Energy Efficiency and Load Balancing in Next-Generation Wireless Cellular Networks

Transcription

1 Energy Efficiency and Load Balancing in Next-Generation Wireless Cellular Networks Kemal Davaslioglu Advisor: Professor Ender Ayanoglu Center for Pervasive Communications and Computing Department of Electrical Engineering and Computer Science University of California, Irvine November 20th, 2015

2 Outline: 1. Introduction 2. Energy-Efficient Resource Allocation in Heterogeneous Networks System Model Formulate the Energy Efficiency Maximization Problem Single-Tier Networks Two-Tier Heterogeneous Networks Propose Iterative Water-Filling Algorithms with Pricing 3. Simulation Results 4. Conclusions

3 Introduction - Motivation The global mobile data traffic grew 69% in It is expected to grow nearly tenfold between 2014 and 2019 Cisco Systems, Inc., Cisco Visual Networking Index: Global mobile data traffic forecast update, , White Paper, Feb /30

4 Introduction - Motivation The global mobile data traffic grew 69% in It is expected to grow nearly tenfold between 2014 and 2019 The number of mobile-connected devices exceeded the world s population by 2014 Cisco Systems, Inc., Cisco Visual Networking Index: Global mobile data traffic forecast update, , White Paper, Feb /30

5 Introduction - Motivation The global mobile data traffic grew 69% in It is expected to grow nearly tenfold between 2014 and 2019 The number of mobile-connected devices exceeded the world s population by 2014 We need energy efficient solutions while satisfying their connectivity and rate demands! 1/30

6 Introduction - Motivation The global mobile data traffic grew 69% in It is expected to grow nearly tenfold between 2014 and 2019 The number of mobile-connected devices exceeded the world s population by 2014 We need energy efficient solutions while satisfying their connectivity and rate demands! Enabling technologies 1/30

7 Introduction - Heterogeneous Networks An example illustrating the deployment of macrocell base stations on towers 2/30

8 Introduction - Heterogeneous Networks Outdoor picocell deployments 2/30

9 Introduction - Heterogeneous Networks Indoor femtocell deployments Sprint s Femtocell (launched on Sept. 2007): it can support up to six 3G users. 2/30

10 Introduction - Heterogeneous Networks Heterogeneous cellular network deployment 2/30

11 Introduction Some Questions to Answer: 3/30

12 Introduction Some Questions to Answer: How are the resources shared in the multi-tier cellular architecture? 3/30

13 Introduction Some Questions to Answer: How are the resources shared in the multi-tier cellular architecture? How should users be associated with base stations? 3/30

14 Introduction Some Questions to Answer: How are the resources shared in the multi-tier cellular architecture? How should users be associated with base stations? Constraints such as power, access, backhaul, etc. 3/30

15 Introduction Some Questions to Answer: How are the resources shared in the multi-tier cellular architecture? How should users be associated with base stations? Constraints such as power, access, backhaul, etc. Spatial and temporal variations in traffic 3/30

16 Introduction Some Questions to Answer: How are the resources shared in the multi-tier cellular architecture? How should users be associated with base stations? Constraints such as power, access, backhaul, etc. Spatial and temporal variations in traffic Spectrum allocation between macrocell and picocell tiers 3/30

17 Introduction Some Questions to Answer: How are the resources shared in the multi-tier cellular architecture? How should users be associated with base stations? Constraints such as power, access, backhaul, etc. Spatial and temporal variations in traffic Spectrum allocation between macrocell and picocell tiers Power allocation of each sector 3/30

18 Introduction Some Questions to Answer: How are the resources shared in the multi-tier cellular architecture? How should users be associated with base stations? Constraints such as power, access, backhaul, etc. Spatial and temporal variations in traffic Spectrum allocation between macrocell and picocell tiers Power allocation of each sector Centralized or distributed processing? 3/30

19 Introduction Some Questions to Answer: How are the resources shared in the multi-tier cellular architecture? How should users be associated with base stations? Constraints such as power, access, backhaul, etc. Spatial and temporal variations in traffic Spectrum allocation between macrocell and picocell tiers Power allocation of each sector Centralized or distributed processing? How much gain can be achieved? 3/30

