Data Propagation Algorithmsin Wireless Sensor Networks (Smart Dust)

Sotiris Nikoletseas

University of Patras

and Computer Technology Institute

Greece

Invited Talk

32nd SOFSEM Conference

Prague, Czech Republic, January 2006

1

Overview of this talk

• technological description / leading projects / applications

• new, critical challenges

• models / assumptions

• 2 problems: data propagation and energy balance

• 5 representative protocols:

- tree-structured (Directed Diffusion)

- hierarchical (LEACH)

- local optimization (LTP)

- probabilistic redundance (PFR)

- energy balance (EBP)

2

• performance evaluation:

- analysis

- simulation

• conclusions / more problems /future directions

3

Work presented in this talk

- C. Intanagonwiwat, R. Govindan and D. Estrin, “Directed Diffusion: A Scalable and

Robust Communication Paradigm for Sensor Networks”, in MOBICOM 2000.

- W. Heinzelman, A. Chandrakasan and H. Balakrishnan, “Energy-Efficient Communi-

cation Protocol for Wireless Microsensor Networks”, in HICSS 2000.

- I. Chatzigiannakis, P. Spirakis and S. Nikoletseas, “Efficient and Robust Protocols for

Local Detection and Propagation in Smart Dust Networks”, in the Journal of Mobile

Networks (MONET), 2005. Also, in the ACM Principles of Mobile Computing (POMC)

2002.

- I. Chatzigiannakis, T. Dimitriou, S. Nikoletseas, and P. Spirakis, “A Probabilistic Al-

gorithm for Efficient and Robust Data Propagation in Smart Dust Networks”, in the

Journal of Ad-Hoc Networks, 2005. Also, in the European Wireless (EW) Conference,

2004.

- C. Efthymiou, S. Nikoletseas and J. Rolim, “Energy Balanced Data Propagation in

Wireless Sensor Networks”, in the Wireless Networks (WINET) Journal, 2005. Also,

in WMAN 2004.

4

What is a Wireless Sensor Network

• very large number of tiny “smart” sensors

• severe limitations

• densely / randomly deployed in an area

• self-organization

• co-operation

• locality

an “ad-hoc” wireless network for:

• sensing crucial events

• data propagation

5

A smart dust “cloud”

Sensor nodes Sensor field

Control Center

6

“Smart” Sensor Nodes

• very small size

• operate on a small battery

• multifunctional sensing

• wireless communication

• limited computing power / small memory

• low cost

7

Technical Specifications of Current Sensors

Dimensions 3cm × 3cm × 2cmCPU Speed 4 MHzMemory ROM: 128Kb FLASHPower Supply 2X AA batteriesPower Consumption 0.75 mWProcessor Current Draw 5.5 mA (active current)

< 20 µA (sleep mode)Network Wireless 4 Kbits/sec at 916 MHz (radio)

Radio range depends on antennae configuration

Figure 1: MPR300CB Specifications (Berkeley motes)

Software Footprint 3.4 KbInactive State < 25 µAPeak Load 20 mATypical CPU Usage < 50 %Prog. Language Used NesCComponents provided Sensor Interface

Very Basic RoutingRadio Communication

Figure 2: TinyOS Key Facts

8

9

Smart vs Traditional Sensors

In traditional sensors:

• sensors perform only sensing

• sensor positions are carefully engineered

• deployed far from the area of interest

• data processing only at some central nodes

• larger in size

• constant power supply

10

Some Leading Projects (I)

• “Smart Dust” (Berkeley)

– primary goal: a few mm3 size

– laser communication

– motes development

– TinyOS

(http://robotics.eecs.berkeley.edu/∼pister/SmartDust/)

11

Some Leading Projects (II)

• WINS (“Wireless Intergrated Network Sensors”) (UCLA)

– focus : RF communication

– development of low power MEMS

(http://www.janet.ucla.edu/WINS/)

• “Ultralow Power Wireless Sensor” (MIT)

