Intelligent Packet Dropping for Optimal Energy-Delay Tradeoffs for Wireless Michael J. Neely University of Southern California ( full paper to appear in WiOpt 2006 ) A(t)  (p(t), s(t)) Delay Energy *Sponsored by NSF OCE Grant

A(t)  (P(t), S(t)) Assumptions: 1)Random Arrivals A(t) i.i.d. over slots. (Rate  bits/slot) 2) Random Channel states S(t) i.i.d. over slots. 3) Transmission Rate Function P(t) --- Power allocation during slot t S(t) --- Channel state during slot t t … Time slotted system (t {0, 1, 2, …}) rate  power P  (P(t), S(t)) Good Med Bad

Fundamental Energy-Delay Tradeoff Theory and the Berry-Gallager Bound: A(t)  (P(t), S(t)) Avg. Power Avg. Delay  ( ) = Min. Avg. Energy Required for Stability [Berry 2000, 2002]

Fundamental Energy-Delay Tradeoff Theory and the Berry-Gallager Bound: Avg. Power Avg. Delay In terms of a dimensionless index parameter V>0: VV O(1/V) [Berry 2000, 2002]

Fundamental Energy-Delay Tradeoff Theory and the Berry-Gallager Bound: Avg. Power Avg. Delay VV O(1/V) [Berry 2000, 2002] In terms of a dimensionless index parameter V>0:

Fundamental Energy-Delay Tradeoff Theory and the Berry-Gallager Bound: Avg. Power Avg. Delay VV O(1/V) [Berry 2000, 2002] In terms of a dimensionless index parameter V>0:

Fundamental Energy-Delay Tradeoff Theory and the Berry-Gallager Bound: Avg. Power Avg. Delay VV O(1/V) [Berry 2000, 2002] In terms of a dimensionless index parameter V>0:

Fundamental Energy-Delay Tradeoff Theory and the Berry-Gallager Bound: Avg. Power Avg. Delay VV O(1/V) [Berry 2000, 2002] In terms of a dimensionless index parameter V>0:

Fundamental Energy-Delay Tradeoff Theory and the Berry-Gallager Bound: Avg. Power VV O(1/V) Avg. Delay Berry-Gallager Bound Assumes: 1.Admissibility criteria 2.Concave rate-power function 3.i.i.d. arrivals A(t) 4. No Packet Dropping

 (P(t), S(t)) Our Formulation: Intelligent Packet Dropping Control Variables: Goal: Obtain an optimal energy-delay tradeoff Subject to: Admitted rate >=  A(t) (rate  (1-  )  ( 0 <  < 1 )

Energy-Delay Tradeoffs with Packet Dropping…  * =  (  ) = New Min. Average Power Expenditure (required to support rate  ). Avg. Power Avg. Delay VV O(1/V) A(t) (rate  (1-  )  ?

 * =  (  ) = New Min. Average Power Expenditure (required to support rate  ). Avg. Power Avg. Delay VV O(1/V) A(t) (rate  (1-  )  ? Energy-Delay Tradeoffs with Packet Dropping…

 * =  (  ) = New Min. Average Power Expenditure (required to support rate  ). Avg. Power Avg. Delay VV O(1/V) A(t) (rate  (1-  )  ? Energy-Delay Tradeoffs with Packet Dropping…

 * =  (  ) = New Min. Average Power Expenditure (required to support rate  ). Avg. Power Avg. Delay VV O(1/V) A(t) (rate  (1-  )  ? Energy-Delay Tradeoffs with Packet Dropping…

An Example of Naïve Packet Dropping: Random Bernoulli Acceptance with probability  Avg. Power V O(1/V) A(t) (rate  (1-  )   * =  (  ) Consider a system that satisfies all criteria for the Berry-Gallager bound, including i.i.d. arrivals every slot. After random packet dropping, arrivals are still i.i.d…. Avg. Delay V

An Example of Naïve Packet Dropping: Random Bernoulli Acceptance with probability  Avg. Power V O(1/V) A(t) (rate  (1-  )   * =  (  ) Consider a system that satisfies all criteria for the Berry-Gallager bound, including i.i.d. arrivals every slot. After random packet dropping, arrivals are still i.i.d., and hence performance is still governed by Berry-Gallager square root law. Avg. Delay V

But here we consider Intelligent Packet Dropping: Avg. Power V O(1/V) A(t) (rate  (1-  )   * =  (  ) Avg. Delay V achievable! Thus: The square root curvature of the Berry Gallager bound is due only to a very small fraction of packets that arrive at innopportune times.

Algorithm Development: A preliminary Lemma: Lemma: If channel states are i.i.d. over slots: For any stabilizable input rate, there exists a stationary randomized algorithm that chooses power P*(t) based only on the current channel state S(t), and yields: *This is an existential result: Constructing the policy could be difficult and would require full knowledge of channel probabilities.

Algorithm 1: (Known channel probabilities) The Positive Drift Algorithm: Step 1 -- Emulate a finite buffer queueing system: A(t) U(t) Q = max buffer size

(where  <  < 1) rate rate  0  max Q Positive drift! Step 2 -- Apply the stationary policy P*(t) such that:

(where  <  < 1) rate rate  0  max Q Positive drift! Step 2 -- Apply the stationary policy P*(t) such that: Choose:  = O(1/V), Q = O(log(V))

Algorithm 2: (Unknown channel probabilities) Constructing a practical Dynamic Packet Dropping Algorithm: 0  max Q Define the Lyapunov Function: U(t) L(U) = e  (Q-U) 0 Q U L(U) …but we still want to maintain  av at least  … rate  (P(t), S(t))

Use the “virtual queue” concept for time average inequality constraints [Neely Infocom 2005] A(t) (rate  U(t)  (P(t), S(t))   av < Want to ensure: X(t)  (P(t), S(t))  )A(t)

Let Z(t) := [U(t); X(t)] Form the mixed Lyapunov function: Define the Lyapunov Drift: Lyapunov Optimization Theory [Neely, Modiano 03, 05]: Similar to concept of “stochastic gradient” applied to a flow network -- [Lee, Mazumdar, Shroff 2005]

The Dynamic Packet Dropping Algorithm: Every timeslot, observe: Queue values U(t), X(t) and Channel State S(t) 1. Allocate power P(t) that solves: 2. Iterate the virtual queue X(t) update equation with 3. Emulate the Finite Buffer Queue U(t).

Avg. Power V O(1/V)  * =  (  ) Avg. Delay V achievable! Theorem: For the Dynamic Packet Dropping Alg.

Conclusions: The Dynamic Algorithm does not require knowledge of channel probabilities, and yields a logarithmic power-delay tradeoff. Intelligent Packet Dropping Fundamentally improves the Power-delay tradeoff (from square root law to logarithm). Further: For a large class of systems, the [O(1/V), O(log(V))] tradeoff is necessary! Energy-Delay Tradeoffs for Multi-User Systems [Neely Infocom 06] “Super-fast” flow control for utility-delay tradeoffs [Neely Infocom 06]