1. Korea Advanced Institute of Science and Technology Network Systems Lab. Cross-layer Control of Wireless Networks: From Theory to Practice Professor Song Chong Network Systems Laboratory EECS, KAIST songchong@kaist.edu

2. Korea Advanced Institute of Science and Technology Network Systems Lab. Multi-user Opportunistic Communication Multi-user diversity In a large system with users fading independently, there is likely to be a user with a very good channel at any time. Long-term total throughput can be maximized by always serving the user with the strongest channel.

3. Korea Advanced Institute of Science and Technology Network Systems Lab. Capacity Region: A Realization of Channel Consider a single channel realization CDMA downlink with two users θ: orthogonality factor in [0,1] Capacity region [Kum03] User 1 User 2 ConvexNonconvex

4. Korea Advanced Institute of Science and Technology Network Systems Lab. Long-term Capacity Region Time-varying achievable rate region Long-term rate region

5. Korea Advanced Institute of Science and Technology Network Systems Lab. Convexity Proof [Stol05] Case of finite channel states and scheduling policies Notation S: finite set of channel states Sequence of channel states s(t) ∈ S, t=0,1,... forms an irreducible Markov chain with stationary distribution K(s): set of all possible scheduling decisions for given channel state s ∈ S r i s (k)≥0: rate allocated to user i for channel state s ∈ S and scheduling decision k ∈ K(s) r s (k): rate vector, i.e., r s (k)=[r i s (k), ∀ i] For each channel state s, a probability distribution φ s =[ φ sk, ∀ k ∈ K(s)] is fixed, i.e.,

6. Korea Advanced Institute of Science and Technology Network Systems Lab. Convexity Proof [Stol05] Rate vector for a set of distributions φ =[ φ s, ∀ s ∈ S] If we interpret φ sk as the long-term average fraction of time slots when the channel state is s and the rate allocation is k, then R( φ ) is the corresponding vector of long-term average service rates F The long-term rate region F is defined as the set of all average service rate vectors R( φ ) corresponding to all possible φ F The convexity of F immediately follows as it is a convex hull of all possible instantaneous rates Consider which is a convex combination of all possible rate vectors r s (k), ∀ k ∈ K(s), ∀ s ∈ S

7. Korea Advanced Institute of Science and Technology Network Systems Lab. Long-term NUM Utility function [Mo00] Network Utility Maximization (NUM) α → 0: throughput maximization α=1: proportional fairness (PF) α → ∞: max-min fairness

8. Korea Advanced Institute of Science and Technology Network Systems Lab. Maximization of sum of weighted rates Both problems yield an unique and identical solution if we set, where is the optimal solution of the long- term NUM problem. Sum of Weighted Rates (SWR)

9. Korea Advanced Institute of Science and Technology Network Systems Lab. Gradient-based Scheduling Assuming stationarity and ergodicity, we have The long-term NUM problem can be solved if we solve with at each state s The resource allocation problem during slot t where R i (t) is the average rate of user i up to time t and is the replacement of R i * which is unknown a priori Convergence of R i (t) to R i * can be proved by stochastic approximation theory [Kush04] or fluid limit technique [Stol05].

10. Korea Advanced Institute of Science and Technology Network Systems Lab. Gradient-based Scheduling This coincides with the optimality condition given by directional derivative Consider The optimality condition is given by The optimal solution to the following problem is R * Thus we set

11. Korea Advanced Institute of Science and Technology Network Systems Lab. HDR PF Scheduler PF scheduler is a special case of gradient- based scheduler Logarithmic utility function Feasible region (TDMA) PF scheduler serves user i * such that

12. Korea Advanced Institute of Science and Technology Network Systems Lab. Opportunistic Communication in OFDMA Downlink Exploit multi-user diversity in time and frequency In a large system with users fading independently, there is likely to be a user with a very good channel at some carrier frequency for each time. Long-term total throughput can be maximized by always serving the user with the strongest channel. Challenge is to share the benefit among the users in a fair way. User M Mobile User 1 frequency Channel gain frequency Channel gain Fading channel

13. Korea Advanced Institute of Science and Technology Network Systems Lab. Frequency Selectivity in Channel Frequency response in multipath environment Delay spread Coherence bandwidth B c Frequency separation at which the attenuation of two frequency-domain samples becomes decorrelated For given delay spread, Frequency-selective channel if B>>B c Frequency-flat channel if B<<B c BcBc B freq. gain B

14. Korea Advanced Institute of Science and Technology Network Systems Lab. Long-term NUM Problem in OFDMA Downlink User 1 User 2 User 3 User 4 User 3 Frequency (subcarrier) Time slot Power allocation Subcarrier allocation (user selection) Dynamic subcarrier and power allocation achieving

15. Korea Advanced Institute of Science and Technology Network Systems Lab. Joint Optimization Consider M mobile users and N subcarriers Joint optimization of subcarrier and power allocation at each time slot t Mixed integer nonlinear programming

