Cooperative Broadcast for Cooperative Broadcast for Maximum Network LifetimeMaximum Network Lifetime

Ivana Maric and Roy YatesIvana Maric and Roy Yates

Wireless Wireless MultihopMultihop Network BroadcastNetwork Broadcast

• N nodes

• Source transmits at rate R

• Messages are to be delivered to all the nodes

• Nodes can choose transmit powers

source

System Model: Orthogonal ChannelsSystem Model: Orthogonal Channels

• Each link is an AWGN channel with bandwidth W

• Each transmission in an orthogonal channel

• Nodes can listen to all the channels

• Motivation: Sensor networks

• Low-powered nodes, very low data rates

• Large bandwidth resources

• Objective: Energy-efficient network broadcast protocols

• Minimum-energy broadcast

• Maximum-lifetime broadcast

MinimumMinimum--energy broadcastenergy broadcast

• Problem: Broadcast at rate R to all nodes using minimum total power

• Formulated as broadcast tree problem [J. Wieselthier, G. Nguyen, A. Ephremides]

• Wireless multicast advantage:all the nodes in the range hear a transmission

• Problem is NP-complete [M. �agalj et al., Ahluwalia et al., W. Liang]

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Maximum network lifetime Maximum network lifetime

• Problem: Maximize the amount of time until the first node battery dies [J.H. Chang and L. Tassiulas] • Performs load balancing: distributes traffic more evenly among the nodes

• Static solution given by a broadcast tree• Based on the initial battery energy levels

• Dynamic solution consists of a series of broadcast trees[R.J. Marks et al., I.Kang et al.]

• suboptimal

Accumulative broadcastAccumulative broadcast

• Conventional broadcast:• No interference• A node forwards only when reliable• Each node retransmits the same message• A node receives message from only one transmission as specified by a tree

• Accumulative broadcast• Old Idea: Exploit Overheard (side) Information• Allow nodes to collect energy of unreliably received signals• As the message is forwarded, a node collects multiple unreliable copies until

it accumulates energy needed for reliable reception

Accumulative broadcastAccumulative broadcast

Wireless advantageWireless advantage Accumulative broadcastAccumulative broadcast

• Allow nodes to collect energy of unreliably received signals

Reliable forwardingReliable forwarding

• More energy-efficient than conventional broadcast because it captures more radiated energy

•Reliable or unreliable forwarding?

• Any node can decide to forward as soon as it receives an unreliable copy

•Problem formulation?

•• A node can forward a message only after reliable decodingA node can forward a message only after reliable decoding

• Suboptimal • Benefits:

• Simplifies the system architecture• Still allows for unreliable overheard information

• After K nodes retransmit a codeword X:

Maximum achievable rate at node m:

rm = W log2(1 +� hmk pk / NoW)

• For a given broadcast rate at the source

r = W log2(1+PT / NoW)

• Node m reliable when

�k hmk pk � PT

Relays use Relays use ““Repetition CodingRepetition Coding””

p1

p2

p3

pK

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�

•• All the nodes use the same codebook: relays resend the same codeAll the nodes use the same codebook: relays resend the same codewordword

YY

XX

XX

XX

XX

m

k=1

K

• As W → �,rm → � hmk pk / Noln2

• MAC upper bound:

CMAC = W � log2(1 + hmk pk / NoW)

→ � hmk pk / Noln2

• Orthogonal channels preclude the coherent combining gain

Repetition is OK for Large WRepetition is OK for Large W

source

• Given fixed powers {p1,…pK } and reliable forwarding, the maximum rate achievable from the source to any destination is achieved by therepetition coding in the limit of large W.

•• How do we solve the accumulative broadcast problem?How do we solve the accumulative broadcast problem?

m

p1

p2

p3

pK

�

kk

kk

Network lifetimeNetwork lifetime

• A lifetime of a node i - transmission time until node battery is fully drained

Ti (pi ) = ei / pi

ei - initial battery energy pi - transmit power

• Network lifetime - duration of a data session until the first node battery is fully drained

Tnet (p)= min Ti (pi )

• Choose a transmission schedule• An order in which nodes become reliable• For each node, schedule specifies a subset of nodes that contribute to its

reliable decoding

• Represent a schedule with matrix X

xij=

• xij indicates that node i collects energy from a transmission by node j

• Define a gain matrix H(X): [H(X)]ij = hij xij

• Problem defined as:

Transmission scheduleTransmission schedule

min max { pi /ei }H(X)p � 1PT

p � 0

1 node i scheduled to transmit after node j

0 otherwise

min max { pi /ei }h21 p1 � PT

h31 p1 + h32 p2 � PT

h41 p1 + h42 p2 + h43 p3 � PT

h51 p1 + h52 p2 + h53 p3 + h54 p4 � PT

p1, p2, p3, p4 � 0

• Network of N = 5 nodes• Consider a schedule [1 2 3 4 5]

• Problem is defined as

Maximum network lifetime problemMaximum network lifetime problem

1

45

23

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X=

0 0 0 0 01 0 0 0 01 1 0 0 01 1 1 0 01 1 1 1 0

• Different node batteries � use normalized node powers

• Problem becomes

• Maximum network lifetime LP

min max pi

H(X)p � 1PT

p � 0

LP for Transmit PowersLP for Transmit Powers

pi = pi e1 / ei

q*(X) = min qH(X)p � 1PT

p � 1q p � 0

• Minimum Total Power

• Maximum Lifetime

• Min Total Power is NP-complete• Independently shown by [Y-W. Hong & A. Scaglione]

- different physical model• Finding the best schedule is hard

• Max Lifetime is easy – Why?