– primary goal : extremely low power consumption

– power levels : 10 µW to 10 mW

– RF communication

(http://www-mtl.mit.edu/∼jimg/project top.html)

12

Applications (I)

Sensors for a wide variety of conditions:

• temperature

• motion

• noise levels

• light conditions

• humidity

• object presence

• mechanical stress levels on attached objects

13

Applications (II)

• continuous sensing

• detection of a crucial event

• location sensing (including motion tracking)

• local control of actuators (ambient intelligence)

• micro-sensing

14

Applications (III)

• Enviromental applications

– forest fire detection

– flood detection (US Projects ALERT, COUGAR, DataSpace)

– precision agriculture

– weather forecast

– planetary exploration

• Health applications

– telemonitoring of human physiological data (“Health Smart Home”,Grenoble, France)

– tracking and monitoring doctors/patients

– drug administration in hospitals

15

Applications (IV)

• Home applications

– home automation (local/remote management of home devices)

– smart enviroments (adapt to the people’s needs)(“Residential Laboratory”, Georgia Institute of Technology)

• More applications (partial list)

– inventory control

– security / military

– vehicle tracking and detection

– interactive museum (San Francisco Exploratorium)

– monitoring material fatigue

16

Sensors vs Ad-hoc Wireless Networks

• much larger number of devices

• more dense deployment / interactions / complexity

• prone to failures

• more limited capabilities

• more frequently/more dynamically changing topology

• less global knowledge

17

New/Critical Challenges (I)

• Scalability

– huge number of sensor nodes

– high densities of nodes

– many complex interactions

How does protocol performance scale with size?

Even correctness may be affected by size.

18

New/Critical Challenges (II)

• Fault-tolerance

Sensors may

– fail (temporarily or permanently)

– be blocked / removed

– cease communication

due to various reasons

– physical damage

– power exhaustion

– interference

– power saving mechanisms

Can the network tolerate failures?

19

New/Critical Challenges (III)

• Efficiency

– energy spent

– time (for data propagation)

20

New/Critical Challenges (IV)

• Inherent trade-offs (e.g. energy vs time)

• Competing goals / various aspects:

- minimizing total energy spent in the network

- maximising the number of “alive” sensors over time

- combining energy efficiency and fault-tolerance

- balancing the energy dissipation

• Application dependence

• Dynamic changes / heterogeneity

• variety of protocols needed / hybrid combinations

• adaptive protocols, locality

• simplicity, randomization, distributedness

21

The Problem of Local Detection and DataPropagation

A single sensor, p, senses a local event E . The general data propagationproblem is the following:

“How can sensor p, via cooperation with the rest of the cloud, prop-agate information about event E to the control center?”

22

Representative data propagation protocols

• Directed Diffusion (DD): a tree-structure protocol (suitable for low dy-namics)

• LEACH: clustering (suitable for small area networks)

• Local Target Protocol (LTP): local optimization (best for dense net-works)

• Probabilistic Forwarding Protocol (PFR): redundant optimized trans-missions (good efficiency / fault-tolerance trade-offs, best for sparse net-works)

• Energy Balanced Protocol (EPB): guaranteeing same per sensor energy(prolong network life-time)

23

Directed Diffusion

• requires some coordination between sensors

• creates / maintains some global structure (set of paths)

• a paradigm / suite of several protocols

• here: a tree-based version

24

Directed Diffusion elements

• Interest messages (issued by the control center)

• Gradients (towards the control center)

• Data messages (by the relevant sensors)

• Reinforcements of gradients (to select “best” paths)

25

Interests

• An interest contains the description of a sensing task.

• Task descriptions are named e.g. by a list of attribute-value pairs.

• The description specifies an interest for data matching the attributes.

type = wheeled vehicle // detect vehicle location

interval = 10 ms // send events every 10 ms

duration = 10 minutes // for the next 10 minutes

rect = [-100, 100, 200, 400] // from sensors within rectangle

Example of an interest

• Interests are injected into the network at the control center

26

Gradients

For each interest message received, a gradient is created

Gradients are formed by local interaction of neighboring nodes, establishinga gradient towards each other.

