Network Calculus. A General Framework for Interference Management and Resource Allocation. Martin Schubert

Transcription

1 Network Calculus A General Framework for Interference Management and Resource Allocation Martin Schubert Fraunhofer Institute for Telecommunications HHI, Berlin, Germany Fraunhofer German-Sino Lab for Mobile Communications (MCI) Heinrich Hertz Chair for Mobile Communications Technical University of Berlin HHI Fraunhofer Institut Nachrichtentechnik Heinrich Hertz Institut MCI Fraunhofer German Sino Lab Mobile Communications

2 Outline 1 Interference Functions 2 Application to SIR-Based Utility Sets 3 Application to Game Theory 4 Structure of Interference Functions and Utility Sets 5 Conclusions 2

3 Outline 1 Interference Functions 2 Application to SIR-Based Utility Sets 3 Application to Game Theory 4 Structure of Interference Functions and Utility Sets 5 Conclusions 3

4 Interference in Multiuser Wireless Networks evolution of wireless networks: high-rate services densely populated user environments interference between users puts a limit on how many users per cell can be served at a certain data rate countermeasure: adaptive signal processing and resource allocation/scheduling strategies 4

5 Example: Multiple Access Channel (MAC)... letting transmitted signals interfere with each other (in a controled way) increases capacity provided that the receivers take into account the multiaccess interference. [Verdu, Fifty Years of Shannon Theory ] capacity user 2 log(1 + p 2 h 2 /σ 2 n) optimal interference aware receivers shared resources single user receivers orthogonal resources capacity user 1 log(1 + p 1 h 1 /σ 2 n) 5

6 Discussion objective: modelling of performance tradeoffs caused by interference in the past, results were mainly derived for special system layouts (e.g. MIMO MAC), are there more general principles? generally difficult due to complicated interdependencies between system functionalities ( cross-layer problem ) chosen approach: abstract framework ( network calculus ) focus on core properties rigorous, allows to handle problems analytically provides intuition and roadmap for implementation 6

7 Some Interference Models classical linear model I k (p, σ 2 n) = v T p + σ 2 n interference in a multiuser MIMO channel with optimum antenna combining I k (p, σ 2 n) = h H k 1 ( σ 2 n I + l k p lh l h H l ) 1hk generalization: adaptive receive strategy z k I k (p, σn) 2 ( = min p T v(z k ) + σ 2 ) z k Z k }{{} nn k (z k ), k = 1, 2,..., K }{{} Interference Noise 7

8 Axiomatic Approach: Interference Functions Definition A function I : R K + R + is said to be an interference function if the following axioms are fulfilled: A1 (positivity) I(p) > 0 if p > 0 A2 (scale invariance) I(αp) = αi(p) α R + A3 (monotonicity) I(p) I(p ) if p p 8

9 Another Example: Robust Nullsteering Beamforming 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 uncertainty 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 9

10 Outline 1 Interference Functions 2 Application to SIR-Based Utility Sets 3 Application to Game Theory 4 Structure of Interference Functions and Utility Sets 5 Conclusions 10

11 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... 11

12 Resource Allocation for multiuser systems, the transmission strategy is typically a tradeoff between efficiency and fairness requirements 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 12

13 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, 13

14 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) =

15 Exploiting Concavity consider concave interference functions 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 15

16 Exploiting Concavity: 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 16

17 Convergence Behavior 3.5 power (5 users) improved algorithm that exploits concavity fixed point iteration iterations 17

18 Outline 1 Interference Functions 2 Application to SIR-Based Utility Sets 3 Application to Game Theory 4 Structure of Interference Functions and Utility Sets 5 Conclusions 18

19 Cooperative Bargaining K players try to reach an unanimous agreement on utilities u = [u 1,..., u k ] the utility region U R K ++ is convex, comprehensive, closed, bounded depending on the chosen strategy, the solution outcome ϕ results if the bargaining fails, the disagreement outcome d results u 2 d 2 U d d 1 bargaining game (U, d) solution outcome ϕ(u, d) u 1 19

20 Standard Properties of Utility Sets downward-comprehensive for all u U and u R K ++ u u = u U closed (contains its boundary) convex upper-bounded u 2 U u u 1 20

