Simulation Results

We consider a 6-node sensornetwork shown inFigure 2.7. It can be easily seen that

φ _1,0 = φ _3,0 = φ _5,0 = 1

^{, node}

0

^{being the common} destination for all the packets generated in the

12

Itis,hence,shownthattheproposeddistributedalgorithmprovidestheoptimal delaythatis achievable

basedontracdynamics.

network. The other available routes could also be usedfor nodes1, 3, and 5. But, we were

moreinterestedinclearlydemonstratignthetracsplit androutingprobabilities. Therouting

algorithm thus has to nd appropriate valueof

φ _2,5

^and

φ _4,3

^{in order thatthe trac ow in}

the networkcorrespondsto aWardrop equilibrium.

3 4

0 1

5 2

λ1 λ4

λ2 φ25

φ43

φ41

φ21

λ5 λ3

Figure2.7: Network Simulated for Stability

Apart froma demonstration of theconvergence of theproposed algorithm, we will seein

this section that the data sampling rates that a network can support using the Open

archi-tecture is very small. This is essentially because of the stability constraints on the channel

access rates. On the other hand, the Closed systemcan support higher datasampling rates

because of the fact that itis essentiallyself-regulating, guaranteed to be stable while

main-taining large data sampling rates; this is because a node generates a new packet only if it

has no other packet inthe queue. This however does not mean that the Closed system can

supportarbitrary datasamplingrates.

We have implemented the Open and Closed system as an application layer module in

TinyOS [52]. TinyOS is an open-source operating system designed for embedded WSNs. It

features acomponent-based architecture whichenables rapid innovation andimplementation

while minimizing code size asrequired bythe severememory constraints inherent inWSNs.

The sensor network model under consideration is shown in Figure 2.8. The sensor nodes

sample the data at a predened rate,

λ ⁰ _i s.

^{The sampled data is sent to the MAC queue for}

both open and closed system according to the explanation given earlier inSection 2.4. The

transmitqueueofnode

i

^{canhaveatmostonepacketinthetransmitqueuethatwasgenerated}

at node

i

^{. It can however have multiple packets in thetransmit queue to be forwarded, i.e.,}

those packets that were initially generated at some other node, and have arrived at node

i

to be forwarded to some other node. Therefore, we need not implement two-queues at the

MAC layer for sensor nodes for prioritizing trac. At simulation start up, the nodes learn

the network topology and built routes toward the fusion center (sink, node

0

^{). The fusion}

center isalso asensornode which has

0

^{samplingrate. Thislearning process,which depends}

onthe network topology for the given network inFigure 2.7, can take up to

50 − 70 seconds

for larger topologies. The routing layeris initiated withthe minimum-hop routing, which is

updated during the network lifetime according to the algorithm proposed in Section 2.6. In

this section, we present the numerical results once the neighbors are discovered and routes

areestablishedtoward thefusion center. We have utilized theTOSSIMsimulator of TinyOS

to validate our proposals. All simulation run for

1000 seconds

^{. The results presentedin this}

section arethe averageoverseveralsimulation runs.

Routing Routing Routing

ROUTING

MAC

PHY APPLICATION

Source Destination

(Fusion Center)

Forwarders

...

...

...

Figure 2.8: Sensor network architecture.

→

^{represents the ow of packets from the source}

to the destination. The forwarding sensor network receives a packet and queues into the

forwarding queueat the MAClayer. The routinglayerdoesnot buer theforwarding trac.

2.7.1 Open System Stability

InFigures. 2.9and2.10weplot,againsttheslotnumber,theaveragedelaysonthefourroutes

2 → 5 → 0

^,

2 → 1 → 0

^,

4 → 3 → 0

^,and

4 → 1 → 0

^{for the open system. Thedata sampling}

rates weresetat

λ ₁ = λ ₂ = λ ₃ = λ ₄ = λ ₅ = 0.2

^{. Notethatthedatasampling ratesaresmall.}

We were forced to select small data rates in order to guarantee stability of thenodes inthe

network. Thechannelaccessrates weresetto

α _i ≤ 0.2

^for

i = 1, ..., 5

^.

Observations

1. The delays on routes

2 → 5 → 0

^and

2 → 1 → 0

^{arevery close to each other, with a}

very fastconvergence. Similarlyfor routes

4 → 3 → 0

^and

4 → 1 → 0

^{. Thisshows that}

thealgorithm succeedsinachievinga Wardrop equilibrium.