20 Introduction Some Questions to Answer: How are the resources shared in the multi-tier cellular architecture? How should users be associated with base stations? Constraints such as power, access, backhaul, etc. Spatial and temporal variations in traffic Spectrum allocation between macrocell and picocell tiers Power allocation of each sector Centralized or distributed processing? How much gain can be achieved? 3/30

21 Energy-Efficient Power Control for Heterogeneous Networks Contributions: 4/30

22 Energy-Efficient Power Control for Heterogeneous Networks Contributions: Formulated the multi-cell multi-carrier power control problem for energy efficiency maximization in Heterogeneous Networks 4/30

23 Energy-Efficient Power Control for Heterogeneous Networks Contributions: Formulated the multi-cell multi-carrier power control problem for energy efficiency maximization in Heterogeneous Networks Decomposed the problem into distributively solved subproblems 4/30

24 Energy-Efficient Power Control for Heterogeneous Networks Contributions: Formulated the multi-cell multi-carrier power control problem for energy efficiency maximization in Heterogeneous Networks Decomposed the problem into distributively solved subproblems These subproblems can be solved independently at each sector using limited information exchange 4/30

25 Energy-Efficient Power Control for Heterogeneous Networks Contributions: Formulated the multi-cell multi-carrier power control problem for energy efficiency maximization in Heterogeneous Networks Decomposed the problem into distributively solved subproblems These subproblems can be solved independently at each sector using limited information exchange Obtained closed-form expressions for the power updates and proposed iterative water-filling algorithms with pricing 4/30

26 Energy-Efficient Power Control for Heterogeneous Networks Contributions: Formulated the multi-cell multi-carrier power control problem for energy efficiency maximization in Heterogeneous Networks Decomposed the problem into distributively solved subproblems These subproblems can be solved independently at each sector using limited information exchange Obtained closed-form expressions for the power updates and proposed iterative water-filling algorithms with pricing Demonstrated that significant gains can be achieved 4/30

27 Multi-cell Energy Efficiency Maximization Problem with Power Constraints The energy efficiency of a sector s: ( ) η s(p) = Rs(p) f log p s (n) χ (n) k n N = P Macro,s P Macro,s (bits/sec/watts) or (bits/joule) G. Auer et al., How much energy is needed to run a wireless network? IEEE Wireless Commun., vol. 18, no. 5, pp , Oct /30

28 Multi-cell Energy Efficiency Maximization Problem with Power Constraints The energy efficiency of a sector s: ( ) η s(p) = Rs(p) f log p s (n) χ (n) k n N = P Macro,s P Macro,s (bits/sec/watts) or (bits/joule) where the channel-to-interference-plus-noise ratio (CINR) is given by χ (n) k = g (n) k,s σ 2 + I (n) = k g (n) k,s ( σ 2 + s s,s S ) (n) p(n) s g (n) k,s G. Auer et al., How much energy is needed to run a wireless network? IEEE Wireless Commun., vol. 18, no. 5, pp , Oct /30

29 Multi-cell Energy Efficiency Maximization Problem with Power Constraints The energy efficiency of a sector s: ( ) η s(p) = Rs(p) f log p s (n) χ (n) k n N = P Macro,s P Macro,s (bits/sec/watts) or (bits/joule) where the channel-to-interference-plus-noise ratio (CINR) is given by χ (n) k = g (n) k,s σ 2 + I (n) = k g (n) k,s ( σ 2 + s s,s S ) (n) p(n) s g (n) k,s We consider the linearized power consumption model which captures the contributions of the power amplifier, radio frequency (RF) transceiver parts, baseband unit, power supply, and cooling devices. P Macro,s = {P 0,m + M n N p(n) s P sleep,m n N p(n) s > 0 p s (n) = 0 for all n N G. Auer et al., How much energy is needed to run a wireless network? IEEE Wireless Commun., vol. 18, no. 5, pp , Oct /30

30 Multi-cell Energy Efficiency Maximization Problem with Power Constraints Our objective is to maximize the sum of all sector energy efficiencies in the network. This problem can be formulated as ( ( ) ) P : max f log p s (n) χ (n) k /P Macro,s s S n N s.t. p s (n) P Total,s for all s S n N P max,s (n) p s (n) 0 for all n N and for all s S 6/30

