optimal power control scheduling and routing in...

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Optimal Power Control Scheduling and Routing in

UWB Networks

ÉCOLE POLYTECHNIQUE FÉDÉRALE DE LAUSANNE

MICSMobile Information and Communication Systems

Božidar Radunovicjoint work with

Jean-Yves Le Boudec

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Contents

1. Goal of our Research and Main Findings2. Assumptions on the Network3. Mathematical Modeling4. Detail Findings5. Solution Method6. Conclusion

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Goal of our Research

• Components of network design:§ Power control§ Scheduling§ Routing

• Optimal is to jointly design all components.

Goal: define design objectives for multi-hop UWB networks.

• We are interested in design objectives; specific implementation are out of scope

• UWB is emerging technology and its network design is not fully understood

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Main Findings

Optimal power control, scheduling and routing in UWB networks1. It is optimal to have exclusion region around destination; nodes

inside the region are silent, nodes outside can send in parallel2. The optimal size of the exclusion region depends only on the

transmission power, and not on link length nor on positions of other nodes

3. Each node sends with maximum power when sending.4. It is optimal to route over minimum energy route5. Design of optimal MAC is independent of choice of routingFor narrow-band networks, (2), (4) and (5) do not hold.

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Contents

1. Goal of our Research and Main Findings2. Assumptions on the Network3. Mathematical Modeling4. Detailed Findings5. Solution Method6. Conclusion

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Ultra Wide Band

• Radio transmission technology that uses the full spectrum§ over 500 MHz§ in practice uses a 3 GHz band§ unlicensed§ potential very low cost, 10 to 100 meters range

Energy densityfrom [DiBenedetto, SRI 2003]

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Specificity of UWB

• Several existing proposals: OFDM, PPM, CDMA

General properties of all UWB physical layers• Linear rate function:

achievable rate is linear function of SNR• Rate adaptation:

repetition coding, convolutional codes• Arbitrary level of interference is possible:

codes are adapted to interference level

δ‘0’ ‘1’PPM example

[WinScholtz00]

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Linear Rate Function

• Information Theory results [Verdu02, Telatar00]:

rate = K * SNR

• Hint: rate = W log(1+P/(NW)) → P/N• Linear rate function holds for any efficient UWB system

within the operational limits. Does not hold for narrow-band.• An example on [WinScholtz00]:

• Rate: rate = 1/NsTf

• Bit-error rate [Durisi02]:

−= )(1

2erfc

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0

δγNEN

BER xs

SNR fixed by hardware

‘0’ ‘0’ ‘0’

Ns repetitions

Tf

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Rate adaptation

• Source can adapt rate to arbitrary level of interference at receiver and maintain BER fixed.

• Examples: repetition coding [WinScholtz00], convolutional coding• Consequence: no collisions

• Counterexample: 802.11 has fixed rate. If interference is too high, packet is lost.

• Rate adaptation is not specificity only of UWB systems

S1 D1

S2

D2

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UWB channel model

• Very high frequency yields many multi-path which implies no fast fading.

• Constant attenuation hl as a function of link length l:

γ−== lPhPP lrcv

PPrcv

lS D

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UWB Power Constraints

Average packet sending power P on link l is limited:

Example of derivation of power constraint on [WinScholtz00]:

• Peak power Ex/Tc = P MAX peak – constrained by hardware• Average power Ex/Tf = P MAX avg – constrained by regulations• Constraint on average power P= Ex/Tf is

MAXll PP ≤

Ex

Tf Tc

Ex

},min{ avgMAXl

f

cpeakMAXl

MAXll P

TT

PPP =≤

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MAC Layer

• MAC layer controls access to wireless medium• Traditional approaches in wireless LANs are based on exclusions§ TDMA, CSMA/CA

• MAC model (pn – power allocation in slot n, an – relative frequency of slot n)

a1, p1 a2, p2 a3, p3 a4, p4 …

Slot 1:p1 = (P1,P2), a1=1

Slot 1:p1 = (P1,0), a1=0.5

Single slot Two slots (exclusions)

Slot 2:p2 = (0,P2), a2=0.5

Examples:

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Routing

• We consider single-path routing policies:§ DIR (direct routing) – direct link with no relaying§ MER (minimum energy router) – route that minimizes Σ hl

-1

§ Intermediate – nodes are subsets of MER or MELR routes

MER

Intermediate

DIR

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Contents


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Performance Objectives

• Focus on rate based performance objectives§ power control used to maximize rates not to minimize energy consumption

• Several existing rate performance metrics [RaduLB04]:§ Maximizing total rate – known to be very unfair§ Max-min fairness – known to be very inefficient§ Proportional fairness – good compromise between efficiency and fairness

• Log-utility of a flow of rate f: U(f) = log(f)• Total log-utility of a network: Σ log(f)• Proportionally fair rate allocation is the one that maximizes total

log-utility of a network.

