Cross-layer Control of Wireless Networks: From Theory to Practice

1. Cross-layer Control of Wireless Networks:From Theory to Practice Professor Song Chong Network Systems Laboratory EECS, KAIST songchong@kaist.edu

2. 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. Capacity Region: A Realization of Channel User 2 User 1 Convex Nonconvex • Consider a single channel realization • CDMA downlink with two users • θ: orthogonality factor in [0,1] • Capacity region [Kum03]

4. Long-term Capacity Region • Time-varying achievable rate region • Long-term rate region

5. 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 • ris(k)≥0: rate allocated to user i for channel state s∈S and scheduling decision k∈K(s) • rs(k): rate vector, i.e., rs(k)=[ris(k),∀i] • For each channel state s, a probability distribution φs=[φsk,∀k∈K(s)] is fixed, i.e.,

6. Convexity Proof [Stol05] • Rate vector for a set of distributions φ=[φs,∀s∈S] • If we interpret φskas 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 • The long-term rate region F is defined as the set of all average service rate vectors R(φ) corresponding to all possible φ • 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 rs(k), ∀k∈K(s), ∀s∈S

7. Network Utility Maximization (NUM) • Long-term NUM • Utility function [Mo00] α→0: throughput maximization α=1: proportional fairness (PF) α→∞: max-min fairness

8. Sum of Weighted Rates (SWR) • 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.

9. 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 Ri(t) is the average rate of user i up to time t and is the replacement of Ri* which is unknown a priori • Convergence of Ri(t) to Ri* can be proved by stochastic approximation theory [Kush04] or fluid limit technique [Stol05].

10. 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. 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. Opportunistic Communication in OFDMA Downlink Channel gain frequency Channel gain frequency • 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. Mobile User 1 Fading channel User M

13. Frequency Selectivity in Channel • Frequency response in multipath environment • Delay spread • Coherence bandwidth Bc • Frequency separation at which the attenuation of two frequency-domain samples becomes decorrelated • For given delay spread, • Frequency-selective channel if B>>Bc • Frequency-flat channel if B<<Bc Bc gain B B freq.

14. Long-term NUM Problem in OFDMA Downlink Frequency (subcarrier) User 3 Power allocation User 4 User 3 User 3 User 2 User 1 Time slot Subcarrier allocation (user selection) Dynamic subcarrier and power allocation achieving

15. 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. 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 Initialization Equal power allocation Subcarrier allocation for given power allocation While subcarrier allocation is changing Power allocation for given subcarrier allocation

17. Subproblem I: Opportunistic Scheduling • Find x for a fixed power vector p0 • Separable w.r.t. subcarriers • For each subcarrier j, select user ij* such that

18. Subproblem II: Water Filling subcarrier • Find p for a fixed subcarrier allocation x0 • Convex optimization • Water filling is optimal • λis a nonnegative value satisfying

19. FPA vs. EPA B=20MHz B=5MHz • 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 MM MM MT MT

20. 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 gm(j)j’s subcarrier

21. Impact of System Bandwidth: Interpretation Channel gain B frequency • Narrowband (less frequency-selective) • Wideband (more frequency-selective) Extreme case (frequency flat) subcarrier Channel gain B subcarrier frequency

22. Impact of SNR Distribution EPA is comparable to FPA only when all the mobiles are located in high SNR regime B=20MHz s=6 Low SNR: gij<5dB MM MT

23. Impact of SNR Distribution: Interpretation • High SNR • Mix of high and low SNR • Low SNR subcarrier Insensitive to power variation Subcarriers with low SNR users are more sensitive to power than high SNR users subcarrier Sensitive to power variation subcarrier

24. 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. Single-carrier Downlink Problem • Assumption • Infinite demands • Infinite backlog at every transport layer queue • Cross-layer control • Joint optimization of flow control and scheduling Flow Control at Source Base Station fading channel Scheduling demands feedback: achievable rates

26. Cross-layer Control • Scheduling at BS • Flow control at source i • Algorithm Performance Stability Optimality

27. Derivation of Cross-layer Control Primal problem Dual problem Dual decomposition

28. Multi-hop Wireless Networks:Cross-layer Control

29. Multi-hop Wireline Network • Network Utility Maximization • Link capacity is given and constant • Rate allocation problem

30. 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. 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. 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. Per-link Queueing Case cA=1 cB=1 User 0 User 2 a is the fraction of time link A is used

34. Lagrange Multipliers

35. Functional Decomposition • Congestion control (sources and nodes) • MAC or scheduling (network)

36. Per-flow Queueing Case User 1 cA=1 cB=1 User 0 User 2 a0 is the fraction of time link A is used for user 0

37. Functional Decomposition • Congestion control (sources) • MAC or scheduling (network) pa1 pb2 x1 μa1 x2 μb2 pa0 pb0 x0 μa0 μb0

38. Interference Model 5 2 4 1 3 • Network connectivity graph G • Conflict graph CG - Links in G = nodes in CG - CG-Edge if links in G interfere with each other node link 5 1 2 3 4

39. Interference Model 5 2 4 1 3 • Maximal independent set model - Only one maximal independent set can be active at a time - - NUM problem CG Maximal independent sets

40. 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. Jointly Optimal Power and Congestion Control r r C Congestion Price Source Node Flow Control Link Power Control Transport layer Physical layer • Flow control at source • Power control at link • Congestion price at link • Interpretation

42. Routing and Network Layer Queueing 1 2 3 5 4 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. 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. 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. 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 Ri(c)(t) to • Routing and resource allocation • Same as previous • Performance • Tradeoff between utility and delay

46. 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.