Mission-based Joint Optimal Resource Allocation in Wireless

Multicast Sensor Networks

Yun Hou

Prof Kin K. Leung

Archan Misra

Existing Congestion Control

Originally, wired networks rate is the only variable to maximize network utility with fixed link capacity

Recently, wireless networks Power defines capacity Power as another variable Alleviating bottlenecks More power on congested nodes Less power on non-congested

nodes Conserving energy

Congestion Control Via Network Utility Maximization Maximize the network utility Utility = U(flow rates)

So far, Congestion Control = Joint optimization (rate, power)

(Kelly, Low) (Chiang)

Issue with Single-radio Wireless Sensor Networks

A node can transmit for one flow at a time

Multiple flows going through the same node

Flows are scheduled one by one

Flow 1

Flow 2

Single-radio Sensor:

All flows ”share” the air-time of the node

Question : how much air-time to spend on each flows?

Motivation – Adaptive airtime-sharing

Equal time sharing= Suboptimal

Biased time sharing= Optimal

More then needed

Less than neededEffective C = C * time fraction

Objective : How to jointly adapt rate, power with airtime-sharing?

Multi-cast networks

Two flows:[1, 2] = sources[3, 4, 5] = forwarding nodes[6, 7, 8, 9] = sinks

One parent has multiple next-hop children nodes

2. Capacity for a transmission (n,f) One parent broadcast to multiple children Bottleneck child defines capacity Capacity of (3,1) = 5

1. (n,f)<-> one multicast transmission

C=5 C=10

C(3,1)=5

Something special with multicast

max ( ) ( )f f n nf F n N

U X g P

,

1( )

n

f

n ff F

X

C

Pfor all ( , ),n f

,,

,

( ) min log 1 r n nn f

r childr k k r

k n

G PC

G P n

P

s.t.

where

challenges:• the non-linear rate constraint –

explicit time fractions sharing scheme

• the non-concavity – High SINR

• Unknown bottleneck child -- known network schedule

Challenges and assumptions

The original problemObjective function = strictly concave

, , ( )f n f n fX C P

,,

,

( ) log r n nn f

r k k rk n

G PC

G P n

P

, 1n

n ff F

: fixed time fraction for transmission (n,f)

: set of flows passing through node n

,n f

nF,

,

fn f

n f

X

C

Problem formulation:

Utility of flows Penalty of power

Capacity is a function of power

Capacity constraint with time sharing

max ( ) ( )f f n nf F n N

U X g P

,,

,

( ) min log 1 r n nn f

r childr k k r

k n

G PC

G P n

Pwhere

Airtime Sharing:Multiple flows passing one node share the airtime of nodeThe time fraction for flow f at n:

,,

fn f

n f

X

C

Congestion control with adaptive air-time sharing (AAS)

s.t.

,

,( )

( )

=1

f n f

n ff Flow n

X C

P

for all ( , ),n f

Decomposition

s.t.where

The Lagrangian of the problem:

The RATE problem is concave by definition

What about the POWER-TIME problem?

, , , , ,,

max ( ) ( ) 1 ( )n

f f n f f n f n f n f n n f n nf n f n f n f F n

L U X X C g P

X,P αP

max ( ) ( )f f n nf F n N

U X g P

, , ( )f n f n fX C P for all ( , ),n f ( , ),

,( , ), ( , )

,

( ) log

n

B n f n nn f

B n f k k B n fk n k A

G PC

G P n

P

• The capacity function Cn,f (P) is concave

– The Hessian matrix of Cn,f < 0

• The capacity function αn,f Cn,f(P) is concave – Relative entropy – Preserves the convexity

( , ), ( , )

( ), , ( , ), 2

( ) ( )

( , ), ( , ) ( , ),

( )

exp( ) exp( )

0

exp( ) exp( )

i R e l j j R e lj ij Sch i

ii e l e l R e l ie Sch i l F e

R e l j j R e l i R e l ij ij Sch i

P G P n

H G

G P n P G

( , ),, , ( , ), 2

( ) ( )

( , ), ( , ) ( , ),

( )

exp( ) exp( )0,

exp( ) exp( )

i R e l k kik e l e l R e l i

e Sch i l F e

R e l j j R e l i R e l ij ij Sch i

P G PH G k i

G P n P G

For any given vector V

Concavity of POWER-TIME

H is definite negative

, , , ,,

, with low noisen f n f n f n fn f

C C

PP

( )ii ik ik i

H H

n

2 2( ) 0Tij i j i i

i j i

V V H V V V

H n

Updating the airtime fractions

, , , ,,

max ( ) ( ) ( ) 1n

f f n n n f f n f n f n n ff n n f n f F

L U X g P X C

X,P αP

Review the Lagrangian:

** *, ,*

,

, , and 1, n

fn f n f

f Fn f

Xn f n

C

At the optimum, we have

Towards the optimal , an iterative algorithm to update is: ,n f*,n f

,, , ,

,

( )( 1) ( ) ( )

( )n

n fn f n f n f

n ee F

tt t t

t

The airtime constraint

,,

( )with ( )

( )f

n fn f

X tt

C t

,,

,

( )( )

( )n

n fn f

n ee F

tt

t

Insight:requires local info onlyworks with existing congestion control readily

Insight:More time to saturated flowsLess time to low-demand flows

Adaptive air-time sharing (AAS) with optimal rate and power allocation

The joint rate and power

allocation algorithm (JRPA)

Airtime allocation based on local info. (rate and capacity) only

Distributed

AAS generally work with most kind of rate and power control.

Numerical results -Multicast scenarios

AAS works with multicast as well The joint congestion control converges AAS improves network utilityOptimal time-allocation at nodes can improve flow rates while saving power

1 2

3 4 6

7 8 11 12

X1

X2

5

109

X3

node 1 and 2 node 3 and 5 node 4 and 6

Slot 1 Slot 2 Slot 3

The Network Schedule:

0 500 1000 1500 2000 2500 3000

-10

-5

0

5

Iteration

Net

wor

k U

tility

AAS congestion control

Traditional congestion control

U(ACC) = 1.08

U(TCC) = - 0.55

0

0.2

0.4

0.6

0.8

1

1.2

1.4

X1 X2 X3 AverageF

low

rat

es (b

ps/H

z)

Comparison of flow rates at convergence

With AAS

Without AAS

0

0.5

1

1.5

2

P1 P2 P3 P4 P5 P6 Average

Tra

nsm

issi

on P

ower

(W

att)

Comparison of power at convergence

With AAS

Without AAS

• Formulated a joint rate, power and per-node airtime optimization problem for multicast wireless networks

• Showed the concavity and convergence• Fully distributed AAS working with existing congestion control

algorithms• Optimal airtime sharing improves the congestion control

algorithms

• Future work• Adaptive network schedule • Optimal rate and power allocation with sensor selection

Conclusions and future work