System constraints Average and peak energy Quality of Service (QoS): Maximum delay

1. The Problem: Optimized Management of Resources • Inadequate conventional routers • Battery-powered wireless nodes • Need to take into account for the nature of time-varying wireless channels • System constraints • Average and peak energy • Quality of Service (QoS): • Maximum delay • Data Link Queue Length? TARGET Energy – vs – Queue Length trade-off To design computationalliy efficient schedulers, that optimally allocate the energy for bursty sources over the wireless channels

2. System Architecture (1/5) • Time is slotted • Fading is assumed slowly varying (block fading) Channel • Current value of the channel-state known slot by slot • Channel probability density function known at the controller (the hypothesis will be removed in the following)

3. System Architecture (2/5) • Random variable (r.v.) a(t) with probability density p(a) known at the trasmitter (the hypothesis will be removed in the following) • λ(t) (IU/slot) number of controlled IU arriving at the input of the queue at the end of slot t Arrival process Link state

4. System Architecture (3/5) • Rate-function IU(t) of the considered system • Summarizes: • The coding system • The modulation scheme • The error probability PE • (ex. 16-QAM, RS 2/3) Rate-function of the considered system Arrival process Link state

5. System Architecture (4/5) • Energy constraints: • Average energy for slot: ɛMAX (Joule) • Peak energy for slot: ɛP (Joule) Rate-function Arrival process Energy constraints Link state

6. System Architecture (5/5) Given the energy constraints (ɛMAX and ɛP) and the traffic patterns (p(a),λ), how much energy must be radiated slot by slot to minimize the avegare queue length SAVE? Rate-function Arrival process Energy constraints Link state

7. Formulation problem (1/2) Transmit buffer VBR - Encoder Wireless Link with Fading Scheduler Cross-layer VBR - Decoder Data Link Layer Physical Layer • probability density of arrivals: Known • average number of arrivals • number of the IUs buffered in the queue at the beginning of slot t

8. Formulation problem (2/2) p(s) depends in an impredictible way unknown on the channel statistics, arrival statistics and service discipline Computationally intractable problem.

9. Unconditional-vs.-Conditional Optimum (1/3) Conditional Problem Unconditional Problem

10. Unconditional-vs.-Conditional Optimum (2/3) Conditional Problem Unconditional Problem Wider energy domain Smaller energy domain (stronger constraint)

11. Unconditional-vs.-Conditional Optimum (3/3) Conditional Problem Wider energy domain Smaller energy domain (stronger constraint)

12. Unconditional-vs.-Conditional Optimum How to generalize the optimal scheduler in the stronger energy domain to the wider domain? Wider energy domain Smaller energy domain (stronger constrait)

13. If is local minimum, such that the following conditions are met: Conditional Approach Conditional scheduler (convex optimization) Objective function Constraints with Lagrange Multiplier: cross-layer parameter

14. Towards the Unconditional Optimal Scheduler (1/2) Conditional Problem Unconditional Problem Constant: No Buffer Depending Buffer Depending To design the scheduler as if the probability density p(s) was known

15. The Unconditional Optimal Scheduler Transmit buffer Wireless Link

16. Unconditional Optimal Multiplier: Real-time computation Transmit buffer Wireless Link