Optimal Adaptive Data Transmission over a Fading Channel with Deadline and Power Constraints Murtaza Zafer and Eytan Modiano Laboratory for Information

Optimal Adaptive Data Transmission over a Fading Channel with Deadline and Power Constraints

Murtaza Zafer and Eytan Modiano

Laboratory for Information and Decision Systems

Massachusetts Institute of Technology

MotivationMotivation

Big Picture (Main issues) –

Deadline constrained data transmission

Fading channel

Energy limitations

Applications –

Sensor with time critical data

Mobile device communicating multimedia/VoIP data

Deep space communication

Transmission energy cost is critical – utilize adaptive rate control


Convexity convexincreasing

r

P

data

in b

uffe

r

deadline time

data

in b

uffe

r

deadline time

Fundamental aspects of the Power-Rate function


Convexity

Channel variations

convexincreasing

r

Pc3c1 c2

improvingchannel

data

in b

uffe

r

deadline time

data

in b

uffe

r

deadline time

Fundamental aspects of the Power-Rate function

Problem SetupProblem Setup

B units of data, deadline T

The transmitter can control the rate

2

( ( ))( )

| ( ) |

g r tP t

h t

Transmitter

Bc(t)

receiver

Tx. Power,




Transmission power,

g(r) – convex increasing, and, is the channel state

( ( ))( )

( )

g r tP t

c t

2|)(|)( thtc

2

( ( ))( )

| ( ) |

g r tP t

h t

Transmitter

Bc(t)

receiver

Tx. Power,




Transmission power,

g(r) – convex increasing, and, is the channel state

For this work, (Monomials)( ) , , 1, 0ng r kr n n k

( ( ))( )

( )

g r tP t

c t

2|)(|)( thtc

2

( ( ))( )

| ( ) |

g r tP t

h t

Transmitter

Bc(t)

receiver

Tx. Power,

Channel model – General Markov process

Transition rate, c →ĉ, is , total rate out of c,

Simplified representation

Define

Define Z(c) as,

Channel transitions at rate . New state is,

cJc

cccˆ

ˆ

c

ccpwpwcccZ 1..,1..,ˆ)( ˆ

cc sup

ccZc )(ˆ

12cc

c1 c2

21cc

cc ˆ



Problem Summary

Transmit B units of data by deadline T over a fading channel

Channel state is a Markov process

Objective: Minimize transmission energy cost


Problem Summary

Transmit B units of data by deadline T over a fading channel



Continuous-time approach

Transmitter controls the rate continuously over time

Yields closed form solutions


Problem Summary

Transmit B units of data by deadline T



Optimal solution - preview

Tx. rate at time t = (amount of data left) * (urgency at t)

Depends on channel and time


Problem Summary

Transmit B units of data by deadline T



Optimal solution - preview


Two settings

No power limits

Short-term expected power limits

System state is (x,c,t)

Transmission policy r(x,c,t)

Sample path evolution – PDP process

Stochastic FormulationStochastic Formulation

Buffer dynamics

time

time

channel

x(t)

c0

c1

c2

),),(()(

0 tctxrdt

tdx ),),((

)(1 tctxrdt

tdx

t1

t2

),),(()(

tctxrdt

tdx

x – amount of data in the queue at time t

c – channel state at time t


Expected energy cost starting in state (x,c,t) is,

T

t s

ssr c

dsscxrgEtcxJ

)),,((),,(

Minimum cost function J(x,c,t) is,

(.)( , , ) inf ( , , )r

rJ x c t J x c t

Objective :

Obtain J(x,c,t) among policies with x(T)=0

Policy r*(x,c,t) (optimal policy)


Expected energy cost starting in state (x,c,t) is,

T

t s

ssr c

dsscxrgEtcxJ

)),,((),,(

Minimum cost function J(x,c,t) is,

(.)( , , ) inf ( , , )r

rJ x c t J x c t

Objective : Obtain J(x,c,t) for policies with x(T)=0

Policy r*(x,c,t) (optimal policy)

Optimality ConditionsOptimality Conditions

Consider a small interval [t,t+h] and apply Bellman’s principle

),,()),,((

min),,((.)

htcxEJdsc

scxrgEtcxJ htht

ht

t s

ss

r

With some algebra and taking limits h → 0, we get

0),,()(min),0[

tcxAJc

rgr

),,(),)(,(),,( tcxJtccZxEJx

Jr

t

JtcxAJ

Optimality ConditionsOptimality Conditions

Optimality conditions (HJB equation)

Boundary conditions

(0, , ) 0

( , , ) , 0

J c t

J x c T x

0),,()(min),0[

tcxAJc

rgr

),,(),)(,(),,( tcxJtccZxEJx

Jr

t

JtcxAJ

Optimal rate r*(.) depends on the channel state, , through

Optimal rate r*(.) is linear in x, with slope (“urgency” of transmission at t)

Optimal PolicyOptimal Policy

ic

Theorem (Optimal Transmission Policy)