21 Utility Sets and Interference Calculus Theorem Every compact comprehensive utility set from R K ++ can be expressed as a sub-level set U = {u R K ++ : C(u) 1} depending on an interference function C(u). The sub-level set U is convex if and only if C(u) is a convex interference function. 21

22 Example: The SIR Feasible Set 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 ) C(γ) satisfies A1 A3 every SIR region is a level set of an interference function 22

23 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 outcome is also the solution of the smaller set. SYM Symmetry: If the region is symmetric, then the outcome does not depend on the identities of the users. STC Scale Transformation Covariance. component-wise scale-invariant. The outcome is 23

24 The Nash Product For convex comprehensive sets the unique Nash bargaining solution fulfilling the axioms WPO, IIA, SYM, STC is obtained as the solution of K max (u k d k ) {u U:u>d} k=1 Often, the choice of the zero of the utility scales does not matter, so we can choose d = 0 max u U K k=1 u k 24

25 Nash Bargaining 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 closed comprehensive and bounded, then symmetric Nash bargaining and proportional fairness are equivalent 25

26 Bargaining over SIR Feasible Sets for wireless systems, an important performance measure is the signal-to-interference ratio SIR k (p) = p k I k (p) useful power interference (+noise) power ( ) γ indicator of feasibility: C(γ) = inf p>0 max k I k (p) k p k the SIR region S = {γ R K + : C(γ) 1} γ 2 infeasible C(γ) > 1 is generally not convex, so results from classical bargaining theory cannot be applied directly feasible C(γ) 1 γ 1 26

27 Convex log-sir Region if the underlying interference functions are log-convex, then the SIR region is log-convex SIR 2 SIR region log SIR 2 log-sir region SIR 1 log SIR 1 SIR region has special properties which can be exploited for bargaining (closed, comprehensive, log-convex) 27

28 Extension of the Classical Nash Bargaining Framework If the region U is strictly convex after a log-transformation ( log-convex ), then the Nash axioms WPO, IIA, SYM, STC characterize a single-valued solution outcome ϕ(u) = arg max u U K log u k. k=1 SIR max-min fairness proportional fairness Nash curve Σ log SIR = SIR 1 28

29 Outline 1 Interference Functions 2 Application to SIR-Based Utility Sets 3 Application to Game Theory 4 Structure of Interference Functions and Utility Sets 5 Conclusions 29

30 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} 30

31 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 an optimum of a utility/cost resource allocation problem 31

32 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 + 32

33 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 33

34 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. 34

35 Interpretation of Concave Interference Functions I(p) = min w N 0 (I) k=1 K w k p k the set N 0 (I) is closed, convex, and upward-comprehensive any concave interference function can be interpreted as the solution of a loss/cost minimization problem w 2 p region N 0 (I) w 1 35

36 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 + 36

37 Example: Robustness An 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 37

38 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. 38

39 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 downward-comprehensive any convex interference function can be interpreted as the solution of a utility maximization problem w 2 W 0 (I) p w 1 39

40 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 40

41 Example: SIR Set Based on Log-Convex Interference Let I 1,..., I K be log-convex interference functions, then the SIR-balancing optimum ( C(γ) = inf p>0 max 1 k K γ k I k (p) p k ) is a log-convex interference function, i.e., C(exp q) is a log-convex (thus convex) function. the SIR feasible set S = {γ : C(γ) 1} is convex on a logarithmic scale this hidden convexity can be exploited for designing resource allocation algorithms 41

42 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 42

43 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) 43

44 Categories of Interference Functions convex interference functions concave interference functions log-convex interference functions general interference functions 44

45 Outline 1 Interference Functions 2 Application to SIR-Based Utility Sets 3 Application to Game Theory 4 Structure of Interference Functions and Utility Sets 5 Conclusions 45

46 Conclusions the framework of interference functions is applicable to different areas in wireless communications: physical layer design medium access control resource allocation and utility optimization results provide intuition about the behavior of coupled multiuser systems useful for characterizing operating points of the system, design of algorithms many interesting open questions 46