Gradient-based scheduling and resource allocation in OFDMA systems

Transcription

1 Gradient-based scheduling and resource allocation in OFDMA systems Randall Berry Northwestern University Dept. of EECS Joint work with J. Huang, R. Agrawal and V. Subramanian CTW 2006 R. Berry (NWU) OFDMA Scheduling CTW / 22

2 Downlink Scheduling and Resource Allocation fading base station time users Key component of most recent wireless data systems e.g. CDMA 1xEVDO, HSPDA, IEEE Dynamically schedule users based on channel conditions/qos. Cross-layer approach. Use frequent channel quality feedback & adaptive modulation/coding. Exploit multi-user diversity. R. Berry (NWU) OFDMA Scheduling CTW / 22

3 Gradient-based Scheduling Scheduler needs to balance users QoS and global efficiency. Many approaches accomplish this via gradient-based scheduling. Assign each user a utility, U i ( ), depending on delay, throughput, etc. Scheduler choosea rate r = (r 1,..., r N ) T to solve: max U(X(t)) r = max U i (X i (t))r i, r R(e) r R(e) i Myopic policy, requires no knowledge of channel or arrival statistics. R. Berry (NWU) OFDMA Scheduling CTW / 22

4 Gradient-based Scheduling Examples α-fairness: utility function of average throughput W i : { ci U i (W i ) = α (W i) α, α 1, α 0. c i log(w i ), α = 0 α = 0 Prop. fair. α = 1 Max. throughput. Utility may also be function of delay/queue size. e.g. Stabilizing policies. R. Berry (NWU) OFDMA Scheduling CTW / 22

5 State-dependent Feasible Rate Regions Optimization is over feasible rate region R(e t ). Region depends on: Available channel quality info e t, Physical layer resource allocation, MAC layer multiplexing. R. Berry (NWU) OFDMA Scheduling CTW / 22

6 State-dependent Feasible Rate Regions Optimization is over feasible rate region R(e t ). Region depends on: Available channel quality info e t, Physical layer resource allocation, MAC layer multiplexing. E.g. TDMA systems/full CSI R(et ) = simplex with max rate r i for each user i. Gradient-policy schedule users with max Ui (X i )r i. R. Berry (NWU) OFDMA Scheduling CTW / 22

7 State-dependent Feasible Rate Regions Optimization is over feasible rate region R(e t ). Region depends on: Available channel quality info e t, Physical layer resource allocation, MAC layer multiplexing. E.g. TDMA systems/full CSI R(et ) = simplex with max rate r i for each user i. Gradient-policy schedule users with max Ui (X i )r i. In many systems, additional multiplexing within a time-slot. e.g. CDMA (HSDPA), OFDMA (802.16). Requires allocating physical layer resources among scheduled users. R. Berry (NWU) OFDMA Scheduling CTW / 22

8 OFDMA systems Frequency band divided into N subcarriers/tones. Resource allocation: assignment of tones to users allocation of power across tones. R. Berry (NWU) OFDMA Scheduling CTW / 22

9 OFDMA rate region Initially, allow users to time-share each subchannel In practice, one user/tone. Assume rate/subchannel = log(1 + SNR). Rate region (similar to [Li,Goldsmith], [Wang, et. al]): { R(e) = r : r i = ( x ij log 1 + p ) ije ij, p ij P, x ij j ij } x ij 1, j, (x, p) X, where i X := {(x, p) 0 : xij 1, i, j}. xij = fraction of subchannel j allocated to user i. p ij = power allocated to user i on subchannel j. e ij = received SNR/unit power. R. Berry (NWU) OFDMA Scheduling CTW / 22

10 Model Variations 1 Maximum SINR constraint: s ij (limit on modulation order) Let X := { (x, p) 0 : 0 x ij 1, 0 p ij x ijs ij e ij } i, j. R. Berry (NWU) OFDMA Scheduling CTW / 22

11 Model Variations 1 Maximum SINR constraint: s ij (limit on modulation order) Let X := { (x, p) 0 : 0 x ij 1, 0 p ij x ijs ij e ij } i, j. 2 Sub-channelization (bundle tones to reduce overhead) Possible channelizations: Interleaved ( standard mode) Adjacent (Band AMC mode) Random (e.g. frequency hopped) Can accommodate by letting x ij = allocation of subchannel j. View eij as average SNR/subchannel. R. Berry (NWU) OFDMA Scheduling CTW / 22

12 Model Variations 1 Maximum SINR constraint: s ij (limit on modulation order) Let X := { (x, p) 0 : 0 x ij 1, 0 p ij x ijs ij e ij } i, j. 2 Sub-channelization (bundle tones to reduce overhead) Possible channelizations: Interleaved ( standard mode) Adjacent (Band AMC mode) Random (e.g. frequency hopped) Can accommodate by letting x ij = allocation of subchannel j. View eij as average SNR/subchannel. 3 Self-interference: SINR ij = e ij p ij x ij + αe ij p ij. R. Berry (NWU) OFDMA Scheduling CTW / 22

13 Optimal Scheduling algorithm The optimal gradient-based scheduling algorithm must solve: max V (x, p) := x ij,p ij X i subject to: i,j w i p ij P, and i j ( x ij log 1 + p ) ije ij x ij x ij 1, j N, (OPT) w i = U i. Need to re-solve every scheduling interval. We consider optimal and suboptimal algorithms for this. R. Berry (NWU) OFDMA Scheduling CTW / 22