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Mathematical Formulation

• We want to find power allocations pn, slot frequencies an, and routing matrix R that maximize total utility U.

• Goal is to solve the convex optimization problem• This model implies perfect exchange of information within network;

we look for design criteria and we do not consider implementation details.

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Mathematical Formulation

Slot 1:p1 = (P1,P2), a1=1

Slot 1:p1 = (P1,0), a1=0.5


Slot 2:p2 = (0,P2), a2=0.5

Examples:

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Contents


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Power Control is 0 - PMAX

Theorem: It is always optimal for node, when sending, to send at maximum power, regardless of the network, routing or scheduling.

Slot 1:p1 = (PMAX,PMAX), a1=1

Slot 1:p1 = (PMAX,0), a1=0.5


Slot 2:p2 = (0,PMAX), a2=0.5

Examples:

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Exclusion Regions

• Finding: in the optimal scheduling, there is exclusion region around destination of active link. All nodes in exclusion region remain silent; nodes outside transmit in parallel. Each destination adapts codes to interference.

• How to calculate size of the exclusion region s:§ N(s) – number of nodes in exclusion region§ I(s) – interference from nodes outside of the exclusion region

§ Rate is approximately:

• Heuristic for optimal size of exclusion region:§ N(s) = density × s2p§ I(s) = s-?

S1 D1

S2

D2

S3D3

s

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0 −=

−

γ

γ

NsP MAX

)()( 0 sINP

sNK

xMAX

+=

SNRexclusions

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Minimum Energy Routing

• Finding: It is optimal to use Minimum Energy Routing (MER)• MER routing minimizes Σ hl

-1

hl-1 – attenuation on link

• Direct routing (DIR) has no routing protocol overhead.We introduce routing overhead Cr: fr = (1 - Cr) fnr

• For large routing overheads, it is better to use DIR

MER

Intermediate

DIR

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Narrow-band Design is Different

• In narrow-band multi-hop networks:§ Power control 0-PMAX might not be optimal§ Size of exclusion region depends on link length and other nodes§ Optimal MAC design depends on choice of routing

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Contents


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Power Control is 0 - PMAX

• Sketch of a proof:§ Consider set of all feasible rates F and

feasible rates FE achievable only with {0,PMAX} power allocations.§ Fix all an and pl

n and R to arbitrary values, except from p1

1.

§ Consider mappingfor arbitrary vector µ§Mapping L(p1

1) is convex, hence maximum is for p1

1 = 0 or p11 = PMAX.

§ Suppose there exists rate f∈F s.t. f is not achievable with {0,PMAX} rule (f∉ FE).§Contradiction: There exists hyper plane

(µ,b) s.t. µTf > b and µTg < b for all g∈F.

∑→ iiL fp µ1

1

Theorem: It is always optimal for node, when sending, to send at maximum power, regardless of the network, routing or scheduling.

FE

F

f(µ,b)

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Optimal Scheduling and Routing:techniques for solving the problem

• Solving the optimization problem for arbitrary network is extremely complex task. Previously solved for max. 6-10 nodes.

• We first solve it in case of static symmetric ring§ n equally spaced nodes on a circle § Each node talks to its right-hand

neighbor d hops away.§ Routing is the same for all nodes

(DIR or MER)§ Powers of all nodes limited by same Pmax.

• We apply numerical solutions from ring case to random networks in plane. We do not explore full state space, but compare to other commonly used strategies.

n = 6, d = 2

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Compared strategies

• Scheduling strategies:§ Global: optimize size of exclusion region for each node separately, given

positions of other nodes.§ Local: uses approximately optimal size of exclusion region§ TDMA: only one node sends at a time§ All-at-once: everybody sends all the time

• Routing strategies:§ MER: minimum energy routing§ DIR: direct routing

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Example Numerical Results - Routing

• One random network instance• 50 randomly distributed nodes and 25

randomly distributed flows on unit square.

• For each routing strategy find the power allocation and the scheduling strategy that maximize log utility.

• MER routing here is better than DIR

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Example Numerical Results - Scheduling

• One random network instance• 50 randomly distributed nodes and 25

randomly distributed flows on unit square.