)(tfi

1 ( )if t

amount of data left at t

urgency of tx. at t

ODE solved offline numerically with boundary conds.,

No channel variations, gives, (simple drain policy)


( ) 0, '( ) 1,i if T f T i

Theorem (Optimal Transmission Policy)

0tT

xtcxr

),,(*

Example – Gilbert-Elliott ChannelExample – Gilbert-Elliott Channel

Good-bad channel model (Gilbert-Elliott channel)

Two states “good” and “bad”

Channel transitions with rate

)(),,(*

tf

xtcxr

ggood

)(),,(*

tf

xtcxr

bbad

Example – Constant Drift ChannelExample – Constant Drift Channel

Constant Drift Channel

)(1

)1(exp1

)1(

)1()( tT

n

ntf

,)(

1

cZ

E

cccZE

cE

)(

1ˆ1

Optimal Transmission Policy

1))((),,(

,)(

),,(*

n

n

tfc

xtcxJ

tf

xtcxr

independent of c

where,

ccZc )(ˆ Since, we have,

Problem Setup – Power LimitsProblem Setup – Power Limits

The interval [0,T] is partitioned into L partitions

0 TL

2TL

( 1)L TL

T T



Let P be the short term expected power limit

0 TL

2TL

( 1)L TL

T

(kth partition constraint)

T

( 1)

( ( , , ))

( )

kT

Ls s

k T

L

g r x c s PTE ds

c s L



Let P be the short term expected power limit

0 TL

2TL

( 1)L TL

T

(kth partition constraint)

T

Penalty cost at T = transmission in time window [ , ]T T

( )

TxgE

c T

(Penalty cost function)

( 1)

( ( , , ))

( )

kT

Ls s

k T

L

g r x c s PTE ds

c s L


Problem Statement

(.)

0

( ( , , ))min

( ) ( )

TTs s

r

g xg r x c sE ds

c s c T

subject to,

( 1)

( ( , , )), 1, 2, ,

( )

kT

Ls s

k T

L

g r x c s PTE ds k L

c s L

(objective function)

(L constraints)

Stochastic optimization

Continuous-time – minimization over a functional space

Solution Approach – We will take a Lagrangian duality approach


Problem Statement

(.)

0

( ( , , ))min

( ) ( )

TTs s

r

g xg r x c sE ds

c s c T

subject to,

( 1)

( ( , , )), 1, 2, ,

( )

kT

Ls s

k T

L


c s L


(L constraints)





Problem Statement

(.)

0

( ( , , ))min

( ) ( )

TTs s

r

g xg r x c sE ds

c s c T

subject to,

( 1)

( ( , , )), 1, 2, ,

( )

kT

Ls s

k T

L


c s L


(L constraints)




Duality ApproachDuality Approach

1. Form the Lagrangian using Lagrange multipliers

2. Obtain the dual function

3. Strong duality (maximize the dual function)

Basic steps in the approach –


1) Lagrangian function

Let be the Lagrange multipliers for the L constraints,0),,,,( 21 L

1. Form the Lagrangian using Lagrange multipliers

2. Obtain the dual function

3. Strong duality (maximize the dual function)

Basic steps in the approach –


2) Dual function

Dual function is the minimum of the Lagrangian over the unconstrained set


2) Dual function

Dual function is the minimum of the Lagrangian over the unconstrained set

Consider the minimization term in the equation above,

This we know how to solve from the earlier formulation – except two changes

1) Cost function has a multiplicative term,

2) Boundary condition is different

interval,1)(1 thk kss

Dual FunctionDual Function

Theorem (Dual function and the minimizing r(.) function)

0 ( 1)L TL

TL

T

Bx(t)



0 ( 1)L TL

TL

T

Bx(t)



0 ( 1)L TL

TL

T

Bx(t)

where over the kth interval is the solution of the following system of ODE


3) Maximizing the dual function (Strong Duality)

Theorem – Strong duality holds

is the optimal cost of the primal (original constrained) problem

is the initial amount of data ( = B)

is the initial channel state


3) Maximizing the dual function (Strong Duality)

Theorem – Strong duality holds

If is the optimal policy for the original problem, then,

is the maximizing

Since the dual function is concave, the maximizing can be easily obtained

offline numerically

Simulation ExampleSimulation Example

Simulation setup

Two state (good-bad) channel model

Two policies – Lagrangian optimal and Full power

P is chosen so that for B ≤ 5, Full power Tx. empties the buffer over all sample paths

0 2 4 6 8 1010

-1

100

101

102

103

Initial data, B

Exp

ecte

d t

ota

l co

st

FullPOptimal

Summary

Deadline constrained data transmission

Continuous-time formulation – yielded simple optimal solution

Future Directions

Multiple deadlines

Extensions to a network setting

Summary & Future WorkSummary & Future Work


Thank you !!

Papers can be found at – web.mit.edu/murtaza/www

Documents

Optimal Adaptive Data Transmission over a Fading Channel with Deadline and Power Constraints Murtaza Zafer and Eytan Modiano Laboratory for Information