Intelligent Packet Dropping for Optimal Energy-Delay Tradeoffs for Wireless

Michael J. NeelyUniversity of Southern Californiahttp://www-rcf.usc.edu/~mjneely/(full paper to appear in WiOpt 2006)

A(t)

(p(t), s(t))Delay

Ener

gy

*Sponsored by NSF OCE Grant 0520324

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

t0 1 2 3 …

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

. Pow

er

Avg. Delay

() = Min. Avg. Energy Required for Stability

[Berry 2000, 2002]

Fundamental Energy-Delay Tradeoff Theory and the Berry-Gallager Bound:

Avg

. Pow

er

Avg

. Del

ayIn terms of a dimensionless index parameter V>0:

V V

O(1/V)

[Berry 2000, 2002]

Fundamental Energy-Delay Tradeoff Theory and the Berry-Gallager Bound:

Avg

. Pow

er

Avg

. Del

ay

V V

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

. Pow

er

Avg

. Del

ay

V V

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

. Pow

er

Avg

. Del

ay

V V

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

. Pow

er

Avg

. Del

ay

V V

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

. Pow

er

V V

O(1/V)

Avg

. Del

ay

Berry-Gallager Bound Assumes: 1. Admissibility criteria2. Concave rate-power function3. 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

. Pow

er

Avg

. Del

ay

V V

O(1/V)

A(t)(rate

(1-)

?

* = () = New Min. Average Power Expenditure (required to support rate ).

Avg

. Pow

er

Avg

. Del

ay

V V

O(1/V)

A(t)(rate

(1-)

?

Energy-Delay Tradeoffs with Packet Dropping…

* = () = New Min. Average Power Expenditure (required to support rate ).

Avg

. Pow

er

Avg

. Del

ay

V V

O(1/V)

A(t)(rate

(1-)

?

Energy-Delay Tradeoffs with Packet Dropping…

* = () = New Min. Average Power Expenditure (required to support rate ).

Avg

. Pow

er

Avg

. Del

ay

V V

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

. Pow

er

V

O(1/V)

A(t)(rate (1-)

* = ()

Consider a system that satisfies all criteria for the Berry-Gallagerbound, including i.i.d. arrivals every slot.After random packet dropping, arrivals are still i.i.d….

Avg

. Del

ay

V

An Example of Naïve Packet Dropping:Random Bernoulli Acceptance with probability

Avg

. Pow

er

V

O(1/V)

A(t)(rate (1-)

* = ()

Consider a system that satisfies all criteria for the Berry-Gallagerbound, 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

. Del

ay

V

But here we consider Intelligent Packet Dropping:A

vg. P

ower

V

O(1/V)

A(t)(rate (1-)

* = ()

Avg

. Del

ay

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

0maxQ

Positive drift!

Step 2 -- Apply the stationary policy P*(t) such that:

(where < < 1)

rate rate

0maxQ

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:

0maxQ

Define the Lyapunov Function:

U(t)

L(U) = e(Q-U)

0 QU

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

. Pow

er

V

O(1/V)

* = ()A

vg. D

elay

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]