q = {q R K : C(exp q) 1} where ( C(exp q) = inf p>0 max 1 k K exp(q k )I k (p) ) p k log-sir feasible region q2 = log γ2 C(exp{q}) < 1 if I 1,..., I K are log-convex, then C(exp q) is a log-convex interference function the log-sir region is a convex set C(exp{q}) = 1 q 1 = log γ 1

74 Extension to Other QoS Measures Let γ k be the inverse function of φ k, then γ k := γ k (q k ) is the minimum SIR level needed to achieve the target q k assume log-convex interference functions I 1,..., I K. The QoS region is convex for all mappings QoS = φ(sir), for which the inverse function γ k (QoS k ) is log-convex. Examples: capacity in the high SNR regime: φ(sir) = α log(sir), with α R. BER slope approximation: φ(sir) = SIR α, for diversity order α 0....

75 Proportional Fairness for Log-Convex Interf. Functions for utilities u k = p k /I k (p) the problem of proportional fairness can be rewritten as PF (I) = inf K p>0 k=1 log I k(p) p k if I 1,..., I K are log-convex interference functions, then this is a convex optimization problem

76 Weighted Proportional Fairness consider weighting factors α 1,..., α K. The weighted proportionally fair optimum is inf s R K K α k g ( I k (e s )/e s ) k k=1 s.t. e s 1 P max Theorem Let I k (e s ) be log-convex k and g monotonic increasing. Then the problem is convex if and only if g(e x ) is convex on R K

78 A Game Theoretic View: Cooperative Bargaining the service qualities of all K users is modeled by a utility vector u = [u 1,..., u k ], chosen from a region U The players try to reach an unanimous agreement on some outcome u d If they fail, the disagreement outcome or disagreement point d results. the solution outcome ϕ(u, d) is the operating point of the system u 2 d 2 U d d 1 bargaining game (U,d) solution outcome ϕ(u, d) u 1

79 Axiomatic Framework for Symmetric Nash Bargaining WPO Weak Pareto Optimality: The players should not be able to collectively improve upon the solution outcome. IIA Independence of Irrelevant Alternatives: If the feasible set shrinks but the solution outcome remains feasible, then the solution outcome for the smaller feasible set should be the same point. SYM Symmetry: If the region is symmetric, then the outcome does only depend on the employed strategies and not on the identities of the users. Axiom SYM basically means that all users have the same priorities. STC Scale Transformation Covariance: The optimization strategy is invariant with respect to a component-wise scaling of the region.

80 The Symmetric Nash Bargaining Solution Let U be convex and u u, u U implies u U. Then, the unique outcome fulfilling the axioms WPO, IIA, SYM, STC, is called symmetric Nash bargaining solution (SNBS). SNBS is equivalent to the solution of max {u U:u>d} k=1 we can assume d = 0, thus max u U K (u k d k ) K k=1 u k

81 Equivalence between SNBS and Proportional Fairness the product optimization approach is equivalent to proportional fairness [Kelly 98] û = arg max u U K k=1 u k = arg max u U log K k=1 u k = arg max u U K log u k k=1 if the region U is convex and monotonic, then symmetric Nash bargaining and proportional fairness are equivalent

82 Representation of the Convex Set U Every convex utility-region U can be represented as a sub-level set of a convex function U = {u R K + : I U1 (u) 1} where I U1 (p) = max w U 1 k=1 K w k p k, p > 0 and U 1 = {u R K + : I U (u) 1} K and I U (p) = max w k p k, p > 0 w U k=1 user 2: u 2 utility region U user 1: u 1

83 Using the Representation of Log-Convex Interf. Functions the product optimum over U is equivalently obtained by computing f IU (w), with w = [1, 1,..., 1] max u U K u k = k=1 1 f IU (w). this shows that the product optimization problem is closely linked with the log-convex structure of U

84 Mid-Point Dominance ϕ D 1 (U, d) is the dictatorial solution for user 1 ϕ D 2 (U, d) is the dictatorial solution for user 2 u 2 {u : u 1 2 ϕd 1 (U,d) ϕd 2 (U,d)} d ϕ D 2 (U,d) the mid-point dominance axiom requires K ϕ(u, d) 1 K ϕ D k (U, d) k=1 minimal amount of cooperation between users U ϕ D 1 (U,d) u 1

85 Disagreement Point Convexity u 2 ϕ(u, d) d U u 1 the disagreement point convexity axiom requires: d(µ) = (1 µ)d + µϕ(u, d) = ϕ(u, d(µ)) = ϕ(u, d) (0 < µ < 1) this models the impact of the user requirements

86 Example of a Non-Standard Characterization of the SNBS Theorem the symmetric Nash bargaining solution is the only solution that satisfies the axioms mid-point dominance and disagreement point convexity advantage: we can analyze the impact of user cooperation and user requirements

88 Conclusions the framework of interference functions is applicable to different areas physical layer design medium access control resource allocation and utility optimization for wireless systems how to operate a wireless system many interesting open questions