• For each scheduling strategy, find the power allocation and the routing strategy that maximize log utility.

• Global and Local are equal and better than TDMA and All-at-once

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Uniform Static Random Networks with no Routing Overhead

• Consider static network with no routing overhead; Global and MER are referent approach

• Global and Local have the same performance. Scheduling 2 is easier to implement.

• Local is always better than TDMA or All-at-once.

• TDMA is optimal in dense networks with high-power constraints

• All-at-once is optimal in sparse, low-power networks.

• DIR routing is worse than MER for all scheduling strategies.

• Optimal scheduling is independent of the choice of routing

SNR [dB]

SNR [dB]

TDMA, MER All-at-once,

MER

Global, Local,MER

TDMA,DIR

All-at-once,DIR

Global, Local,DIR

refe

renc

e ut

ility

-ac

hiev

ed u

tility

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Non-uniform node and power distribution

TDMA, MER

All-at-once,MER

Local,MER

Global,DIR

TDMA, MER

All-at-once,MER

Local,MER

Global,DIR

Non-uniform distributions:• Power distribution: maximum power varies 100% around average value (avg. is given on x-axis)• Node distribution: left side of unit square has 4 times more nodes than right side of square

The same conclusions hold as in the uniform case!

refe

renc

e ut

ility

-ac

hiev

ed u

tility

SNR [dB]

SNR [dB]

Non-uniform power dist.

Non-uniform node dist.

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Effects of Cost of Routing

Cost of routing:• 50 nodes, 25 flows• Difference of utility: ? U=50• Routing overhead of 85% leads to ? U=50 for 25 flows.• Even for large routing overheads, MER is better than DIR!

SNR [dB]

refe

renc

e ut

ility

-ac

hiev

ed u

tility DIR, 25% loss

DIR, 10% loss

MER, 25% loss

MER, 10% loss

Utility difference between DIR and MELR

50

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Findings Valid Only in UWB

• In narrow-band multi-hop networks:§ Power control 0-PMAX might not be optimal§ Size of exclusion region depends on link length and other nodes§ Optimal MAC design depends on choice of routing

SNR SNR

optim

al li

nk le

ngth

tota

l util

ity

DIR

MER

DIR

MER

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Contents


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Conclusions

• We analyzed the jointly optimal power control, scheduling and routing for UWB networks.

• We analytically found the optimal power control, and used heuristic to derive optimal scheduling and routing.

Main findings are:• Optimal MAC protocol does not depend on choice of routing• MER is always optimal (even for routing protocols with high

overhead). • Optimal MAC protocol§ Exclusion regions around destinations. § Source needs to adapt rate to interference from outside of exclusion region. § When a node is sending, it is optimal to send with full power

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Conclusions

• Our findings hold only for wide-band systems due to linearity of rate function. In particular, in narrow-band system, choice of MAC typically depends on choice of routing.

• Focus of analysis was to find design goals, and not details of implementation

• Implementation guidelines:§ In static case routing can be implemented with existing algorithms using

inverse attenuation as cost of link§ Exclusion region radius depends only on local information;

Derive signaling to enforce exclusions§ For cheap, low complex networks: sufficient to use TDMA or All-at-once§ For low-power UWB networks, no exclusion protocol may be necessary

• Comparison with 802.11 MAC: we allow controlled interference, and adapt rate. Exclusions only around destinations

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Routing and Mobility

• High-level effects of mobility model:§ Packet loss probability on a link due to mobility: q§ Packet loss probability on a route of r hops: 1 - (1-q)r

§ Detailed mobility modeling impossible as we do not implement protocols

• We consider single-path routing policies:§ DIR (direct routing) – direct link with no relaying§ MER (minimum energy router) – route that minimizes Σ hl

-1

§ MELR (minimum energy-loss route) – route that minimizes Σ hl-1 (1-q)r-1

§ Intermediate – nodes are subsets of MER or MELR routes

MER

Intermediate

DIR

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Minimum Energy and Loss Routing

• Finding: It is optimal to use Minimum Energy Loss Routing (MELR)• MELR routing minimizes Σ hl

-1 (1-q)r-1

hl-1 – attenuation on link, q – loss due to mobility, r – number of hops

• Direct routing (DIR) has no routing protocol overhead.We introduce routing overhead Cr: fr = (1 - Cr) fnr

• For large routing overheads, it is better to use DIR• MELR is the same as MER when there is no mobility;

MELR is the same as DIR when there is high mobility

MER

Intermediate

DIR

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