Optimal Energy and Delay Tradeoffs forMulti-User Wireless Downlinks

Michael J. NeelyUniversity of Southern California

http://www-rcf.usc.edu/~mjneely/Infocom 2006, Barcelona, Spain

*Sponsored by NSF OCE Grant 0520324

1 2 N

Avg. Delay

Avg

. Pow

er

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

t0 1 2 3 …

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

t0 1 2 3 …

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]:

UN(t)U1(t) U2(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]:

UN(t)U1(t) U2(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]:

UN(t)U1(t) U2(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]:

UN(t)U1(t) U2(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]:

UN(t)U1(t) U2(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]:

UN(t)U1(t) U2(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]:

UN(t)U1(t) U2(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]:

UN(t)U1(t) U2(t)

Analysis: Use theory of Performance Optimal Lyapunov Scheduling: -Neely, Modiano 2003, 2005-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)

UN(t)U1(t) U2(t)

Analysis: Use theory of Performance Optimal Lyapunov Scheduling: -Neely, Modiano 2003, 2005-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-DelayTradeoffs 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 Optimizationfor 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 Optimizationfor 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:

1) ))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

QUQmax

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

QUQmax

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

QUQmax

drift

L R

Let’s Consider Multi-Dimensional Buffer Partitioning:

Q

U1

U2Case N=2

1 2 N

Q

Let’s Consider Multi-Dimensional Buffer Partitioning:

Q

U1

U2Case N=2

Q

Let’s Consider Multi-Dimensional Buffer Partitioning:

Q

U1

U2Case N=2

(not implementable)

Q

Analysis of the Threshold Algorithm:(exchanging sums over the 2N regions yields…)

iL(t) = Pr[Ui(t) <Q]

iR(t) = 1 - i

L(t)

An Online Algorithm for Optimal Energy-Delay Tradeoffs:

1 2 N

Define the bi-modal Lyapunov Function:

UiQ

Designing “gravity”into the system:

An Online Algorithm for Optimal Energy-Delay Tradeoffs:

1 2 N

Define the bi-modal Lyapunov Function:

UiQ

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]:

Xi(t)Ai(t) + 1i

R(t) i(t) + 1iL(t)

indicator functions

Xi(t+1) = max[Xi(t) - i(t) - 1iL(t), 0] + Ai(t) + 1i

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]:

Xi(t)Ai(t) + 1i

R(t) i(t) + 1iL(t)

indicator functions

Xi(t) Stable i + 1iL > i + 1i

R

To Stabilize Virtual Queues Xi(t) and Actual Queues Ui(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 Xi(t).

Theorem 3 (TOCA Performance): For suitable , Q:

1 2 N

V

h Av.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 thatlogarithmic delay is necessaryand achievable!

Conclusions:1 2 N

V

h Av.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.