16. Korea Advanced Institute of Science and Technology Network Systems Lab. Suboptimal Allocation [Lee08] Iteratively solve two subproblems For fixed p, subcarrier allocation problem Opportunistic scheduling over each subcarrier For fixed x, power allocation problem Convex optimization (water filling) Each subproblem is easy to solve Frequency-selective power allocation (FPA) Equal power allocation Subcarrier allocation for given power allocation Subcarrier allocation for given power allocation Power allocation for given subcarrier allocation Power allocation for given subcarrier allocation While subcarrier allocation is changing Initialization Equal power allocation

17. Korea Advanced Institute of Science and Technology Network Systems Lab. Subproblem I: Opportunistic Scheduling Find x for a fixed power vector p 0 Separable w.r.t. subcarriers For each subcarrier j, select user i j * such that

18. Korea Advanced Institute of Science and Technology Network Systems Lab. Subproblem II: Water Filling Find p for a fixed subcarrier allocation x 0 Convex optimization Water filling is optimal λ is a nonnegative value satisfying subcarrier

19. Korea Advanced Institute of Science and Technology Network Systems Lab. FPA vs. EPA FPA gives significant throughput gain (up to 40%) in OFDMA downlink when Sharing policy becomes more fairness-oriented Delay spread (frequency selectivity) increases System bandwidth becomes wider B=5MHz B=20MHz MT MM MT MM

20. Korea Advanced Institute of Science and Technology Network Systems Lab. Impact of α: Interpretation Efficiency-oriented policy (α=0) Only best user for each subcarrier Fairness-oriented policy (α → ∞) Bad-channel users are also selected FPA ≈ EPA subcarrier High, medium, low g m(j)j ’s

21. Korea Advanced Institute of Science and Technology Network Systems Lab. Impact of System Bandwidth: Interpretation Narrowband (less frequency-selective) Wideband (more frequency-selective) frequency Channel gain B frequency Channel gain B subcarrier Extreme case (frequency flat) subcarrier

22. Korea Advanced Institute of Science and Technology Network Systems Lab. Impact of SNR Distribution EPA is comparable to FPA only when all the mobiles are located in high SNR regime B=20MHz s=6 MT MM Low SNR: g ij <5dB

23. Korea Advanced Institute of Science and Technology Network Systems Lab. Impact of SNR Distribution: Interpretation High SNR Mix of high and low SNR Low SNR subcarrier Sensitive to power variation Insensitive to power variation Subcarriers with low SNR users are more sensitive to power than high SNR users subcarrier

24. Korea Advanced Institute of Science and Technology Network Systems Lab. Throughput-optimal Scheduling and Flow Control Joint scheduling and flow control — Stabilize the system whenever the long-term input (demand) rate vector lies within the capacity region — Stabilize the system while achieving throughput optimality even if the long-term input (demand) rate vector lies outside of the capacity region Long-term NUM for arbitrary input rates [Nee05]

25. Korea Advanced Institute of Science and Technology Network Systems Lab. Single-carrier Downlink Problem Assumption Infinite demands Infinite backlog at every transport layer queue Cross-layer control Joint optimization of flow control and scheduling demands Flow Control at SourceBase Station Scheduling fading channel feedback: achievable rates

26. Korea Advanced Institute of Science and Technology Network Systems Lab. Cross-layer Control Scheduling at BS Flow control at source i Algorithm Performance Stability Optimality

27. Korea Advanced Institute of Science and Technology Network Systems Lab. Derivation of Cross-layer Control Primal problem Dual problem Dual decomposition

28. Korea Advanced Institute of Science and Technology Network Systems Lab. Multi-hop Wireless Networks: Cross-layer Control

29. Korea Advanced Institute of Science and Technology Network Systems Lab. Multi-hop Wireline Network Network Utility Maximization ― Link capacity is given and constant ― Rate allocation problem

30. Korea Advanced Institute of Science and Technology Network Systems Lab. Functional Decomposition Lagrangian function Dual problem Dual decomposition ― Flow control at source ― Congestion price at link TCP is an approximation of this dual decomposition

31. Korea Advanced Institute of Science and Technology Network Systems Lab. Multi-hop Wireless Network Long-term Network Utility Maximization ― Link capacity is time-varying and a function of resource control ― Joint rate, power allocation and link scheduling

32. Korea Advanced Institute of Science and Technology Network Systems Lab. Functional Decomposition For a realization of channels Lagrangian function Dual problem Dual decomposition ― Flow control at source ― Scheduling/power control at link ― Congestion price at link Joint MAC and transport problem Distributed scheduling/power control is a challenge

33. Korea Advanced Institute of Science and Technology Network Systems Lab. Per-link Queueing Case User 0 User 2 c A =1 c B =1  a is the fraction of time link A is used

34. Korea Advanced Institute of Science and Technology Network Systems Lab. Lagrange Multipliers

35. Korea Advanced Institute of Science and Technology Network Systems Lab. Functional Decomposition Congestion control (sources and nodes) MAC or scheduling (network)