31 Multi-cell Energy Efficiency Maximization Problem with Power Constraints Our objective is to maximize the sum of all sector energy efficiencies in the network. This problem can be formulated as ( ( ) ) P : max f log p s (n) χ (n) k /P Macro,s s S n N s.t. p s (n) P Total,s for all s S n N P max,s (n) p s (n) 0 for all n N and for all s S Large problem with variables (57 sectors 600 subcarriers for 10 MHz) Requires a central processor High amount of information exchange is required Prone to delays or failures in the backhaul 6/30

32 Fractional Programming and Dinkelbach s Method Dinkelbach s method solves nonlinear fractional programming problems by successively solving a sequence of simplified nonlinear problems. To solve the following maximization problem R(x) max Q(x) = x S P (x) We introduce an auxiliary variable λ and successively solve until convergence. F (λ l ) = max x S R(x) λ lp (x) and λ l+1 = λ l F (λ l) F (λ l ) = R(x l) P (x l ) 7/30

33 Fractional Programming and Dinkelbach s Method Dinkelbach s method solves nonlinear fractional programming problems by successively solving a sequence of simplified nonlinear problems. To solve the following maximization problem R(x) max Q(x) = x S P (x) We introduce an auxiliary variable λ and successively solve until convergence. F (λ l ) = max x S R(x) λ lp (x) and λ l+1 = λ l F (λ l) F (λ l ) = R(x l) P (x l ) Dinkelbach s Method 1: Initialize λ 0 such that F (λ 0) > 0 and set l = 0. 2: while F (λ l ) > ɛ do 3: Find x that solves F (λ l ) 4: λ l+1 = R(x )/P (x ) 5: l = l + 1 6: end while 7/30

34 Multi-cell Energy Efficiency Maximization Problem with Power Constraints Using the Dinkelbach s method, we can introduce λ s per sector and reformulate the network energy efficiency maximization problem with power constraints as max ( ) f log p s (n) χ (n) k λ sp Macro,s s S n N s.t. p (n) s P Total,s for all s S n N P max,s (n) p s (n) 0 for all n N and for all s S The corresponding Lagrangian can be written as L(p s, λ, µ s, υ l,s, υ u,s) = ( ) f log p s (n) χ (n) k λ sp Macro,s s S n N + µ s P Total,s p s (n) n N + n N ( υ (n) l,s p(n) s ( )) + υ u,s (n) P max,s (n) p (n) s 8/30

35 Multi-cell Energy Efficiency Maximization Problem with Power Constraints We can now take the derivative of L w.r.t. p (n) s L = f p (n) log(2) s χ (n) k 1+p (n) s χ (n) k f log(2) j k,j K (n) π (n) where the interference pricing terms are expressed as π (n) k,j = γ(n) j γ (n) j + 1 and equate to zero k,j λs M µ s + υ (n) l,s g (n) j,s I (n) j + σ 2 υ(n) u,s = 0, 9/30

36 Multi-cell Energy Efficiency Maximization Problem with Power Constraints We can now take the derivative of L w.r.t. p (n) s L = f p (n) log(2) s χ (n) k 1+p (n) s χ (n) k f log(2) j k,j K (n) π (n) where the interference pricing terms are expressed as π (n) k,j = γ(n) j γ (n) j + 1 and equate to zero k,j λs M µ s + υ (n) l,s g (n) j,s I (n) j + σ 2 Rearranging terms, we obtain the following closed-form expression p (n) s = 1 log(2) (λ s M + µ f s) + π (n) k,j j k,j K (n) 1 χ (n) k P max,s (n) 0 υ(n) u,s = 0, 9/30

37 Multi-cell Energy Efficiency Maximization Problem with Power Constraints The cut-off value with pricing is determined using Ω (n) EE,P = log(2) f (λ s M + µ s) + j k,j K (n) π (n) k,j In the water-filling solution, this cut-off value can be interpreted as the threshold that determines if the subcarrier is used or not. p (n) s > 0 if χ (n) k > Ω (n) EE,P and p(n) s = 0 if χ (n) k Ω (n) EE,P Multi-level (or modified) water-filling solution for energy efficiency maximization where the pricing terms determine the water filling level on each subcarrier. 10/30