2. Notethe highvalueofdelayonroutes

2 → 1 → 0

^and

4 → 1 → 0

^{even forthemoderate}

(or, very small) loadon thesystem.

3. The delays on dierent routes are sensitive to the channel access probabilities. Thus,

there is a need for carefully tuning the channel access probabilities. In Figure 2.9 and

2.10, we also see the convergence to a load-balanced regime (equal delays on all the

possible routes froma particular source).

0 1 2 3 4 5

0 100 200 300 400 500 600 700 800 900 1000

Estimated Delay (msec)

Time -> (Seconds)

Delay on Route 2->5->0 Delay on Route 2->1->0

Figure 2.9: Delays incurred on routes

2 → 5 → 0

^,

2 → 1 → 0

^{for Open System. Where}

λ ₁ = λ ₂ = λ ₃ = λ ₄ = λ ₅ = 0.2

2.7.2 Closed System Stability

Simulation results for the closed system are presented in Figure 2.11 and 2.12. The data

sampling rates were set at

λ ₁ = λ ₂ = λ ₃ = λ ₄ = λ ₅ = 0.2

^{. Nodes were expected to adapt}

theirchannel accessprobabilities basedon theoptimaltrac split usedby node

2

^and

4

^.

Observations

1. The delays on routes

2 → 5 → 0

^and

2 → 1 → 0

^{are very close to each other, with}

a fast convergence. This shows that the algorithm succeeds in achieving a Wardrop

equilibrium.

2. For routes

4 → 3 → 0

^and

4 → 1 → 0

^{, the delays are also close to each other, with a}

fast convergence. This shows that the algorithm is successful in achieving a Wardrop

equilibrium (equaldelays on allthe possibleroutes from aparticular source).

3. Note the small value of delay on routes

2 → 5 → 0

^and

4 → 3 → 0

^{even for moderate}

(or, very small) load on the system. This is to be compared with the corresponding

values shown underthe results for open system where the delays on these routes were

higher even thoughthe averagedatasamplingrates weresignicantly smaller. Thus, in

comparison withtheopen system,theclosed systemprovides better performance.

Wesimulateanother6-nodesensornetworkshowninFigure2.13todemonstratetheresults

onrouting. Theonlydierencewiththerstnetworkisthatwehaveadierentroutingsetup

but its logical representation is the same. It is easily seen that

φ _1,0 = φ _3,0 = φ _5,0 = 1

^{, node}

0

^{being the common} destination for all the packets generated in the network. The routing

0 1 2 3 4 5

0 100 200 300 400 500 600 700 800 900 1000

Estimated Delay (msec)

Time -> (Seconds)

Delay on Route 4->3->0 Delay on Route 2->1->0

Figure 2.10: Delays incurred on routes

4 → 3 → 0

^,

4 → 1 → 0

^{for Open System. Where}

λ ₁ = λ ₂ = λ ₃ = λ ₄ = λ ₅ = 0.2

algorithm thus has to nd appropriate valueof

φ 2,5

^and

φ 4,3

^{in order thatthe trac ow in}

the networkcorrespondsto aWardrop equilibrium.

2.7.3 Open System Routing

InFigure2.14and2.15weplot, againsttheslotnumber,theaveragedelaysonthefourroutes

3 → 2 → 0

^,

3 → 1 → 0

^,

5 → 4 → 0

^{, and}

5 → 1 → 0

^{for the open system. The data}

sampling rates were set at

λ _i ≤ 0.2

^{, for}

i = 1, ..., 7

^{. Note that the data sampling rates are}

small. We were forced to selectsmall data rates in order to guarantee stability of the nodes

inthenetwork. Thechannelaccessrates weresetto

α _i ≤ 0.2

^for

i = 1, ..., 7

^.

Observations fromOpen System

1. The delays on routes

3 → 1 → 0

^and

3 → 2 → 0

^{arevery close to each other, with a}

very fastconvergence. Similarlyfor routes

5 → 4 → 0

^and

5 → 1 → 0

^{. Thisshows that}

thealgorithm succeedsinachievinga Wardrop equilibrium.

2. Note the high value of delay on routes

3 → 1 → 0

^and

3 → 2 → 0

^{even for moderate}

(or, very small) loadon thesystem.

