1. Cross-Layer Optimization in Wireless Networks under Different Packet Delay Metrics Chris T. K. Ng, Muriel Medard, Asuman Ozdaglar Massachusetts Institute of Technology

2. Introduction Reliable communications over unreliable wireless channels. Physical layer: channel coding. Erasure channel: coding across packets. Fundamental tradeoff in coding. Long coding blocks are more effective in mitigating channel variations. But introduce larger decoding delay. End-to-end performance depends on parameters across networks layers: Delay sensitivity, packet coding strategy, transmission SNR target, power allocation among users. 2

3. End-to-end Performance Metrics Physical layer link performance: Instantaneous rate and outage probability. Cannot resolve system-level design choices: Higher rate with greater outage probability, or vice versa? To optimize end-to-end performance, need to additionally consider: i.User decoding delay requirements. ii.How and when the transmitter learns about the outage event. iii.Retransmission or coding strategy that recovers the outage data loss. Cross-layer model to jointly optimize packet level and physical layer parameters. 3

4. Packet Erasure Channel Packet erasure channel with delayed acknowledgment feedback. In-order packet delivery; erasure probability q. ACK/NACK feedback after D time slots. Linear packet coding: Transmitter may combine (encode) source packets to form a coded packet. Coded packet is a linear combination of the source packets. Receiver knows the transmitter’s coding scheme. 4

5. Packet Delay Cost Function Inter-decoding times: Delay cost function: Normalized p-norm of the expected inter-decoding times: Larger p: more sensitive to delay between decoding times. When p=1; expected completion time: When p=∞; per-packet delay: 5

6. Block-by-Block Packet Coding Transmitter sends linearly independent coded packets. Over a block size of k packets until receives ACK. Tradeoff between completion time and per-packet delay: Optimize block size k based on delay sensitivity p: 6 Completion Time Per-Packet Delay

7. Wireless Erasure Channel Fading wireless channel: With additive white Gaussian noise: Packet erasure induced by small-scale channel fading. Shadowing: G can be accurately estimated. Fading: F is a random variable; transmitter knows only its distribution. Transmission outage leads to packet erasure. Transmitter picks SNR target s. Outage/erasure probability: q = Pr{ realized SNR < s }. 7

8. Single-User Wireless Erasure Channel Optimization Target SNR optimization: Jointly optimize packet coding block size k and target SNR s. According to user delay sensitivity p. Convex optimization problem: Change of variables transformation similar to that in geometric programming (GP). Efficiently solved by standard numerical methods (e.g., with the CVX software package). 8

9. Optimal Target SNR and Outage Probability Large p: leads to short coding blocks. More redundancy at packet level allows more aggressive SNR target. Target SNR increases with average SNR, but not as fast; hence outage probability drops. 9 Optimal Target SNR Outage Probability

10. Wireless Packet Network Multiuser wireless erasure channels: M users in the network: Transmission from each user interferes with one another. 10

11. Optimal Transmit Power Allocation Need to optimize power allocation among users. Transmit at maximum power is not necessarily optimal due to interference. Power constraint for each user: Interference is treated as noise. Signal to interference-plus-noise ratio (SINR) at receiver i : Outage probability: q i = Pr{ realized S i < target s i }. Power allocation among the users are coupled in the outage probability constraints on the SINRs: 11

12. Cross-Layer Optimization Minimize global cost function: J(d) jointly convex in d 1,…,d M. Convexity of J(d) penalizes overlong user delays. Independent Rayleigh fading channels: 12

13. Convex Optimization Problem Minimizing the global cost function can be formulated as a convex optimization problem. Transformation similar to the single-user channel optimization previously considered. We assume The optimization formulation is otherwise valid for all ranges of SINR. 13

14. Convexity of the Feasibility Regions In a wireless network (when interference is treated as noise), the feasible rate region is not convex. However, the corresponding feasible delay region is convex. Delay performance metrics: Allows joint optimization over physical layer and packet level parameters. 14 Rate RegionDelay Region

15. Conclusions Packet coding strategy: Tradeoff between expected completion time and per-packet delay. Optimal block size in block-by-block packet coding. Wireless erasure channels: Packet erasure caused by outage induced by small-scaling fading. Joint optimization of packet coding block size and target SNR. Multiuser wireless packet network: Transmission of each user interferes with one another. Optimal power allocation to minimize a global delay cost function. Can be formulated as a convex optimization problem. 15