38 Multi-cell Energy Efficiency Maximization Problem with Power Constraints Closed-form expression for the non-cooperative energy efficiency maximization problem can be expressed as P max,s (n) p (n) s = 1 log(2) (λ s M + µ 1 f s) χ (n) k 0 The corresponding cut-off value is determined by Ω EE,NP = log(2) (λ s M + µ s) f Single-level water-filling solution for energy efficiency maximization without pricing. 11/30

39 Multi-cell Energy Efficiency Maximization Problem with Power Constraints Iterative Water-Filling Algorithm with Pricing for Network Energy Efficiency Maximization 1: Initialize transmit power levels and interference prices, and set t = 0. Solve the following at each sector s 2: while F s(λ ( s) > ɛ and l < l max do ( )) 3: λ s(l) = n N f log p (n) s χ (n) k /P Macro,s 4: Obtain µ s using the bisection method 5: For all n N, solve for p (n) Next using p (n) Next = 1 log(2)/ f ( M λ s(l) + µ s) + π (n) j k,j K (n) k,j 6: Calculate F s(λ s) 7: l = l + 1 8: end while 9: Update the power levels using p (n) s (t + 1) = (1 δ) p (n) s (t) + δ p (n) Next 10: Distribute the interference prices, {π (n) k,j }, among base stations 11: Go to Step 2 and repeat for t = t χ (n) k P max,s (n) 0 12/30

40 Multi-cell Energy Efficiency Maximization Problem with Power Constraints System is warmed up with either using the non-cooperative solution or transmitting at a predefined power level 13/30

41 Multi-cell Energy Efficiency Maximization Problem with Power Constraints Users calculate interference prices and feed them back to base stations 13/30

42 Multi-cell Energy Efficiency Maximization Problem with Power Constraints Distribute interference prices between base stations 13/30

43 Multi-cell Energy Efficiency Maximization Problem with Power Constraints Solve the cooperative resource allocation problem using the iterative water-filling solution with pricing 13/30

44 Multi-cell Energy Efficiency Maximization Problem with Power Constraints in Two-Tier Networks Two-tier Deployments: The total power consumed in a sector is given by ψ s = P Macro,s + P Pico,p, p S P,s where P Pico,p is total power consumption of a picocell base station p and the set S P,s is the set of picocell base stations in sector s. The power consumption at a picocell base station is given by P Pico,p =P 0,p + P n N p (n) p, Base Station Power Consumption Model Parameter Values Base Station P sleep P 0 P max Type (W) (W) (W) Macrocell Base Station Picocell Base Station G. Auer et al., How much energy is needed to run a wireless network? IEEE Wireless Commun., vol. 18, no. 5, pp , Oct /30

45 Multi-Cell Energy Efficiency Maximization with Rate and Power Constraints We can incorporate the minimum rate constraints per user to the energy efficiency maximization problem in two-tier networks as follows: max ( ) f log p s (n) χ (n) k s S n N + ( ) f log p p (n) χ (n) k λ sψ s p S P,s n N s.t. R min,k, for all k K r (n) k n N k n N n N p (n) s P Total,s for all s S p (n) p P Total,p for all p S P,s P max,s (n) p s (n) 0 for all n N and for all s S P max,p (n) p p (n) 0 for all n N and for all p S P,s 15/30

46 Multi-Cell Energy Efficiency Maximization with Rate and Power Constraints The corresponding Lagrangian can be defined as L(p, λ, τ, µ) = ( ) f log p s (n) χ (n) k s S n N + ( ) f log p p (n) χ (n) k λ sψ s p S P,s n N + τ k r (n) k R min,k + µ s P Total,s p (n) s k K s n N k n N + P Total,p p p (n), n N p S p,s µ p where τ = [τ 1,, τ K ] T denotes the vector of Lagrange multipliers associated with the minimum rate constraints 16/30

47 Multi-Cell Energy Efficiency Maximization with Rate and Power Constraints The closed-form expressions including the rate and power constraints are P max,s (n) p (n) s = (1 + τ k ) log(2) (λ s M + µ f s) + (1 + τ j ) π (n) 1 k,j χ (n) k j k,j K (n) 0 P max,p (n) p (n) p = (1 + τ k ) log(2) (λ s P + µ f s) + (1 + τ j ) π (n) 1 k,j χ (n) k j k,j K (n) 0 The corresponding cut-off value is given by Ω (n) EE,P,RC = log(2) (λ s M + µ s) + (1 + τ j )π (n) k,j / (1 + τ k ) f j k,j K (n) 17/30