3. Figure 2.15 shows the delay obtained by varying the channel access rates to

α i = 0.1

for

i = 1, ..., 5

^{, and}

λ ⁰ s

^{remaining the same as earlier. The estimated delays show the}

sensitivity to channel access probabilities. Thus, there is a need to carefully tune the

α ⁰ _i s

^{. InFigure2.15,wealsoseethat}convergencetoaload-balanced regime(equaldelays on all the possible routes froma particular source) isviolated bychanging the

α ⁰ _i s

^{. As}

we will seelater, this is not a problem intheclosed systembecause the systemadapts

0 1 2 3 4 5

0 100 200 300 400 500 600 700 800 900 1000

Estimated Delay -> (msec)

Time -> (Seconds)

Delay on Route 2->5->0 Delay on Route 2->1->0

Figure 2.11: Delays incurred on routes

2 → 5 → 0

^,

2 → 1 → 0

^{for Closed System. Where}

λ ₁ = λ ₂ = λ ₃ = λ ₄ = λ ₅ = 0.2

its channelaccessprobabilities tomeet thetargettrac and thereisno needof further

tuning this parameter.

4. Thedelaysondierentroutesarealsoclose toeachother,withafastconvergence. This

is also reected in the trac split obtained by the algorithm, as in Figure 2.16we see

that node 3 uses node 2 a little less than the other available route because of smaller

delay on

3 → 1 → 0

^{. Similarly, node 5 also use}

5 → 1 → 0

^{more than}

5 → 4 → 0

becauseofsmallerdelayontheformer. Itisalsointeresting tonotethatthetracsplit

obtained inthis gureis proportional to thedelays on dierent routes inthe network,

i.e.,

φ 32

^{is very close to 0.5 due to a smaller dierence in estimated delays on routes}

3 → 1 → 0

^and

3 → 2 → 0

^{, whereas,}

φ ₅₄

^{isnot due to relatively large dierence inthe}

estimated delays on routes

5 → 1 → 0

^,

5 → 4 → 0

^{. Thisis Wardrop} equilibriumwhere a slightly higher delay path is less usedi.e., the

+ve

^{valueof trac on alternate route}

is imposedbythealgorithmto ensure thatallthealternativesareprobedoftenenough

to cope upwitha change intrac patterns.

2.7.4 Closed System Routing

Simulation results for the closed system are presented in Figure 2.17, 2.18, 2.19, and 2.20.

Thedata sampling rates were setat

λ ₁ 0.1, = λ ₂ = 0.2, λ ₃ = 0.1, λ ₄ = 0.005, λ ₅ = 0.1, λ ₆ = 0.1, λ ₇ = 0.1

^{. Nodeswereexpected to adapt their channel access} probabilities based on the optimaltrac splitused bynode

3

^and

5

^.

Observations fromClosed System

1. The delays on routes

3 → 1 → 0

^and

3 → 2 → 0

^{are very close to each other, with}

0 1 2 3 4 5

0 100 200 300 400 500 600 700 800 900 1000

Estimated Delay -> (msec)

Time -> (Seconds)

Delay on Route 4->3->0 Delay on Route 4->1->0

Figure 2.12: Delays incurred on routes

4 → 3 → 0

^,

4 → 1 → 0

^{for Closed System. Where}

λ ₁ = λ ₂ = λ ₃ = λ ₄ = λ ₅ = 0.2

a fast convergence. This shows that the algorithm succeeds in achieving a Wardrop

equilibrium.

2. Forroutes

5 → 1 → 0

^and

5 → 4 → 0

^{,thedelaysarealsoclose toeachother,withafast}

convergence. This is also reected inthe trac split obtained by the algorithm, asin

Figure2.19we seethatnode5uses node1 formost ofitstrac,thusobtainingsmaller

delay. Thisis again Wardrop equilibrium where thehigher delaypath isnot used(the

small

+ve

^{valueoftraconroute}

5 → 4 → 0

^{isimposedbythealgorithm toensurethat}

all thealternativesareprobed oftenenough to copeup withachange inthenetwork).

3. Note the small value of delay on routes

3 → 1 → 0

^and

3 → 2 → 0

^{even for moderate}

(or, very small) load on the system. This is to be compared with the corresponding

values shown underthe results for open system where the delays on these routes were

higher even thoughthe averagedatasamplingrates weresignicantly smaller. Thus, in

comparison withtheopen system,theclosed systemprovides better performance.

4. Figure 2.20 shows that the algorithm is also able to adapt the channel access rates in

a distributed fashion. It can be checked that the values of

α ⁰ _i s

converged-to by the algorithm indeedarejustenough to serve thetrac oered to thedierent nodes.

