System constraints

Average and peak energy Quality of Service (QoS):

Maximum delay

Data Link Queue Length?

Inadequate conventional routers

Battery-powered wireless nodes Need to take into account for the nature of time-

varying wireless channels

To design computationalliy efficient schedulers, that optimally allocate the energy for bursty sources over

the wireless channels

TARGET Energy – vs – Queue Length trade-off

The Problem: Optimized Management of Resources

Time is slotted

Fading is assumed slowly varying (block fading)

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)

( )t

Channel

0( ) , 0t t

0( ) , 0 ( )t t p

System Architecture (1/5)

Random variable (r.v.) a(t) with probability density p(a) known at the trasmitter (the hypothesis will be removed in the following)

λ(t) (IU/slot) number of controlled IU arriving at the input of the queue at the end of slot t

( )a t

Arrival process

0( ) , 0 ( )a t t p a

( )t

Link state

0( ) , 0 ( )t t p

System Architecture (2/5)

Rate-function IU(t) of the considered system

( )IU t

( ) ( ( ); ( )) ( / )IU t R t t IU Slot

Rate-function of the considered

system

( )a t( )t

Arrival process

0( ) , 0 ( )a t t p a

Link state

0( ) , 0 ( )t t p

Summarizes:

The coding system The modulation scheme The error probability PE

(ex. 16-QAM, RS 2/3)

21

3

. : ( ( ); ( )) log 1( )E

cex R t t c

c P

System Architecture (3/5)

( )IU t

( ) ( ( ); ( )) ( / )IU t R t t IU Slot Rate-function

Energy constraints: Average energy for slot: ɛMAX (Joule) Peak energy for slot: ɛP (Joule)

Energy constraints

&MAX P

( )a t( )t

Arrival process

0( ) , 0 ( )a t t p a

Link state

0( ) , 0 ( )t t p

System Architecture (4/5)

( ) ( ( ); ( )) ( / )IU t R t t IU Slot Rate-function

Energy constraints

& ( )AVE P Joule

( )t

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?( )IU t

( )a t( )t

Arrival process

0( ) , 0 ( )a t t p a

Link state

0( ) , 0 ( )t t p

System Architecture (5/5)

( ; )tt se s

( ; )ttr ss

tsSchedulerCross-layer

ts

( )a tTransmit buffer

Physical LayerData Link Layer

VBR - Encoder

VBR - Decoder

Wireless Link with Fading

( ; )ttr ss

( )p a

( )t IUs{ }E ( ) ( / )a t IU slotl º

Formulation problem (1/2)

• probability density of arrivals: Known• average number of arrivals• number of the IUs buffered in the queue at the beginning of slot t

{ }

{ }1

1

0

0

( )

1mi

1s.t. lim

n lim sup

up E

E

s

0 ( ; ) , ,

MAX

t

t

t

t n

nn

t t p t

n

t

r t

t

t

s s

s

e e

e s e s

-

®

-

®¥

¥ =

=

£

å

å

æ ö÷ç ÷çè

£ "

ç ø

"

÷

£

p(s) depends in an impredictible way unknown on the channel statistics, arrival statistics and service

discipline

Computationally intractable problem.

Formulation problem (2/2)

{ }

{ }

0

1

0

1

( )

1s.t. lim sup E

1min lim sup

0 ( ; ) , ,

E

MAX

t

t

t

n

t nn

t t p t t

nr t

t

s

t

s

s

e

e s s

e

e

-

®¥

-

¥ =

=

®

£å

£ £ "

å

"

æ ö÷ç ÷ç ÷çè ø

( , ) ( ) ( )MAX

sp ss p d ds

se s s s e£òò

{ } { }(.)

1(.)

arg min E arg min lim sup E ( )t ttr r

s s s t+ ®¥º

{ }

{ }

{ }

1( )

s.t. E ( ; ) ,

0 ( ; ) ( ; ), ,

( ( ; ) min ; (

min E

; )

|

)

MAX

t p t t

p t t p

tt

t t

t t t

t t

r

s s

r s r s s

r s

s

s

s

s e s

s s s

s s e

e

+×

£ "

£ £ " "

º R

( , ) ( ) MAXt ts p d s

se s s s e£ "ò

Unconditional Problem Conditional Problem

Unconditional-vs.-Conditional Optimum (1/3)

Wider energy domain Smaller energy domain(stronger constraint)

{ }

{ }

0

1

0

1

( )

