Transcript A(t) m(p(t), s(t)) Energy Intelligent Packet Dropping for Optimal Energy-Delay Tradeoffs for Wireless Delay Michael J.

A(t)
m(p(t), s(t))
Energy
Intelligent Packet Dropping for Optimal
Energy-Delay Tradeoffs for Wireless
Delay
Michael J. Neely
University of Southern California
http://www-rcf.usc.edu/~mjneely/
(full paper to appear in WiOpt 2006)
*Sponsored by NSF OCE Grant 0520324
A(t)
m(P(t), S(t))
rate m
Good
Med
Bad
Time slotted system (t
0
1
2
3
{0, 1 , 2, …})
…
Assumptions:
1) Random Arrivals A(t) i.i.d. over slots.
(Rate l bits/slot)
2) Random Channel states S(t) i.i.d. over slots.
3) Transmission Rate Function m(P(t), S(t))
P(t) --- Power allocation during slot t
S(t) --- Channel state during slot t
power P
t
Avg. Power
Fundamental Energy-Delay Tradeoff Theory
and the Berry-Gallager Bound:
A(t)
m(P(t), S(t))
Avg. Delay
F(l) = Min. Avg. Energy Required for Stability
[Berry 2000, 2002]
Avg. Delay
Avg. Power
Fundamental Energy-Delay Tradeoff Theory
and the Berry-Gallager Bound:
O(1/V)
V
V
In terms of a dimensionless index parameter V>0:
[Berry 2000, 2002]
Avg. Delay
Avg. Power
Fundamental Energy-Delay Tradeoff Theory
and the Berry-Gallager Bound:
O(1/V)
V
V
In terms of a dimensionless index parameter V>0:
[Berry 2000, 2002]
Avg. Delay
Avg. Power
Fundamental Energy-Delay Tradeoff Theory
and the Berry-Gallager Bound:
O(1/V)
V
V
In terms of a dimensionless index parameter V>0:
[Berry 2000, 2002]
Avg. Delay
Avg. Power
Fundamental Energy-Delay Tradeoff Theory
and the Berry-Gallager Bound:
O(1/V)
V
V
In terms of a dimensionless index parameter V>0:
[Berry 2000, 2002]
Avg. Delay
Avg. Power
Fundamental Energy-Delay Tradeoff Theory
and the Berry-Gallager Bound:
O(1/V)
V
V
In terms of a dimensionless index parameter V>0:
[Berry 2000, 2002]
Avg. Delay
Avg. Power
Fundamental Energy-Delay Tradeoff Theory
and the Berry-Gallager Bound:
O(1/V)
V
Berry-Gallager Bound Assumes:
1. Admissibility criteria
2. Concave rate-power function
3. i.i.d. arrivals A(t)
4. No Packet Dropping
V
Our Formulation: Intelligent Packet Dropping
A(t) (rate l)
r
m(P(t), S(t))
(1-r)
Control Variables:
Goal: Obtain an optimal energy-delay tradeoff
Subject to:
Admitted rate >= r l
(0<r<1)
Avg. Delay
Avg. Power
Energy-Delay Tradeoffs with Packet Dropping…
O(1/V)
?
V
V
F* = F(lr) = New Min. Average Power Expenditure
(required to support rate rl).
A(t)(rate l)
r
(1-r)
Avg. Delay
Avg. Power
Energy-Delay Tradeoffs with Packet Dropping…
O(1/V)
?
V
V
F* = F(lr) = New Min. Average Power Expenditure
(required to support rate rl).
A(t)(rate l)
r
(1-r)
Avg. Delay
Avg. Power
Energy-Delay Tradeoffs with Packet Dropping…
O(1/V)
?
V
V
F* = F(lr) = New Min. Average Power Expenditure
(required to support rate rl).
A(t)(rate l)
r
(1-r)
Avg. Delay
Avg. Power
Energy-Delay Tradeoffs with Packet Dropping…
O(1/V)
?
V
V
F* = F(lr) = New Min. Average Power Expenditure
(required to support rate rl).
A(t)(rate l)
r
(1-r)
Avg. Delay
Avg. Power
An Example of Naïve Packet Dropping:
Random Bernoulli Acceptance with probability r.
O(1/V)
F* = F(lr)
V
V
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….
A(t)(rate l)
(1-r)
r
Avg. Delay
Avg. Power
An Example of Naïve Packet Dropping:
Random Bernoulli Acceptance with probability r.
O(1/V)
F* = F(lr)
V
V
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.
A(t)(rate l)
(1-r)
r
Avg. Delay
Avg. Power
But here we consider Intelligent Packet Dropping:
O(1/V)
F* = F(lr)
achievable!
V
V
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.
A(t)(rate l)
(1-r)
r
Algorithm Development: A preliminary Lemma:
Lemma: If channel states are i.i.d. over slots: For any
stabilizable input rate l, 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
Step 2 -- Apply the stationary policy P*(t) such that:
(where r < r + e < 1)
rate (r+e)l
rate l
Positive drift!
Q
mmax 0
Step 2 -- Apply the stationary policy P*(t) such that:
(where r < r + e < 1)
rate (r+e)l
rate l
Positive drift!
Q
Choose: e = O(1/V) ,
mmax 0
Q = O(log(V))
Algorithm 2: (Unknown channel probabilities)
Constructing a practical Dynamic Packet Dropping
Algorithm:
rate l
m(P(t), S(t))
U(t)
mmax 0
Q
…but we still want to
maintain mav at least
(r+e)l…
Define the Lyapunov Function:
L(U) =
w
(Q-U)
e
L(U)
0
U
Q
A(t) (rate l)
U(t)
m(P(t), S(t))
Want to ensure:
(r + e)l < mav
Use the “virtual queue” concept for time average
inequality constraints [Neely Infocom 2005]
(r+e)A(t)
X(t)
m(P(t), S(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. Delay
Avg. Power
Theorem: For the Dynamic Packet Dropping Alg.
O(1/V)
F* = F(lr)
V
achievable!
V
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]