48 Multi-Cell Energy Efficiency Maximization with Rate and Power Constraints The dual prices associated with the minimum rate constraints are updated using τ (l+1) k = τ (l) k α(l) + k R min,k, where the step-size is adaptively chosen as ( α (l) βα (l 1) if R = min,k α (l 1) otherwise. n N k r (n) r (n,l) k n N k ) > κ ( R min,k r (n,l 1) k n N k In the simulations, we take the step size as α (0) = , the increment factor β as 2, and the comparison threshold κ as 0.9. ) 18/30

49 Multi-Cell EE-Maximization with Rate and Power Constraints Iterative Water-Filling Algorithm with Rate and Power Constraints 1: Set the initial transmit power levels, interference prices, and dual prices. Set t = 0. At each sector, solve 2: while F s(λ s) > ɛ and l < l max do 3: Determine ( λ s using the following λ s = n N f log 2 (1 + p (n) s χ (n) k ) + p S P,s n N f log 2 (1 + p (n) p χ(n) k )/ψ ) s 4: Obtain µ s using the bisection method 5: For all n N, solve for p (n) Next 6: for all p S P,s do 7: Obtain µ p using the bisection method 8: Solve for p (n) Next,p for all n N 9: end for 10: Update the dual prices, τ k for all k K s 11: Calculate the following F s(λ s) = ( n N f log p (n) s χ (n) k )+ ( ) p SP,s n N f log p (n) p χ(n) k λ sψ s 12: l = l : end while 14: Update the power levels using p (n) s (t + 1) = (1 δ) p (n) s (t) + δ p (n) Next and p (n) p (t + 1) = (1 δ) p(n) p (t) + δ p(n) Next,p p S P,s 15: Distribute the interference prices, {π (n) k,j }, among base stations 16: Go to Step 2 and repeat for t = t /30

50 Simulations: Simulation Setup Network Model: y (km) x (km) Example Layout of an LTE Network We consider 30 users per sector For the single-tier deployment, consider uniform user distribution For the multi-tier deployment, consider clustered user distribution 1 1 3GPP, TR , Further advancements for E-UTRA physical layer aspects (Release 9), Mar /30

51 Simulations: Simulation Setup Simulation Parameters: Parameter Setting Channel bandwidth 10 MHz Total number of data RBs 50 RBs Freq. selective channel model (CM) Extended Typical Urban CM UE to MeNB PL model log 10 (d) UE to pico enb PL model log 10 (d) Effective thermal noise power, N dbm/hz UE noise figures 9 db MeNB and Pico enb antenna gain 14 dbi and 5 dbi UE antenna gain 0 dbi Antenna horizontal pattern, A(θ) min(12(θ/θ 3dB ) 2, A m) A m and θ 3dB 20 db and 70 Penetration loss 20 db Macro- and picocell shadowing 8 db and 10 db Inter-site distance 500 m Minimum macro- to user distance 50 m Minimum pico- to user distance 10 m Minimum pico- to macro- distance 75 m Minimum pico- to pico- distance 40 m Traffic model Full buffer [1] 3GPP, TR , Further advancements for E-UTRA physical layer aspects (Release 9), Mar /30

52 Simulations: Simulation Setup We investigate two problems: Energy Efficiency Maximization (EE-Max) Throughput Maximization (Throughput-Max) We compare the performance of the proposed algorithm with IWF, EE-Max, No-Pricing: C. Isheden, et al., Framework for link-level energy efficiency optimization with informed transmitter, IEEE Trans. Wireless Commun., vol. 11, no. 8, pp , Aug Constant Power Allocation (CPA): K. Davaslioglu, C. C. Coskun, and E. Ayanoglu, Energy-Efficient Resource Allocation for Fractional Frequency Reuse in Heterogeneous Networks, IEEE Trans. Wireless Commun., vol. 14, no. 10, pp , Oct /30

53 Simulations: Numerical Results - Single-Tier (a) (b) Average sector energy efficiency and throughput of a single-tier network using the proposed iterative water-filling algorithms with different initial power levels. Interference pricing brings 40% and 14% improvements over the case without interference pricing in terms of energy efficiency and throughput, respectively. 23/30

