intelligent packet dropping for optimal energy-delay tradeoffs for wireless michael j. neely...
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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
Ene
rgy
*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]