Gradients store a value (data rate) and a direction (towards the sink) for“pulling down” data.

27

Data propagation

• A sensor that receives an interest it can serve, begins sensing.

• As soon as a matching event is detected, data messages are sent to therelevant neighbors using the gradients established.

• A data cache is maintained at each sensor recording recent history.

28

Reinforcement for Path Establishment

The sink initialy repeatedly diffuses an interest for a low-rate event notifica-tion, through exploratory messages.

The gradients created by exploratory messages are called exploratory andhave low data rate.

As soon as a matching event is detected, exploratory events are generatedand routed back to the sink.

After the sink receives those exploratory events, it reinforces one (or more)particular neighbor in order to “draw down” real data.

The gradients that are set up for receiving high data rate information arecalled data gradients.

29

Path Establishment Using Positive Reinforcement

To reinforce a neighbor, the sink re-sends the original interest message witha smaller interval (higher rate).

Upon reception of this message a node updates the corresponding gradientto match the requested data rate.

The selection of a neighbor for reinforcement is based on local criteria, i.e.:

• the neighbor that reported first a new event is reinforced.

• the higher data rate neighbor is reinforced.

• more than one neighbors are reinforced.

The data cache is used to determine which criteria are fulfilled.

30

Negative Reinforcement

Negative reinforcement is achieved by sending an interest with exploratory(low) data rate.

If all outgoing gradients of a node are exploratory the node negatively rein-forces its neighbors.

Negative reinforcement is applied when certain criteria are met i.e. a gradientdoesn’t deliver any new messages for an amount of time, or a gradient has avery low data rate, etc.

31

Summary Evaluation of Directed Diffusion

• improves over flooding a lot

• significant energy savings

• performance drops when network dynamics are high

32

Low Energy Adaptive Clustering Hierarchy(LEACH)

• Network is partitioned in clusters.

• Each cluster has one cluster-head.

• Each non cluster-head sends data to the head of the cluster it belongs.

• Cluster-heads gather the sent data, compress them and send them to thesink directly.

33

Dynamic Clusters

34

Randomized rotation of cluster-heads

- Node n decides with probability T(n) to elect itself cluster-head

T (n) =

P1−P∗(rmod 1

P )if n ∈ G

0 otherwise

P : the desired percentage of cluster heads.r: the current round.G: the set of nodes that have not been cluster-heads in the last 1

P rounds.

- T(n) is chosen so as to get on the average the same number of cluster-heads in each round.

35

Cluster-head advertisement

• Each cluster-head broadcasts an advertisment message to the rest nodesusing a CSMA-MAC protocol.

• Non cluster-head nodes hear the advertisments of all cluster-head nodes.

• Each non cluster-head node decides its cluster-head by choosing the onethat has the stronger singal.

36

Cluster set-up phase

• Each cluster-head is informed for the members of its cluster.

• The cluster-head creates a TDMA schedule

• Cluster-head broadcasts the schedule back to the cluster members.

37

Steady Phase

• Non cluster-heads

– Sense the environment.

– Send their data to the cluster head during their transmission time.

• Cluster-heads

– Receive data from non cluster-head nodes.

– Compress the data received.

– Send their data directly to the base station.

38

Summary Evaluation of LEACH

• reduces energy dissipation through compression of data at cluster-heads.

• in large networks, direct transmissions are very expensive.

• performance drops when the network traffic is high (e.g. many eventsgenerated).

39

Our Local Target Protocol (LTP)

We call network nodes, “grain particles”.

a) Each particle may have two communication modes: a broadcast (radio)beacon mode and a directed to a point transmission mode (laser beam).

b) Each particle may alternate between a sleeping and an awake mode.During sleeping periods grain particles cease any communication.

c) Particles do not move.

d) A smart dust cloud (a set of particles) is spread in a two-dimensionalarea (plane).

