Optimal Energy and Delay Tradeoffs for Multi-User Wireless Downlinks Michael J. Neely University of Southern California Infocom 2006, Barcelona, Spain *Sponsored by NSF OCE Grant N Avg. Delay Avg. Power

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

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

rate  i power P Good Med Bad 1 2 N Control: Allocate Power (P(t)  ) in Reaction to Current Channel State And Current Queue Backlogs. Goal: Stabilize with Minimum Average Power while also Maintaining Low Average Delay.

rate  i power P Good Med Bad 1 2 N Control: Allocate Power (P(t)  ) in Reaction to Current Channel State And Current Queue Backlogs. Goal: Stabilize with Minimum Average Power while also Maintaining Low Average Delay. [ Avg. Power and Avg. Delay are Competing Objectives! ] What is the Fundamental Energy-Delay Tradeoff?

1 2 N Our Previous Work on Minimum Power Scheduling with Delay Tradeoffs [Neely Infocom 2005]: U N (t)U 1 (t) U 2 (t) V P V Av. Delay O(1/V) O(V) P* ( P* = Min Av. Power for Stability )

1 2 N Our Previous Work on Minimum Power Scheduling with Delay Tradeoffs [Neely Infocom 2005]: U N (t)U 1 (t) U 2 (t) V P V Av. Delay O(1/V) O(V) P* ( P* = Min Av. Power for Stability )

1 2 N Our Previous Work on Minimum Power Scheduling with Delay Tradeoffs [Neely Infocom 2005]: U N (t)U 1 (t) U 2 (t) ( P* = Min Av. Power for Stability ) V P V Av. Delay O(1/V) O(V) P*

1 2 N Our Previous Work on Minimum Power Scheduling with Delay Tradeoffs [Neely Infocom 2005]: U N (t)U 1 (t) U 2 (t) ( P* = Min Av. Power for Stability ) V P V Av. Delay O(1/V) O(V) P*

1 2 N Our Previous Work on Minimum Power Scheduling with Delay Tradeoffs [Neely Infocom 2005]: U N (t)U 1 (t) U 2 (t) ( P* = Min Av. Power for Stability ) V P V Av. Delay O(1/V) O(V) P*

1 2 N Our Previous Work on Minimum Power Scheduling with Delay Tradeoffs [Neely Infocom 2005]: U N (t)U 1 (t) U 2 (t) ( P* = Min Av. Power for Stability ) V P V Av. Delay O(1/V) O(V) P*

1 2 N Our Previous Work on Minimum Power Scheduling with Delay Tradeoffs [Neely Infocom 2005]: U N (t)U 1 (t) U 2 (t) ( P* = Min Av. Power for Stability ) V P V Av. Delay O(1/V) O(V) P*

1 2 N Our Previous Work on Minimum Power Scheduling with Delay Tradeoffs [Neely Infocom 2005]: U N (t)U 1 (t) U 2 (t) Analysis: Use theory of Performance Optimal Lyapunov Scheduling: -Neely, Modiano 2003, Georgiadis, Neely, Tassiulas [F&T 2006, NOW Publishers] Achieves: [O(1/V), O(V)] energy-delay tradeoff V P V Av. Delay O(1/V) O(V) P*

1 2 N Our Previous Work on Minimum Power Scheduling with Delay Tradeoffs [Neely Infocom 2005]: V P V Av. Delay O(1/V) O(V) U N (t)U 1 (t) U 2 (t) Analysis: Use theory of Performance Optimal Lyapunov Scheduling: -Neely, Modiano 2003, Georgiadis, Neely, Tassiulas [F&T 2006, NOW Publishers] Achieves: [O(1/V), O(V)] energy-delay tradeoff P*

V P V Av. Delay O(1/V)  ( V ) P* The Fundamental Berry-Gallager Bound for Energy-Delay Tradeoffs in a Single Wireless Downlink: A(t)  (t) =  (P(t), S(t)) Av. Delay >=  ( V ) [Berry, Gallager 2002] Approach Achievability via Technique of Buffer Partitioning.

Precedents for Energy and Delay Optimization for Single Wireless Links: -Berry and Gallager [2002] ( Fundamental Square Root Law ) -Uysal-Biyikoglu, Prabhakar, El Gamal [2002] -Khojastepour and Sabharwal [2004] ( “Lazy Scheduling” and Filter Theory for Static Links ) -Fu, Modiano, Tsitsiklis [2003] -Goyal, Kumar, Sharma [2003] -Zafer and Modiano [2005] ( Dynamic Programming, Markov Decision Theory )

Precedents for Energy and Delay Optimization for Single Wireless Links: -Berry and Gallager [2002] ( Fundamental Square Root Law ) -Uysal-Biyikoglu, Prabhakar, El Gamal [2002] -Khojastepour and Sabharwal [2004] ( “Lazy Scheduling” and Filter Theory for Static Links ) -Fu, Modiano, Tsitsiklis [2003] -Goyal, Kumar, Sharma [2003] -Zafer and Modiano [2005] ( Dynamic Programming, Markov Decision Theory )

Challenging to extend optimal delay results for stochastic systems beyond a single queue because… 1)Parameter Explosion: (cannot practically measure) Number of channel state vectors S grows geometrically with number of links N. Markov Decision Theory and Dynamic Programming requires knowledge of:  S = Pr[ S(t) = S] (for each channel state S ). 2) State Space Explosion: (cannot practically implement) Number of Queueing State Vectors U grows geometrically.

