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
Motivation
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
Motivation
Fundamental aspects of the Power-Rate function
Convexity
convex
increasing
P
r
data in buffer
data in buffer

deadline
time
deadline
time
Motivation
Fundamental aspects of the Power-Rate function

Convexity
c1 c
c
2
3
P
Channel variations
improving
channel
data in buffer
r
data in buffer

convex
increasing
deadline
time
deadline
time
Problem Setup
Transmitter
Tx. Power, P (t ) 
B
c(t)
receiver

B units of data, deadline T

The transmitter can control the rate
g (r (t ))
| h(t ) |2
Problem Setup
Transmitter
Tx. Power, P (t ) 
B
c(t)
g (r (t ))
| h(t ) |2
receiver

B units of data, deadline T

The transmitter can control the rate

Transmission power,
P (t ) 
g (r (t ))
c(t )
g(r) – convex increasing, and, c(t ) | h(t ) |2 is the channel state
Problem Setup
Transmitter
Tx. Power, P (t ) 
B
c(t)
g (r (t ))
| h(t ) |2
receiver

B units of data, deadline T

The transmitter can control the rate

Transmission power,
P (t ) 
g (r (t ))
c(t )
g(r) – convex increasing, and, c(t ) | h(t ) |2 is the channel state

For this work, g (r )  kr n , n , n  1, k  0 (Monomials)
Problem Setup
Channel model – General Markov process
c1c 2
c1
c2
c 2c1
Transition rate, c →ĉ, is ccˆ , total rate out of c,
c   ccˆ
Simplified representation
Define   sup c c
Define Z(c) as,
cˆ c, w. p. ccˆ 
Z (c)  
1,
w. p. 1  c 

Channel transitions at rate . New state is, cˆ  Z (c)c
cˆJ c
Problem 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 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 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
Depends on
channel and time
Tx. rate at time t = (amount of data left) * (urgency at t)
Problem 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
Stochastic Formulation
System state is (x,c,t)
x – amount of data in the queue at time t
c – channel state at time t
Transmission policy r(x,c,t)
Sample path evolution – PDP process
Buffer dynamics
dx(t )
  r ( x(t ), c, t )
dt
c2
c0
channel
c1
time
x(t)
t2
dx(t )
 r ( x(t ), c0 , t )
dt
t1
dx(t )
  r ( x(t ), c1 , t )
dt
time
Stochastic Formulation
Expected energy cost starting in state (x,c,t) is,
T g (r ( xs , cs , s )) ds 
J r ( x, c, t )  E  

c
s
t


Minimum cost function J(x,c,t) is,
J ( x, c, t )  inf J r ( x, c, t )
r (.)
Objective :
Obtain J(x,c,t) among policies with x(T)=0
Policy r*(x,c,t) (optimal policy)
Stochastic Formulation
Expected energy cost starting in state (x,c,t) is,
T g (r ( xs , cs , s )) ds 
J r ( x, c, t )  E  

c
s
t


Minimum cost function J(x,c,t) is,
J ( x, c, t )  inf J r ( x, c, t )
r (.)
Objective :

Obtain J(x,c,t) for policies with x(T)=0

Policy r*(x,c,t) (optimal policy)
Optimality Conditions
Consider a small interval [t,t+h] and apply Bellman’s principle
 t  h g (r ( xs , cs , s ))

J ( x, c, t )  min  E 
ds  EJ ( xt  h , ct  h , t  h)
r (.)
c
s
 t

With some algebra and taking limits h → 0, we get
min  g (r )  AJ ( x, c, t )   0
c
r[ 0 , )

AJ ( x, c, t ) 
J
J
r
  EJ ( x, Z (c)c, t )  J ( x, c, t ) 
t
x
Optimality Conditions
Optimality conditions (HJB equation)
min  g (r )  AJ ( x, c, t )   0
c
r[ 0, )

AJ ( x, c, t ) 
J
J
r
  EJ ( x, Z (c)c, t )  J ( x, c, t ) 
t
x
Boundary conditions
J (0, c, t )  0
J ( x, c, T )  , x  0
Optimal Policy
Theorem (Optimal Transmission Policy)
amount of data left at t
urgency of tx. at t
c i , through f i (t )

Optimal rate r*(.) depends on the channel state,

Optimal rate r*(.) is linear in x, with slope 1 fi (t ) (“urgency” of transmission at t)
Optimal Policy
Theorem (Optimal Transmission Policy)

ODE solved offline numerically with boundary conds., fi (T )  0, fi '(T )  1, i

No channel variations,   0 gives, r * ( x, c, t ) 
x
(simple drain policy)
T t
Example – Gilbert-Elliott Channel
Good-bad channel model (Gilbert-Elliott channel)