40

e) A receiving wall W is a line in the plane. The wall represents the controlcenter.

f) Each particle is aware of the direction towards W .

g) No geolocation abilities assumed

Definitions: Let d (in numbers of particles /m2) be the density of thecloud.

Let R be the maximum (beacon/laser) transmission range of each particle.

41

W p'

beacon circle

Example of the Search Phase

W

p 0

p 1

p 2

a 1

p 3

a 0

a 2

Example of a Transmission

42

Efficiency

Definitions: Let hopt the (optimal) number of “hops” (vertical to W trans-missions) needed to reach W , if particles always exist in pair-wise distancesR towards W .

Let h the actual number of hops (transmissions) taken to reach W . The“hops” efficiency of the data propagation protocol is the ratio

Ch =h

hopt

where hopt =⌈

d(p,W)R

⌉

43

Why studying h, Ch?

When a particle p “looks around” for a particle as close to W as possible topass information, it may not get any particle in the perfect direction (on theline vertical to W passing from p), mainly because:

a) There might never have been any particles in that direction.

b) Particles of sufficient remaining battery power may not be available.

c) Particles available may temporarily “sleep” to save energy.

44

The Local Target Protocol (LTP)

Let d(pi, pj) the (vertical) distance of pi, pj and d(pi,W) the (vertical)distance of pi from W . Let info(E) the info to be propagated. Each p′

receiving info(E) does the following:

• Search Phase: It uses a low energy broadcast of a beacon (angle a aboveand below the vertical line) to discover a particle closer to W (i.e. a p′′

where d(p′′, W) < d(p′, W)).

• Direct Transmission Phase: If found, p′ sends info(E) to p′′ via a directline (laser) transmission.

• Backtrack Phase: If repetitions of the search phase fail to discover aparticle nearer to W , then p′ sends info(E) to the particle it received theinformation from.

45

Simplifying Assumptions for a Rigorous Analysis

• The search phase always finds a p′′ in the semicircle of center p′ andradius R towards W .

This assumption can be relaxed: (a) by repetitions of the search phase (b)we may consider a cyclic sector defined by circles of radiuses R−∆R, R(c) if a search phase ultimately fails, the protocol backtracks.

• The position of p′′ is random uniform in the arc of angle 2a.

• Each target selection is stochastically independent of the others.

46

Lemma: The expected “hops” efficiency of LTP in the a-uniform case isE(Ch) ≃ α

sin α, for large hopt.

Also 1 ≤ E(Ch) ≤ π2 ≃ 1.57.

Proof: A sequence of points is generated,p0 = p, p1, p2, . . . , ph−1, ph where ph−1 is the first particle found withinW ’s range and ph is beyond W . Let αi be the (positive or negative) angleof pi w.r.t. pi−1’s vertical line to W . It is:

h−1∑

i=1d(pi−1, pi) ≤ d(p, W) ≤

h∑

i=1d(pi−1, pi)

The (vertical) progress towards W is

∆i = d(pi−1, pi) = R cos αi. We get:

47

h−1∑

i=1cos αi ≤ hopt ≤

h∑

i=1cos αi

From Wald’s equation, then

E(h − 1) · E(cos αi) ≤ E(hopt) ≤ E(h) · E(cos αi) ⇒

α

sin α≤

E(h)

hopt= E(Ch) ≤

α

sin α+

1

hopt

since

E(cos αi) =∫ α−α cos x ·

1

2α· dx =

sin α

α

Assuming large values for hopt and since for 0 ≤ α ≤ π2 it is 1 ≤ α

sin α ≤ π2)

we get the result.

48

Summary evaluation of LTP

• local, simple, greedy protocol

• no global structure (set of paths) maintained

• good for dense networks

• performance drops in sparse / faulty networks

49

Our Probabilistic Forwarding Protocol (PFR)

We assume that each grain particle has the following abilities:

i) It can estimate the direction of a received transmission (e.g. via the technology of

direction-sensing antennae).

ii) It can estimate the distance from a nearby particle that did the transmission (e.g. via

estimation of the attenuation of the received signal).

iii) It knows the direction towards the sink S. This can be implemented during a set-up

phase, where the sink broadcasts information about itself to all particles.

iv) All particles have a common co-ordinates system.