1s.t. lim sup E

1min lim sup

0 ( ; ) , ,

E

MAX

t

t

t

n

t nn

t t p t t

nr t

t

s

t

s

s

e

e s s

e

e

-

®¥

-

¥ =

=

®

£å

£ £ "

å

"

æ ö÷ç ÷ç ÷çè ø

( , ) ( ) ( )MAX

sp ss p d ds

se s s s e£òò

{ }

{ }

{ }

1( )

s.t. E ( ; ) ,

0 ( ; ) ( ; ), ,

( ( ; ) min ; (

min E

; )

|

)

MAX

t p t t

p t t p

tt

t t

t t t

t t

r

s s

r s r s s

r s

s

s

s

s e s

s s s

s s e

e

+×

£ "

£ £ " "

º R

( , ) ( ) MAXt ts p d s

se s s s e£ "ò

Unconditional-vs.-Conditional Optimum (2/3)Unconditional Problem Conditional Problem

( , ) ?t tr ss =

{ }

{ }

{ }

1( )

s.t. E ( ; ) ,

0 ( ; ) ( ; ), ,

( ( ; ) min ; (

min E

; )

|

)

MAX

t p t t

p t t p

tt

t t

t t t

t t

r

s s

r s r s s

r s

s

s

s

s e s

s s s

s s e

e

+×

£ "

£ £ " "

º R

( , ) ( ) MAXt ts p d s

se s s s e£ "ò

Unconditional-vs.-Conditional Optimum (3/3)Conditional Problem

Wider energy domain Smaller energy domain(stronger constraint)

* *( )t tsm mº* *tm mº

How to generalize the optimal scheduler in the stronger energy domain to the wider domain?

* * ))( (; ;t tt

top s sr s m

( )*; ;t toptr s s m

( )( , ( ; ) ( ) ?s

r s p dp s dsse s s s sò =ò

Unconditional-vs.-Conditional Optimum

Smaller energy domain(stronger constrait)

Wider energy domain

min ( )

( ) 0 t t t

f s

g s s

ìïïíï £ "ïî

Conditional scheduler (convex optimization)

Objective function

Constraints

* *

* *

( , ) 0

( ) 0

s

t t t t

L s

g s s

m

m

ìï Ñ =ïïíï × = "ïïî

*m$If is local minimum, such that the following conditions are met:*s

with ( , ) ( ) ( )t t tt

L s f x g sm m= - ×å

* *( )t tsm mº

Lagrange Multiplier: cross-layer parameter

Conditional Approach

* *nm mº

*

* *

*( , ) 0

0 ) (t t

s

t t

L

g

s

s s

m

m

ìï Ñ =ïïíï × = "ïïî

* ( , ( ; )) ( ) ( ) 0MAXt tts sr p d

se s s s s em -ò "× =

* *( )t tsm mº

* *

**

( ,

)

) 0

0(

s

g

s

s

L m

m

ìï Ñ =ïïíï × =ïïî

* ( , ( ; ) ( ( )) 0)(MAX

sp s dsr s p d

se s s s sm e-òò× =

Conditional Problem Unconditional Problem

Constant:No Buffer

Depending

Buffer Depending

To design the scheduler as if the probability

density p(s) was known

Towards the Unconditional Optimal Scheduler (1/2)

( ; ; )tt sr ms

ts

ts

( )a t Transmit buffer

Wireless Link

( ; ; )tt sr ms

tm m=

( ; ; )tt se ms

1 1( ; ; ) max 0;min ; ; ; ( ; )t tt t

oppt

trs sr es s s

mem -

ì üì üæ öï ïï ïï ï ï ï÷ç= ÷í í ýýç ÷ç ÷çï ï ï ïè øï ï ï ïî þî þR

0m=

0tm=

m m m= +D

m

*optm

*optm m=

0

( ; ; )1 n

AVEt

tt st

e mse=

= å

?AVEe MAXe

( ; )0;t tse s

MAXe£

The Unconditional Optimal Scheduler

*( ; ; )tt tsr ms

ts

ts

( )a t Transmit buffer

Wireless Link

*( ; ; )tt tsr ms

tm m=

*( ; ; )t tt ts me s

**

1 1( ; ; ) max 0;min ; ; ; ( ; )opt

r pt

tt tt ttsr s s sem

em s-ì üì üæ öï ïï ïï ï÷ï ïç ÷= çí í ýý÷ç ÷çï ï ï ï÷è øï ï ï ïî þî þ

R

* 0tm=

1[( 1) ]t tt

t

t

e ee -- × +=

max( )ttm

gm e e= + -

Unconditional Optimal Multiplier: Real-time computation