54 Simulations: Numerical Results - Single-Tier (c) (d) (e) (f) Average sector energy efficiency and throughput of a single-tier network using the proposed iterative water-filling algorithms. 24/30

55 Simulations: Numerical Results - Single-Tier Average power consumption of the proposed iterative water-filling algorithms in a single-tier network. IWF, EE-Max, No Pricing reduces the average transmit power from 46 dbm to 41 dbm IWF, EE-Max, Pricing reduces the average transmit power from 46 dbm to 31 dbm 25/30

56 Simulations: Numerical Results - Heterogeneous Networks Two-tier Heterogeneous Networks (g) (h) Average sector energy efficiency and throughput of a two-tier network using the proposed iterative water-filling algorithms. We consider 4 picocell base stations per sector and clustered user distribution Power control improves energy efficiency and throughput by factors of 2.68 and 1.77, respectively. Interference pricing brings 39% and 29% improvement in energy efficiency and throughput, respectively, compared to the case without pricing. 26/30

57 Simulations: Numerical Results - Heterogeneous Networks Average transmit power consumption of the proposed iterative water-filling algorithms with and without pricing in a two-tier network. On average, macrocell base station transmissions are reduced to 20 dbm (compare with 31 dbm for the single-tier) Picocell base station transmissions are not reduced significantly. They are already in the energy-efficient regime. 27/30

58 Simulations: Numerical Results - Heterogeneous Networks Incorporating Minimum Rate Constraints in Two-Tier Networks (i) (j) Average sector energy efficiency and sector throughput for different minimum rate requirements. Higher minimum rate requirements result in reduced energy efficiency solution (i.e., a larger loss compared to the case without rate constraints) 28/30

59 Simulations: Numerical Results - Heterogeneous Networks (k) (l) (a) The outage probability of various minimum rate requirements and (b) the cumulative distribution function of user rates for the minimum rate requirement of 512 kbits/sec. Using IWF and Interference Pricing, outage can be reduced from an unacceptable level (24%) to a tolerable outage (7%) 29/30

60 Energy-Efficient Power Control for Heterogeneous Networks Summary: The multi-cell multi-carrier energy-efficient power control problem in Heterogeneous Networks is studied We incorporated the load-adaptive base station power consumption model in our formulation We decomposed the centralized network energy maximization problem into distributively solved subproblems These subproblems can be solved independently at each sector using limited information exchange Simulation results demonstrate significant energy efficiency improvements 30/30

61 Publications Preprints: 1. K. Davaslioglu, and E. Ayanoglu, Iterative Water-Filling Algorithms with Pricing for Network Energy Efficiency Maximization, under review at IEEE Trans. Wireless Communications. 2. C. C. Coskun, K. Davaslioglu, and E. Ayanoglu, Distributed Interference Pricing Algorithm for Energy-Efficient Heterogeneous Network with Rate Constraints, under review at IEEE Trans. Wireless Communications. Journals: 1. K. Davaslioglu, C. C. Coskun, and E. Ayanoglu, Energy-Efficient Resource Allocation for Fractional Frequency Reuse in Heterogeneous Networks, IEEE Trans. Wireless Communications, vol. 14, no. 10, pp , Oct K. Davaslioglu and E. Ayanoglu, Efficiency and Fairness Trade-offs in SC-FDMA Schedulers, IEEE Transactions on Wireless Communications, vol. 13, no. 6, pp , June K. Davaslioglu and E. Ayanoglu, Quantifying Potential Energy Efficiency Gain in Green Cellular Wireless Networks, IEEE Communications Surveys and Tutorials, vol. 16, no. 4, pp , 4th Quarter, 2014.

62 Publications Conferences: 1. C. C. Coskun, K. Davaslioglu, and E. Ayanoglu, An Energy-Efficient Resource Allocation Algorithm with QoS Constraints for Heterogeneous Networks, accepted to IEEE Globecom K. Davaslioglu and E. Ayanoglu, Interference-Based Cell Selection in Heterogeneous Networks, in Proc. Information Theory and Applications Workshop, San Diego, CA, Feb [Online]. Available arxiv: [cs.ni]. 3. K. Davaslioglu and E. Ayanoglu, Common Rate Maximization in Two-Layer Networks, in Proc. IEEE Globecom 2012 Multicell Cooperation Workshop, Anaheim, CA, pp , Dec [Online]. Available arxiv: [cs.ni]. 4. K. Davaslioglu and E. Ayanoglu, Power Control in Multi-Layer Cellular Networks, Mar Available arxiv:1203:6131 [cs.ni].