Note that GPS information is not needed for this protocol. Also, there is no need to know

the global structure of the network.

50

Propagation Protocol Properties

• Correctness. Π must guarantee that data arrives to the sink S, giventhat the network is operational.

• Robustness. Π must guarantee that data arrives at enough points in asmall interval around S, in cases where part of the network has becomeinoperative.

• Efficiency. Π should have a small ratio of the number k of activatedparticles over the total number N of particles r = k

N . Thus r is an energyefficiency measure of Π.

51

The basic idea of PFR

PFR probabilistically favors transmissions towards the sink within a thinzone of particles around the line connecting the particle sensing the event Eand the sink.

S

E

Particles that particiapate in

forwarding path

Thin Zone of particles

52

The Forwarding Probability

Data is propagated with a suitable chosen probability p, while it is notpropagated with probability 1 − p.

The favor near-optimal transmissions we choose

Pfwd =φ

π

S

E

1

2 p 1

p 2

53

The two phases of PFR

Phase 1: The “Front” Creation Phase. Initially we build (by using alimited, in terms of rounds, flooding) a sufficiently large “front” of particles,to guarantee the survivability of the data propagation process. Each particlehaving received the data, deterministically broadcasts them towards the sink.

Phase 2: The Probabilistic Forwarding Phase.

Each particle p receiveng info(ǫ), broadcasts it to all its neighbors withprobability IPfwd (or it does not propagate any data with probability 1 −IPfwd) defined as follows:

IPfwd ={1

φπ

if φ ≥ φthreshold = 134o

otherwise

54

The Correctness of PFR

Lemma: PFR always succeeds in sending the information from E to Swhen the whole network is operational.

In the proof, we use geometry (i.e. we cover the network area by unit squaresand show that there are always particles “close enough” to the optimal line,i.e. with φ > 134o, that deterministically broadcast).

55

The Energy Efficiency of PFR• We consider particles that are active but as far as possible from ES

S E

n 0

The particles inside the LQ Area

• We approximate w by the following random walk

E S

y

y’

x

x

x o

x o

0

The random walk W

56

By using stochastic dominance by a continuous time “discouraged arrivals”birth-death process, we prove:

Theorem: The energy efficiency of the PFR protocol is Θ((n0n )2) where

n0 = |ES| and the total numbr of particles in the network is N = n2. Forn0 = o(n), this is o(1).

57

The Robustness of PFR

Here we consider particles very near the ES line.

We study the case when some of these particles (at angles > 134o) are notoperating.

The probability that none of them transmits is very small.

We show:

Lemma: PFR manages to propagate the crucial data across lines parallelto ES, and of constant distance, with fixed nonzero probability.

58

Experimental Comparison of LTP and PFR

• C++

• 2D geometry data types of LEDA

• a variety of sensor fields in a 100m×100m square area:

– we drop randomly n ∈ [100, 3000] particles (i.e. 0.01 ≤ d ≤ 0.3)

– fixed radio range R = 5m

– search angle a = 90◦

• we repeated each experiment for more than 5000 times to get good av-erage results

59

Ratio of activated particles over density

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0.01 0.03 0.05 0.07 0.09 0.11 0.13 0.15 0.17 0.19 0.21 0.23 0.25 0.27 0.29

Particle Density (d)

Rat

io o

f A

ctiv

e P

arti

cles

ove

r T

ota

l Par

ticl

es (

r)

PFR LTPe LTPa LTPr

• PFR behaves very well (r ≤ 0.3) for low densities (d ≤ 0.07)

• PFR’s energy dissipation increases with density

• LTP performs best in dense networks

60

Backtracks over density

0

20

40

60

80

100

120

140

160

0.01 0.03 0.05 0.07 0.09 0.11 0.13 0.15 0.17 0.19 0.21 0.23 0.25 0.27 0.29

Particle Density (d)

Nu

mb

er o

f H

op

s to

rea

ch S

ink

PFR LTPe LTPa LTPr

• For very low densities (i.e. d ≤ 0.12), LTP backtracks a lot.