Intheaboveanalysis,wehaveonlypresentedresultsontheperformanceofourdistributed

routingalgorithm forboth open andclosed systems. We, now, considera randomly deployed

sensornetworkwith50 sensornodes. There isonly corner-sink which istherepresentative of

datacollectioninthe network. Sensorstransmit theirreadingsinamultihop fashion towards

this sink. The sampling rate of all the nodes is a random variable uniformly distributed

between 0and 0.2. Thechannelaccessrateofallthenodesissetaccordingtotheir sampling

0 1

φ31

φ32 φ51

φ54

λ1

2 3

4 5

λ2 λ3

λ4 λ5

Figure 2.13: Network Simulated for Routing

rates. Thesimulationrunsfor3000sand thesamplingvector ischanged every100sto seethe

impact of trac pattern change on the performance of closed system. Figure 2.21 displays

individual node delays over time w.r.t change in the sampling rate. It can be easily seen,

for each node in the network, that the delay does not change much due to a change in the

trac pattern overtime. The coupling intheclosed systemautomatically regulates thedelay

by adapting its own sampling process to the change in network dynamics. The last degree

of freedom, i.e., the channel access rates are also adapted using the proposed optimization

criteria. TheCDF oftheestimateddelayinthenetwork ispresentedinFigure2.22.

2.7.5 Closed System with Two Transmit Queues

We,again, considerthe6-nodesensornetwork showninFigure2.13. We considerthis simple

network toclearly demonstrate the stability region in closed system withtwo transmit queues.

The transmit queue of node

i

^{can have multiple packets in thetransmit queue (both}

Q _i

^,i.e,

selfgenerated,and

F _i

^{,i.e.,thosepacketsthatwereinitiallygenerated atsomeothernode,and}

havearrivedatnode

i

^{to beforwardedtosomeothernode). Therefore,weneedtoimplement}

two-queuesattheMAClayerforsensornodesforprioritizingtrac(basedontheappropriate

weights given to

Q _i

^and

F _i

^{). We have} implemented the Closed system with two-queues as a cross-layer (application-mac) module in TOSSIM [52]. The routing layer is initiated with

the minimum-hop routing, which is updated during the network lifetime according to the

algorithm proposedinSection 2.6. Inthis section, we presentthe simulation resultsoncethe

neighbors are discovered and routes are established toward the fusion center. All simulation

runsfor

10 ⁸

^,seconds.

We present in Table 2.3, the results on stability region and throughput for sensors 1, 2,

and4 assensors3 and5 donot forward anytrac and

y _i

^for

i = 3, 5

^{is setto 1.}

In order to demonstrate theresults on delay-and-stability together using a closed-system

with two-queues, we have implemented a 50-nodes sensor network with a common sink. In

Figure 2.23 we plot, against the slot number, the average delays for our closed-system with

0 5 10 15 20 25 30 35 40 45

0 1e+07 2e+07 3e+07 4e+07 5e+07 6e+07 7e+07 8e+07 9e+07 1e+08

Estimated Delay (s)

Time ->

Delay on Route 3 -> 1 -> 0 Delay on Route 3 -> 2 -> 0 Delay on Route 5 -> 1 -> 0 Delay on Route 5 -> 4 -> 0

Figure2.14: Delays incurred on routes

3 → 1 → 0

^,

3 → 2 → 0

^,

5 → 1 → 0

^,

5 → 4 → 0

^for

open system.

α ₁ = 0.2, α ₂ = 0.15, α ₃ = 0.1, α ₄ = 0.2, α ₅ = 0.2

^,

λ ₁ = 0.01, λ ₂ = 0.01, λ ₃ = 0.04, λ ₄ = 0.05, λ ₅ = 0.05.

two-queues and single-queue system. The datasampling rates were set at

λ _i ≤ 0.1

^,

∀ i

^{. Note}

thatthedatasampling rates are small. We were forcedto select smalldata rates inorder to

guaranteestabilityof thenodesinthenetwork.

Observations from the Simulations: The average delays on routes in two-queues

closed systemareverysmall comparedto single-queuesystem. Thisisduetotheappropriate

choice of weights given to both

F _i

^and

Q _i

^{(as discussed in Section 2.4) compared to the}

single queue system where we do not have the service dierentiation. The routing schemes

(Section2.6) allows both systems to pickthe shortest-delaypaths based on delayestimates.

Theseresults complywithour motivationthatservicedierentiation at MAClayer results in

better overall performanceof thesystemand canhelpstudy theimpact of dierentnetwork

parameters on its performance.

Dans le document Cross layer optimization of wireless sensor and sensor-actuator networks (Page 56-64)