Idea: Combine Techniques of Buffer Partitioning and Performance Optimal Lyapunov Scheduling. 1 2 N V P V Av. Delay P* Goals: 1)Establish the fundamental energy-delay curve for multi-user systems (extend Berry-Gallager to this case). 2)Design a dynamic algorithm to achieve optimal energy-delay tradeoffs. (Must overcome the complexity explosion problem).

Specifically: Define a general power cost metric h( P ) : 1 2 N Define average power cost: Define: h* = Min. avg. power cost for network stability (Push h arbitrarily close to h*, with optimal delay tradeoff…)

Theorem 1: (Characterize h*) Assume . The min average power cost h* is given as the solution to: Define  ( ) = min. avg. power cost h* above. Corollary: For each, there is a stationary randomized alg. such that: 

The Fundamental Energy-Delay Tradeoff:    mild admissibility assumptions Theorem 2 (Multi-User Berry-Gallager Bound): If Then if avg. cost satisfies: We necessarily have: 1 2 N V h V Av. Delay h*  ( V ) O(1/V)

Achieving Optimal Tradeoffs via Buffer Partitioning… Recall the Berry-Gallager threshold algorithm for single queues:  (t) =  (P(t), S(t)) U(t)  max Q U Q drift    L R [Requires full knowledge of channel probs  S ]

Achieving Optimal Tradeoffs via Buffer Partitioning… Recall the Berry-Gallager threshold algorithm for single queues:  (t) =  (P(t), S(t)) U(t)  max Q U Q drift    L R

Achieving Optimal Tradeoffs via Buffer Partitioning… Recall the Berry-Gallager threshold algorithm for single queues:  (t) =  (P(t), S(t)) U(t)  max Q U Q drift    L R

Let’s Consider Multi-Dimensional Buffer Partitioning:         Q U1U1 U2U2 Case N=2 1 2 N Q

Let’s Consider Multi-Dimensional Buffer Partitioning:         Q U1U1 U2U2 Case N=2 Q

Let’s Consider Multi-Dimensional Buffer Partitioning:         Q U1U1 U2U2 Case N=2 (not implementable) Q

Analysis of the Threshold Algorithm: (exchanging sums over the 2 N regions yields…)  i L (t) = Pr[U i (t) <Q]  i R (t) = 1 -  i L (t)

An Online Algorithm for Optimal Energy-Delay Tradeoffs: 1 2 N Define the bi-modal Lyapunov Function: UiUi Q Designing “gravity” into the system:

An Online Algorithm for Optimal Energy-Delay Tradeoffs: 1 2 N Define the bi-modal Lyapunov Function: UiUi Q Designing “gravity” into the system: “Usually” creates proper drift direction…

1 2 N *Key inequality that holds with equality for the stationary threshold algorithm. Need to strengthen the drift guarantees… Want to also ensure for all i {1, 2, …, N}

Need to strengthen the drift guarantees… Want to also ensure for all i {1, 2, …, N} Use Virtual Queue Concept from [Neely Infocom 2005]: X i (t) A i (t) +  1 i R (t)  i (t) +  1 i L (t) indicator functions X i (t+1) = max[X i (t) -  i (t) -  1 i L (t), 0] + A i (t) +  1 i R (t)

Need to strengthen the drift guarantees… Want to also ensure for all i {1, 2, …, N} Use Virtual Queue Concept from [Neely Infocom 2005]: X i (t) A i (t) +  1 i R (t)  i (t) +  1 i L (t) indicator functions X i (t) Stable  i +  1 i L > i +  1 i R

To Stabilize Virtual Queues X i (t) and Actual Queues U i (t):

The Tradeoff Optimal Control Algorithm (TOCA): 1) Every slot t, observe channel state S(t) and queue backlogs U(t), X(t). Allocate power P(t) = P, where P solves: 2) Transmit with rate  i (t) =  i (P(t), S(t)). 3) Update the Virtual Queues X i (t).

Theorem 3 (TOCA Performance): For suitable , Q: 1 2 N V hAv. Delay  ( ) V O(1/V)  ( V )

Beyond the Berry-Gallager Bound: Logarithmic delay! If the Minimum Energy function  ( ) is peicewise linear (not strictly concave), then under suitable , Q, TOCA yields:  ( ) (shown in 1 dimension)

Further, logarithmic delay in this scenario is optimal! Simple One Queue Example: P(t)  ={0, 1} Watt. Two Equally Likely Channel States (GOOD, BAD): U(t)  (t)=  (P(t),S(t))  ( ) Can show that logarithmic delay is necessary and achievable!

Conclusions: 1 2 N V hAv. Delay  ( ) V O(1/V)  ( V ) -Extend Berry-Gallager Square Root Law to Multi-User Systems. -Novel Lyapunov Technique for Achieving Optimal Energy-Delay Tradeoffs. -Overcome the Complexity Explosion Problem. -Channel Statistics, Traffic Rates not Required. -Superior Tradeoff via a Logarithmic Delay Law in exceptional (piecewise linear) cases.