• As density increases, the number of backtracks of LTP reduces fast and almost reaches

zero.

61

Comparison of PFR, LEACH

Let K be the total number of events and k the number of events successfully reported to

the sink.

Let Is be the injection rate of events.

Let Ei be the available energy for particle i.

• We define the success rate:

IPs =k

K

• We define the average energy per particle Eavg as:

Eavg =∑n

i Ei

n

where n is the number of particles deployed.

• Let hA be the number of “active” particles. Particle i is active–alive if Ei > 0.

62

Success Rate over Network Size

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 200 400 600 800 1000 1200 1400 1600

Network Size

Su

cces

s R

ate

PFR 0.05

PFR 0.8

H-TEEN 0.05

H-TEEN 0.8

• For small networks, 0.85 ≤ IPs ≤ 1.

• For large networks, success rate drops:

– For SW-PFR IPs → 0.7.

– For H-TEEN IPs → 0.8 for Is = 0.05, and IPs → 0.3 for Is = 0.8.

63

Success Rate over Injection Rate

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

Event Rate

Su

cces

s R

ate

PFR 500x500

PFR 1500x1500

H-TEEN 500x500

H-TEEN 1500x1500

Is ∈ [0.05, 0.8]

• For large networks the success rate of H-TEEN drops significantly when Is increases.

• SW-PFR seems to be less affected by changes in Is.

64

Average Energy over Time

0.5

0.6

0.7

0.8

0.9

1

0 1000 2000 3000 4000 5000 6000 7000 8000

Simulation Rounds

Ave

rag

e E

ner

gy

per

Par

ticl

e (J

)

PFR 500x500

PFR 1000x1000

H-TEEN 500x500

H-TEEN 1000x1000

• For small network size, H-TEEN is more energy efficient.

• For large network size, the energy consumption of H-TEEN increases dramatically.

• The energy consumption of SW-PFR is almost identical in both cases.

65

Number of Alive Particles over Time

1100

1350

1600

1850

2100

2350

2600

0 3000 6000 9000 12000 15000

Simulation Rounds

Aliv

e P

arti

cles

PFR 500x500

PFR 1000x1000

H-TEEN 500x500

H-TEEN 1000x1000

• For small network size SW-PFR “drains” particles rapidly.

• For large network size H-TEEN “drains” particles rapidly.

66

Conclusions

• SW-PFR behaves quite well for all network sizes and injection rates.

• SW-PFR tends to strain the particles which lie close to the sink more.

• H-TEEN behaves very well on small network size and low injection ratesettings, while it is “expensive” on large network areas and high injectionrate.

67

Our Energy Balance Protocol (EPB)

All protocols tend to “strain” some specific nodes in the network.

• In a hop-by-hop scheme the nodes closer to the sink tend to be overused.

• In a direct transmission scheme the distant nodes tend to be overused.

“How can we achieve equal energy dissipation per node in order to prolongthe network lifetime by avoiding early network disconnection?”

68

Our Solution

• Direct transmission cost is much larger than that of one-hop transmissionand exhaust distant nodes.

• One-hop transmissions are cheap but tend to overuse nodes that lie closerto the sink.

“Each node chooses randomly whether to propagate the event one-hop closerto the sink or to transmit it directly to the sink”.

The goal is to “balance” the two types of transmissions and achieve equalaverage energy dissipation per sensor.

69

The Protocol (I)

• Network Partition:

– Partition the network into n sectors, “slices”, of width R.

70

The Protocol (II)

• Data Propagation:

Each node in sector i propagates its messages according to the follow-ing rule:

- Propagate the message to sector i − 1 with probabilty pi.