Min Power vs. Max LifetimeMin Power vs. Max Lifetime

min �i pi

H(X)p � 1PT

p � 0

min max { pi /ei }H(X)p � 1PT

p � 0

Max lifetime Max lifetime

• Identifying one best schedule is not crucial

• Solution: Power p*=minX q*(X) and a schedule for which p* is feasible

• Power p*=minX q*(X) feasible for a set of schedules X*

• To identify X* : use a simple procedure that, for any power p, finds the schedules for which pis feasible

The The ASAP(ASAP(pp) distribution) distribution

• Use the observation:

• The ASAP(p) distribution:•during a broadcast with power p, a node transmits with pas soon as possible � as soon as it becomes reliable

As soon as one node transmits with p:every reliable node can use p with no impact on the network lifetime

The The ASAP(ASAP(pp) distribution) distribution

source

power p

reliable

reliable

• Source transmits with power p� any node can transmit with p

no impact on the network lifetime

• At each stage:Set of nodes that became reliable

in the previous stage transmit with p

• If p is large enough, ASAP(p) is a feasible broadcast :message is delivered to all nodes

• All relays transmit with power p

• Otherwise, ASAP(p) stalls

ASAP TheoremASAP Theorem

• Theorem: If p is a feasible power for a schedule X, then ASAP(p) is a feasible broadcast.

• ASAP(p) finds all the schedules for which p is feasible

• For the optimum power p* , ASAP(p*) is feasible

• If p* were known, we could broadcast with ASAP(p*)

Maximum Lifetime Accumulative BroadcastMaximum Lifetime Accumulative Broadcast(MLAB)(MLAB)

• Finds the power p* to maximize the network lifetime• Then, broadcasting with ASAP(p*) maximizes the network lifetime

• Finds p* through a series of ASAP(p) distributions

• Start with the smallest possible power p=PT / h21

• If ASAP(p) stalls at stage µ(p): • Find the minimum increase �* for which ASAP( p+�*) doesn’t stall at µ(p)• Set p= p+ �* and perform ASAP(p)

• MLAB finishes when ASAP(p) makes all nodes reliable

MLAB MLAB –– the ASAP( the ASAP( pp**) distribution) distribution

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power p

reliable

reliable

1. Initialize power: p=PT / h21

2. Apply ASAP(p)3. If ASAP(p) stalls:4. For all j unreliable find �j:

PT = (p+�j) � hjk

set: �* =min �j

increase: p ← p+�*go to 2.

ASAP(p) stallspower p reliablereliable

MLAB finds the optimal powerMLAB finds the optimal power

Theorem 2:The MLAB algorithm finds the optimum power p* such that ASAP(p*) maximizes the network lifetime.

Conventional Broadcast ComparisonConventional Broadcast Comparison

• Throw N nodes in a square (100 trials)• Compared with [I. Kang & R. Poovendran]

• static problem solution: MST and MSNL• dynamic problem solution: WMSTSW

Accumulative broadcast enables load balancingAccumulative broadcast enables load balancing

• Conventional broadcast:Network lifetime determined by node with the most disadvantaged child

• Accumulative broadcast:Nodes cooperatively transmit to increasethe shortest lifetime in the network

All relay nodes have the same lifetime

network lifetime

transmitted energy

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Accumulative broadcast enables load balancingAccumulative broadcast enables load balancing

2

transmitted energy

network lifetime

3

45

6

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• Conventional broadcast:Network lifetime determined by node with the most disadvantaged child

• Accumulative broadcast:Nodes cooperatively transmit to increasethe shortest lifetime in the network

All relay nodes have the same lifetime

Total powerTotal power

• Min energy algorithms:Conventional broadcast

• BIP [J. Wieselthier, G. Nguyen & A. Ephremides]

Accumulative broadcast• Greedy filling algorithm

Maximum powerMaximum power

• Min energy algorithms:Conventional broadcast

• BIP [J. Wieselthier, G. Nguyen & A. Ephremides]

Accumulative broadcast• Greedy filling algorithm

ConclusionConclusion

• Accumulative broadcast: Nodes collect energy of unreliably received signals

• For maximum lifetime problem: ASAP distribution is optimal

• MLAB finds min power ASAP distribution

• Performs load balancing

• No need for updates: solves the static and dynamic problem

• Distributed implementation

• When ASAP stalls: no need to restart

• Reliable nodes retransmit with power increment �*