Download Jointly optimal source-flow, transmit-power and sending

The Problem: Optimized Management of Resources
Inadequate conventional routers
System constraints
Battery-powered wireless nodes
Need to take into account for the nature of timevarying wireless channels
Average and peak energy
Quality of Service (QoS):
 Maximum delay
Data Link Queue Length?
TARGET
Energy – vs – Queue Length trade-off
To design computationalliy efficient schedulers, that
optimally allocate the energy for bursty sources over
the wireless channels
System Architecture (1/5)
Time is slotted
 (t )
Fading is assumed slowly varying (block
fading)
Channel
 (t ) 0 , t  0
 (t)  ,

0
t  0  p( )
Current value of the channel-state known slot
by slot
Channel probability density function known at
the controller (the hypothesis will be removed
in the following)
System Architecture (2/5)
 (t )
Random variable (r.v.) a(t) with probability
density p(a) known at the trasmitter (the
hypothesis will be removed in the following)
a (t )
λ(t) (IU/slot) number of controlled IU
arriving at the input of the queue at the end
of slot t
Arrival process
a(t)  ,

0
Link state
t  0  p(a)
 (t)  ,

0
t  0  p( )
System Architecture (3/5)
Rate-function IU(t) of the considered system
 (t )
a (t )
IU (t )
Summarizes:



The coding system
The modulation scheme
The error probability PE
(ex. 16-QAM, RS 2/3)
Arrival process
a(t)  ,

0
t  0  p(a)
Rate-function of the considered
system
IU (t )  R( (t );  (t ))
Link state
 (t)  ,

0
t  0  p( )
 ex. :


( IU / Slot )

c   
R( (t );  (t ))  c1 log 1 2
 

(
c
P
)
3 E 

System Architecture (4/5)
Energy constraints:
 Average energy for slot:
 (t )
 Peak energy for slot:
ɛMAX (Joule)
ɛP (Joule)
a (t )
IU (t )
Arrival process
a(t)  ,

0
Link state
t  0  p(a)
 (t)  ,

0
t  0  p( )
Rate-function
IU (t )  R( (t );  (t ))
Energy constraints
 MAX
&
P
( IU / Slot )
System Architecture (5/5)
Given the energy constraints
(ɛMAX and ɛP) and the traffic
patterns (p(a),λ), how much
energy must be radiated
slot by slot to minimize the
avegare queue length SAVE?
 (t )
a (t )
IU (t )
 (t )
Arrival process
a(t)  ,

