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
MotivationMotivation
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
MotivationMotivation
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,
Problem SetupProblem Setup
B units of data, deadline T
The transmitter can control the rate
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,
Problem SetupProblem Setup
B units of data, deadline T
The transmitter can control the rate
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 SetupProblem Setup
Problem SetupProblem Setup
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 SetupProblem Setup
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
Continuous-time approach
Transmitter controls the rate continuously over time
Yields closed form solutions
Problem SetupProblem Setup
Problem Summary
Transmit B units of data by deadline T
Channel state is a Markov process
Objective: Minimize transmission energy cost
Optimal solution - preview
Tx. rate at time t = (amount of data left) * (urgency at t)
Depends on channel and time
Problem SetupProblem Setup
Problem Summary
Transmit B units of data by deadline T
Channel state is a Markov process
Objective: Minimize transmission energy cost
Optimal solution - preview
Tx. rate at time t = (amount of data left) * (urgency at t)
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
Stochastic FormulationStochastic Formulation
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)
Stochastic FormulationStochastic Formulation
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)
Optimal PolicyOptimal 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
Problem Setup – Power LimitsProblem Setup – Power Limits
The interval [0,T] is partitioned into L partitions
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
Problem Setup – Power LimitsProblem Setup – Power Limits
The interval [0,T] is partitioned into L partitions
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 SetupProblem Setup
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 SetupProblem Setup
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 SetupProblem Setup
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
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 –
Duality ApproachDuality 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 –
Duality ApproachDuality Approach
2) Dual function
Dual function is the minimum of the Lagrangian over the unconstrained set
Duality ApproachDuality Approach
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)
Dual FunctionDual Function
Theorem (Dual function and the minimizing r(.) function)
0 ( 1)L TL
TL
T
Bx(t)
Dual FunctionDual Function
Theorem (Dual function and the minimizing r(.) function)
0 ( 1)L TL
TL
T
Bx(t)
where over the kth interval is the solution of the following system of ODE
Optimal PolicyOptimal Policy
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
Optimal PolicyOptimal Policy
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
Tx. rate at time t = (amount of data left) * (urgency at t)
Thank you !!
Papers can be found at – web.mit.edu/murtaza/www