36. Korea Advanced Institute of Science and Technology Network Systems Lab. Per-flow Queueing Case User 0 User 1 User 2 c A =1 c B =1  a0 is the fraction of time link A is used for user 0

37. Korea Advanced Institute of Science and Technology Network Systems Lab. Functional Decomposition Congestion control (sources) MAC or scheduling (network) x0x0 μ a0 μ b0 x1x1 μ a1 x2x2 μ b2 p a0 p b0 p a1 p b2

38. Korea Advanced Institute of Science and Technology Network Systems Lab. Interference Model 1234 5 node link 52 13 4 Network connectivity graph G Conflict graph CG - Links in G = nodes in CG - CG-Edge if links in G interfere with each other

39. Korea Advanced Institute of Science and Technology Network Systems Lab. Interference Model Maximal independent set model - Only one maximal independent set can be active at a time - - NUM problem 52 13 4 CG Maximal independent sets

40. Korea Advanced Institute of Science and Technology Network Systems Lab. Jointly Optimal Power and Congestion Control NUM at particular state s [Chiang05] is a nonconcave function of p Assuming high SINR regime, i.e, can be converted into a concave function of p through a log transformation (geometric programming) Joint optimization of congestion control and power control

41. Korea Advanced Institute of Science and Technology Network Systems Lab. Jointly Optimal Power and Congestion Control Flow control at source Power control at link Congestion price at link Interpretation Physical layer rC Transport layer Source Node Flow Control Link Power Control Congestion Price r

42. Korea Advanced Institute of Science and Technology Network Systems Lab. Routing and Network Layer Queueing 12 3 45 transport layer network layer = set of commodities in the network = the amount of new commodity c data that exogenously arrives to node i during slot t = the amount of commodity c data allowed to enter the network layer from the transport layer at node i during slot t = the backlog of commodity c data stored in the network layer queue at node i during slot t

43. Korea Advanced Institute of Science and Technology Network Systems Lab. Dynamic Control for Network Stability The stabilizing dynamic backpressure algorithm [Tassiulas92] - An algorithm for resource allocation and routing which stabilizes the network whenever the vector of arrival rates lies within the capacity region of the network Resource allocation - For each link, determine optimal commodity and optimal weight by - Find optimal resource allocation action by solving

44. Korea Advanced Institute of Science and Technology Network Systems Lab. Dynamic Control for Network Stability Routing - For each link such that, offer a transmission rate of to data of commodity. The algorithm requires in general knowledge of the whole network state. However, there are important special cases where the algorithm can run in a distributed fashion with each node requiring knowledge only of the local state information on each of its outgoing links. Interpretation - The resulting algorithm assigns larger transmission rates to links with larger differential backlog, and zero transmission rates to links with negative differential backlog.

45. Korea Advanced Institute of Science and Technology Network Systems Lab. Dynamic Control for Infinite Demands Assumption Infinite backlog at every transport layer queue Cross-layer control Flow control at node i Each time t, set R i (c) (t) to Routing and resource allocation Same as previous Performance Tradeoff between utility and delay

46. Korea Advanced Institute of Science and Technology Network Systems Lab. References [Kum03] K. Kumaran and L. Qian, “Uplink Scheduling in CDMA Packet-Data Systems,” IEEE INFOCOM 2003. [Mo00] J. Mo and J. Walrand, “Fair End-to-End Window-Based Congestion Control,” IEEE/ACM Trans. Networking, Vol. 8, No. 5, pp. 556-567, Oct. 2000. [Kush04] H. J. Kushner and P. A. Whiting, “Convergence of Proportional-Fair Sharing Algorithms Under General Conditions,” IEEE Trans. Wireless Comm., vol., no., 2004. [Stol05] A. L. Stolyar, “On the Asymptotic Optimality of the Gradient Scheduling Algorithm for Multiuser Throughput Allocation,” Operations Research, vol. 53, no. 1, pp. 12-25, Jan. 2005. [Lee08] H. W. Lee and S. Chong, "Downlink Resource Allocation in Multi-carrier Systems: Frequency-selective vs. Equal Power Allocation", IEEE Trans. on Wireless Communications, Vol. 7, No. 10, Oct. 2008, pp. 3738-3747. [Nee05] M. J. Neely et al., “Fairness and Optimal Stochastic Control for Heterogeneous Networks,” IEEE INFOCOM 2005. [Chiang05] M. Chiang, “Balancing Transport and Physical Layers in Wireless Multihop Networks: Jointly Optimal Congestion Control and Power Control,” IEEE J. Sel. Areas Comm., vol. 23, no. 1, pp. 104-116, Jan. 2005. [Tassiulas92] L. Tassiulas and A. Ephremides, “Stability Properties of Constrained Queueing Systems and Scheduling Policies for Maximum Throughput in Multihop Radio Networks,” IEEE Trans. Automatic Control, vol. 37, no. 12, Dec. 1992.