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

Time is slotted Fading is assumed slowly varying (block fading) 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) Channel System Architecture (1/5)

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 System Architecture (2/5)

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

Rate-function Energy constraints:  Average energy for slot: ɛ MAX (Joule)  Peak energy for slot: ɛ P (Joule) Energy constraints Arrival process Link state System Architecture (4/5)

Rate-function Energy constraints Given the energy constraints ( ɛ MAX and ɛ P ) and the traffic patterns (p(a),λ), how much energy must be radiated slot by slot to minimize the avegare queue length S AVE ? Arrival process Link state System Architecture (5/5)

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

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

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

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

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

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

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

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

Transmit buffer Wireless Link The Unconditional Optimal Scheduler

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