14 Optimal algorithm Scheduling problem (OPT) is convex and has no duality gap. Consider Lagrangian: L(x, p, λ, µ) := i w i ( + λ P i,j j ( x ij log 1 + p ) ije ij x ij p ij ) + j µ j ( 1 i x ij ). Associated dual function: L(λ, µ) = By duality, solution to (OPT) is: V = max L(x, p, λ, µ) (x,p) X min L(λ, µ) (λ,µ) 0 R. Berry (NWU) OFDMA Scheduling CTW / 22

15 Dual Function Can explicitly solve for the dual function. Fixing x, λ, µ, optimizing over p ij water-filling like solution. pij = x [ (wi ij e ij e ij λ 1 ) + sij ]. R. Berry (NWU) OFDMA Scheduling CTW / 22

16 Dual Function Can explicitly solve for the dual function. Fixing x, λ, µ, optimizing over p ij water-filling like solution. Given optimum p ij, pij = x [ (wi ij e ij e ij λ 1 ) + sij ]. L(x, p, λ, µ) = ij x ij (µ ij (λ) µ j ) + j µ j + λp Optimizing over xij [0, 1] is now easy. L(λ, µ) = ij (µ ij(λ) µ j ) + + j µ j + λp R. Berry (NWU) OFDMA Scheduling CTW / 22

17 Minimizing the dual function Dual function: L(λ, µ) = ij (µ ij (λ) µ j ) + + j µ j + λp. First minimize over µ: L(λ) := min µ 0 L(λ, µ) = λp + j max µ ij (λ). i Requires one sort of users per subchannel. R. Berry (NWU) OFDMA Scheduling CTW / 22

18 Minimizing the dual function Dual function: L(λ, µ) = ij (µ ij (λ) µ j ) + + j µ j + λp. First minimize over µ: L(λ) := min µ 0 L(λ, µ) = λp + j max µ ij (λ). i Requires one sort of users per subchannel. L(λ) is convex function of λ. Can minimize using iterated 1-D search (e.g. golden section). R. Berry (NWU) OFDMA Scheduling CTW / 22

19 Optimal Primal Values. Given λ, µ, let (x, p ) = arg max (x,p) X L(x, p, λ, µ ). (*) If (x, p ) are primal feasible and satisfy complimentary slackness, they are an optimal scheduling decision. Can find these as before, except multiple µ ij s may be tied at the maximum value. Multiple x ij s can be > 0. Not all choices result in feasible primal solutions. R. Berry (NWU) OFDMA Scheduling CTW / 22

20 Breaking ties - optimal time-sharing When ties occur, can show L(λ) is not differentiable. Each (x, p ) that satisfy (*) and complimentary slackness give a subgradient of L(λ). Simple sort can find max and min subgradients (one user/subchannel). Time-sharing between these gives a primal optimal solution. At most 2 users/subchannel. R. Berry (NWU) OFDMA Scheduling CTW / 22

21 Single User per Subchannel Heuristic In practice typically restricted to one user/subchannel. If no ties in optimal dual solution, this will be satisfied. When ties occurs, selecting one user involved in the tie corresponds to choosing one subgradient. In simulations, we choose the user that corresponds to the smallest negative subgradient. Other heuristics also possible. Resulting power constraint may not be tight. R. Berry (NWU) OFDMA Scheduling CTW / 22

22 Re-optimizing the power allocation Given a feasible x, consider max V (x, p) s.t. p:(p,x) X ij p ij P solution again given by water-filling like power allocation with a given Lagrange multiplier λ. Optimal λ can be shown to satisfy fixed point equation λ = f (λ), f (λ) is increasing, finite-valued (piece-wise constant). finite time algorithm for finding λ. R. Berry (NWU) OFDMA Scheduling CTW / 22

23 Single Sort Heuristic Optimal subchannel assignment is to user with max µ ij (λ). Requires iterating to find optimal λ. Instead consider single-sort using metric w ij Rij, R ij = log[1 + (s ij (e ij P/N))]. Motivated by e.g. [Hoo, et al.]. Then optimally allocate power as before. Also looked at other heuristics. R. Berry (NWU) OFDMA Scheduling CTW / 22

24 Numerical Results Simulation set-up: Single cell, M = 40 users. e ij = (fixed location-based term) (frequency selective fast fading) Fixed term = empirical distribution. frequency selective term = block fading in time (2msec coh. time); standard ref. mobile delay spread (1 µsec ). 5 MHz BW, 512 tones. Initially adjacent channelization, 8 tones/subchannel. use α-utility functions. Simulate full algorithm (with one user/subchannel) and single sort. R. Berry (NWU) OFDMA Scheduling CTW / 22

25 Different choices of α α Algorithm Utility Log U Rate(kbps) Num. 0.5 FULL MO-w R FULL MO-w R FULL MO-w R R. Berry (NWU) OFDMA Scheduling CTW / 22

26 User throughput CDFs 1 User Throughput CDF α= FULL GOLDEN 1 WEIGHTED 1 MO wr Throughput in Kbps α = 0.5. R. Berry (NWU) OFDMA Scheduling CTW / 22

27 Different channelization schemes Chan. Algorithm Utility Log U Rate (kbps) Num. Adj. FULL Adj. MO-w R Ran. FULL Ran. MO-w R Int. FULL Int. MO-w R Upperbound on rate/channel; looser for interleaved/random case. R. Berry (NWU) OFDMA Scheduling CTW / 22

28 Conclusions Presented optimal and sub-optimal algorithms for gradient-based scheduling in OFDM systems. Can accommodate different channelizations and max. SINR constraints. Subchannel allocation is based on a sort metric that depends on power constraint Lagrange multiplier. Can solve dual problem with geometric rate of convergence. Given suchannel allocation, can optimize power in finite time. Simple sort has near optimal performance. Can extended the model to include self-interference. R. Berry (NWU) OFDMA Scheduling CTW / 22