Two states “good” and “bad”

Channel transitions with rate 
r * ( x, cgood , t ) 
x
f g (t )
r * ( x, cbad , t ) 
x
f b (t )
Example – Constant Drift Channel
Constant Drift Channel
 1 
E
  ,
Z
(
c
)


Since, cˆ  Z (c)c we have,
independent of c
 1  
1 
E   E


ˆ
c 
 Z ( c )c  c
Optimal Transmission Policy
r * ( x, c, t ) 
x
,
f (t )
xn
J ( x, c, t ) 
c ( f (t )) n 1
where,
f (t ) 
(n  1) 
  (   1)

1  exp  
(T  t )  
 (   1) 
n 1


Problem Setup – Power Limits
0

T
L
2T
L
( L  1)T
The interval [0,T] is partitioned into L partitions
L
T T 
Problem Setup – Power Limits
0
T
L
2T
L
( L  1)T

The interval [0,T] is partitioned into L partitions

Let P be the short term expected power limit
 kT

L
g (r ( xs , cs , s))  PT

E 
ds  
c
(
s
)
L
 ( k L1)T

L
T T 
(kth partition constraint)
Problem Setup – Power Limits
0
T
L
2T
L
( L  1)T
L

The interval [0,T] is partitioned into L partitions

Let P be the short term expected power limit
 kT

L
g (r ( xs , cs , s))  PT

E 
ds  
c
(
s
)
L
 ( k L1)T


T T 
(kth partition constraint)
Penalty cost at T = transmission in time window [T , T   ]

 xT

 g
  

E 

c
(
T
)






(Penalty cost function)
Problem Setup
Problem Statement
 T g ( r ( xs , cs , s ))
g  xT   
min E  
ds  

r (.)
c
(
s
)
c
(
T
)
0

(objective function)
 kT

L
g (r ( xs , cs , s))
PT


E 
ds  
, k  1, 2,
c( s )
L
( k 1)T


 L

, L (L constraints)
subject to,
Stochastic optimization
Continuous-time – minimization over a functional space
Solution Approach – We will take a Lagrangian duality approach
Problem Setup
Problem Statement
 T g ( r ( xs , cs , s ))
g  xT   
min E  
ds  

r (.)
c
(
s
)
c
(
T
)
0

(objective function)
 kT

L
g (r ( xs , cs , s))
PT


E 
ds  
, k  1, 2,
c( s )
L
( k 1)T


 L

, L (L constraints)
subject to,

Stochastic optimization

Continuous-time – minimization over a functional space
Solution Approach – We will take a Lagrangian duality approach
Problem Setup
Problem Statement
 T g ( r ( xs , cs , s ))
g  xT   
min E  
ds  

r (.)
c
(
s
)
c
(
T
)
0

(objective function)
 kT

L
g (r ( xs , cs , s))
PT


E 
ds  
, k  1, 2,
c( s )
L
( k 1)T


 L

, L (L constraints)
subject to,

Stochastic optimization

Continuous-time – minimization over a functional space
Solution Approach – We will take a Lagrangian duality approach
Duality Approach
Basic steps in the approach –
1.
Form the Lagrangian using Lagrange multipliers
2.
Obtain the dual function
3.
Strong duality (maximize the dual function)
Duality Approach
Basic steps in the approach –
1.
Form the Lagrangian using Lagrange multipliers
2.
Obtain the dual function
3.
Strong duality (maximize the dual function)
1) Lagrangian function
Let   (1 , 2 ,, L ),   0, be the Lagrange multipliers for the L constraints
Duality Approach
2) Dual function
Dual function is the minimum of the Lagrangian over the unconstrained set
Duality 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, 1   ( s )  1   k , s  k th interval
2)
Boundary condition is different
Dual Function
B
x(t)
0
T
L
( L  1)T
L
Theorem (Dual function and the minimizing r(.) function)
T
Dual Function
B
x(t)
0
T
L
( L  1)T
L
Theorem (Dual function and the minimizing r(.) function)
T
Dual Function
B
x(t)
0
T
L
( L  1)T
L
T
Theorem (Dual function and the minimizing r(.) function)
where over the kth interval
is the solution of the following system of ODE
Optimal 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 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
offline numerically
can be easily obtained
Simulation Example
3
10
Expected total cost
FullP
Optimal
2
10
1
10
0
10
-1
10
0
2
4
6
8
10
Initial data, B
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
Summary & Future Work
Summary

Deadline constrained data transmission

Continuous-time formulation – yielded simple optimal solution
Tx. rate at time t = (amount of data left) * (urgency at t)
Future Directions

Multiple deadlines

Extensions to a network setting
Thank you !!
Papers can be found at – web.mit.edu/murtaza/www