0
Link state
t  0  p(a)
 (t)  ,

0
t  0  p( )
Rate-function
IU (t )  R( (t );  (t ))
( IU / Slot )
Energy constraints
 AVE
&
P
( Joule)
Formulation problem (1/2)
VBR - Encoder
a (t )
Transmit buffer
st
r (s t ; st )
e (s t ; st )
Scheduler
Cross-layer
VBR - Decoder
r (s t ; st )
Data Link Layer
st
Wireless Link
with Fading
Physical Layer
• p ( a ) probability density of arrivals: Known
• l º E {a ( t )} ( IU / slot ) average number of arrivals
• st ( IU ) number of the IUs buffered in the queue at the beginning of slot t
Formulation problem (2/2)
æ
ö
1 t- 1
ç
min ç lim sup å E {sn }÷
÷
÷
r ( t ) èt ® ¥
ø
t n= 0
s.t. lim sup
t® ¥
1
t- 1
t
n= 0
å E {en }£ e MAX
0 £ e (s t ; st ) £ e p , " st , " s t
p(s) depends in an impredictible way unknown on
the channel statistics, arrival statistics and service
discipline
Computationally intractable problem.
Unconditional-vs.-Conditional Optimum (1/3)
Conditional Problem
Unconditional Problem
æ
ö
1 t- 1
ç
min ç lim sup å E {sn }÷
÷
÷
r ( t ) èt ® ¥
ø
t n= 0
1
t- 1
t
n= 0
min E {st+ 1 | st }
r (×)
s.t. E s
{e (s ; st )}£ eMAX , " st
å E {en }£ e MAX
0 £ r ( s t ; st ) £ rp ( s t ; st ), " st , " s t
0 £ e (s t ; st ) £ e p , " st , " s t
(rp ( s t ; st ) º min {st ; R ( s t ; e p )})
s.t. lim sup
t® ¥
ò ò e (s , s ) p (s ) p ( s )d s ds £ eMAX
s s
ò e (s , st ) p(s )d s £ eMAX " st
s
arg min E {st+ 1 st }º arg min lim sup E {s(t )}
r (.)
r (.) t ® ¥
Unconditional-vs.-Conditional Optimum (2/3)
Conditional Problem
Unconditional Problem
æ
ö
1 t- 1
ç
min ç lim sup å E {sn }÷
÷
÷
r ( t ) èt ® ¥
ø
t n= 0
1
t- 1
t
n= 0
min E {st+ 1 | st }
r (×)
s.t. E s
{e (s ; st )}£ eMAX , " st
å E {en }£ e MAX
0 £ r ( s t ; st ) £ rp ( s t ; st ), " st , " s t
0 £ e (s t ; st ) £ e p , " st , " s t
(rp ( s t ; st ) º min {st ; R ( s t ; e p )})
s.t. lim sup
t® ¥
ò ò e (s , s ) p (s ) p ( s )d s ds £ eMAX
s s
Wider energy domain
ò e (s , st ) p(s )d s £ eMAX " st
s
Smaller energy domain
(stronger constraint)
Unconditional-vs.-Conditional Optimum (3/3)
Conditional Problem
min E {st+ 1 | st }
r (×)
s.t. E s
{e (s ; st )}£ eMAX , " st
0 £ r ( s t ; st ) £ rp ( s t ; st ), " st , " s t
r (s t , st ) = ?
(rp ( s t ; st ) º min {st ; R ( s t ; e p )})
ò e (s , st ) p(s )d s £ eMAX " st
s
Wider energy domain
Smaller energy domain
(stronger constraint)
Unconditional-vs.-Conditional Optimum
How to generalize the optimal scheduler in the stronger energy domain to the wider domain?
r opt (s t ; st* ; m( st* ))
Wider energy domain
Smaller energy domain
(stronger constrait)
r opt (s t ; st ; m*)
mt* º m*
mt* º m( st* )
ò ò e (s , r (s ; s ) p (s ) p ( s ) d s ds =
s s
?
Conditional Approach
Conditional scheduler (convex optimization)
ìïï min f ( s)
í
ïïî gt ( st ) £ 0 " st
Objective function
Constraints
*
If s is local minimum, $ m* such that the following conditions are met:
ìï Ñ s L( s * , m* ) = 0
ï
í *
ïï m ×g ( s* ) = 0 " s
t
î t t t
with
L( s , m) = f ( x ) -
å
mt ×gt ( st )
t
mt* º m( st* )
Lagrange Multiplier:
cross-layer
parameter
Towards the Unconditional Optimal Scheduler (1/2)
Conditional Problem
Unconditional Problem
ìï Ñ s L( s * , m* ) = 0
ï
í *
ïï m ×g ( s * ) = 0
î
ìï Ñ s L( s * , m* ) = 0
ï
í *
ïï m ×g ( s* ) = 0 " s
t
î t t t
mt* ×( ò e (s , r (s ; st )) p(s ) d s - eMAX ) = 0 " st
s
Buffer
*
Depending mt
º m( st* )
m* ×( ò ò e (s , r (s ; s ) p(s ) p( s )d s ds - eMAX ) = 0
s s
mn* º
To design the scheduler
as if the probability
density p(s) was known
Constant:
* Buffer
mNo
Depending
The Unconditional Optimal Scheduler
ìï
ìï - 1 æ 1 ö
üï üï
ï
ï
ï
ï
÷
ç
r (s t ; st ; m) = max í 0; min í er çs t ; ÷
;
s
;
R
(
s
;
e
)
ýý
t
t
p
÷
çè m÷
ïï
ïïî
ïïþïï
ø
î
þ
*
opt
mopt
a (t )
Transmit buffer
r (s t ; st ; m)
m
st
m= 0
e AVE
1 n
= å e (s t ; st ; m)
t t= 0
e AVE ?
eMAX
e(s t ; stt;;m
0)
st
Wireless
Link
0
mt = m
m = m+ D m
£ eMAX
*
mopt
=m
r (s t ; st ; m)
Unconditional Optimal Multiplier: Real-time computation
a (t )
Transmit buffer
r (s t ; st ; mt* )
st
[(t - 1) ×et- 1 + et ]
et =
t
m = m+
g
t
(emax - et )
et (s t ; st ; mt* )
st
Wireless
Link
0
mt* == m
r (s t ; st ; mt* )
ìï
üï
ì
ü
æ
ö
ï
ï
1
ïï
÷
r opt (s t ; st ; mt* ) = max ïí 0;min íï er- 1 ççs t ; * ÷
;
s
;
R
(
s
;
e
)
ýý
t
t
p
çè m ÷
ïï
ïï
ïï ïï
÷
ø
t
î
þþ
î