63 Thank you for listening! Any Questions?

64 Appendix

65 Optimality and Dinkelbach s Method Dinkelbach s method has superlinear convergence, i.e., lim l x l+1 L x l L = 0 Theorem: F (λ ) = max { R(x) λ P (x) } = 0 if and only if λ = R(x ) P (x ) = max R(x) x S P (x) Proof: Let x S be an optimal solution of such that we have max R(x) x S λ P (x) R(x ) λ P (x ) = 0 and R(x) λ P (x) R(x ) λ P (x ) = 0, x S. Since P (x ) > 0, we have λ = R(x ) P (x ) R(x) P (x) for all x S. Thus, λ is the maximum of max { R(x) λp (x) } and x is the optimal solution of max { R(x)/P (x) x S }.

66 Convergence and Dinkelbach s Method The convergence rate of Dinkelbach s method is 1 ψs(p ) ψ s(p i ). Superlinear Convergence: x k+1 L lim = µ k x k L Superlinear convergence, µ = 0. E.g., a n = n n Sublinear convergence µ = 1. E.g., a n = 1/n Q-Convergence: lim k x k+1 L q = µ, such that µ > 0 x k L Quadratic convergence (q = 2) Cubic convergence (q = 3)

67 Return Multi-cell Energy Efficiency Maximization Problem with Power Constraints Bisection Method for the Iterative Water-Filling Algorithm 1: Let ɛ denote the tolerance and l max be the maximum number of iterations 2: Initialize µ s,l = 0 and µ s,u = 1 3: Calculate p (n) s (µ s,u). 4: while n N p(n) s (µ s,u) > P Total,s do 5: µ s,u = 2 µ s,u 6: end while 7: while µ s,u µ s,l > ɛ do 8: µ s,mid = (µ s,l + µ s,u)/2 9: Calculate p (n) s (µ s,mid ) 10: if sign( n N p(n) s (µ s,mid ) P Total,s ) = sign( n N p(n) s (µ s,l ) P Total,s ) then 11: µ s,l = µ s,mid 12: else 13: µ s,u = µ s,mid 14: end if 15: end while 16: if n N p(n) s (µ s,mid ) < P Total,s then 17: µ s,mid = 0 18: end if

68 Energy-Efficient Power Control for Heterogeneous Networks Energy Efficiency (kbits/joule) β ε The sector energy efficiency region is shown for given interference conditions and prices. Red circles depict the gradient ascent solutions at each iteration.

69 Mercury Water-Filling Mercury Water-Filling: The Shannon capacity gives us the theoretical achievable rate for an ideal Gaussian input. For an arbitrary input modulation scheme, the rate function is described by the mutual information such that r k (p n) =I(s n; p nχ n k sn + nn) and MMSEn(ρn) = dr k(ρ n), dρ n [ where MMSE n(ρ n) = E sn sn ŝ n 2]. drk n dp n λ s m = 0, s Then, we obtain p n s =p n s MMSE n(p n s χ n k ) = λs M χ n k { 1 p n s = χ n MMSE 1 n (ξ) if ξ < 1 k 0 otherwise, where ξ = λ s m/χ n k. In the water-filling solution, 1/χn k is replaced by Γn(ξn)/χn k where { 1/ξ MMSE 1 n (ξ) if χ < 1 Γ n(ξ n) = 1 otherwise For an ideal Gaussian input signal, Γ n = 1

70 Simulations: Simulation Setup Network Model: E1 Macro: B Pico: A,C,D E3 C1 Macro: A Pico: C,D Macro: A Pico: B,D C3 Macro: A Pico: B,C Macro: D Pico: A,B,C Power εpm PM A B f C2 Power E2 Macro: C Pico: A,B,D P C P C D f Power A B C D P E P A C D f We consider 30 users per sector For the single-tier deployment, consider uniform user distribution For the multi-tier deployement, consider clustered user distribution 2 2 3GPP, TR , Further advancements for E-UTRA physical layer aspects (Release 9), Mar