- Propagate the message directly to the sink with probability 1− pi.

The choice of pi is made such as the average per sensor energy dissipation isthe same for all sensors in the network.

71

Average per Sensor Energy Dissipation

• Si: the area size of sector i.

• Ei : the total energy dissipated sector i.

• Smart-Dust nodes are spread random uniformly in the network area.

E[Ei]

Si=

E[Ej]

Sj∀i, j ∈ {1, . . . n}

72

An Instance of the Execution

There is a message on node ”i”.

73

The Model & Assumptions

The events occur in random uniform positions in the network.

Let ǫij a random variable that measures the energy that dissipates the sectori so as to handle message j.

ǫij =

cR2 with probability pi

c(iR)2 with probability 1 − pi

Clearly,

E[ǫij] = [i2 − pi(i2 − 1)]cR2

74

Computation of pi (message handling)

• Let hi the number of messages that handles sector i.

• Let fi the number of messages that were forwarded to sector i.

• Let gi the number of messages that were generated in sector i.

Clearly:hi = fi + gi

By linearity of expectation:

E[hi] = E[fi] + E[gi]

75

Computation of pi

Lemma. The following relationship holds:

E[fi] = pi+1E[hi+1] = pi+1 · (E[fi+1] + E[gi+1])

Clearly:

pi+1 =E[fi]

E[fi+1] + E[gi+1]

The Energy Balance Property:

E

∑hik=1 ǫik

Si

= E

∑

hjk=1 ǫjk

Sj

∀i, j ∈ {1, . . . n}

76

A recurrence relation for pi

a(i+1)E[fi+1]−(d(i)+a(i))E[fi]+d(i−1)E[fi−1] = a(i)E[gi]−a(i+1)E[gi+1]

where

a(i) =i2

2i − 1d(i) =

(i + 1)2 − 1

2i + 1

Initial conditions:E[fn] = 0 E[f0] = n

77

Computation of pi

Structure of the rest of solution evaluation.

• First we transform the recurrency into a simpler recurrence with onlytwo successive terms (whose coefficient is 1).

• Having solved the latter recurrency, we solve the initial one.

• Having computed the values of E[fi] for i = 1, . . . , n we compute theexact values of probabilites pi.

78

The solution

- The solution:

E[fi] = −n−i∑

k=1

∏n−i+1j=k a(n − j)∏n−ij=k d(n − j)

·

n−k∑

j=1(a(j)E[gj] − a(j + 1)E[gj+1]) + a(1) · E[f1]

- wherei−1∏

ia(i) = 1

- Easily computed in a repetitive manner

- Thus

pi+1 =E[fi]

E[fi+1] + E[gi+1]

79

A closed solution for pi

If E[fi] ≃ E[fi−1] then

pi = 1 −3x

(i + 1)(i − 1)for 3 ≤ i ≤ n

p2 = x and can be set to 12

E[fi] ≃ E[fi−1] because:

• Si ≃ Si−1

• pi ≃ pi−1.

80

Some Conclusions

• A new fascinating field, relevant to emerging technology

• New critical issues and challenges

• Several aspects: hardware, software, algorithms, systems, applications

• Protocol design: an urgent need for complementary use of rigorous analy-sis and large scale simulation

81

Other Models

• random geometric graphs: a random plane network is constructedby picking randomly points (sensors) by a Poisson process, with densityd points per unit area, and joining two points by a line if they are atdistance less than r (transmission range)

• random sector graphs: each point (sensor) chooses an angle and aeuclidean distance (a cyclic sector for a beacon transmission).

• random intersection graphs: each vertex (sensor) randomly pickselements from a universe, and two vertices are adjacent when they pickat least one element in common.

• stochastic models: (Markov chains, martingales, dynamical systems)for energy dissipation.

• computational models: for evolving artificial living systems.

82

References for Models

- M. D. Penrose, “Random Geometric Graphs”, Oxford University Press, 2003.

- Diaz, J., Penrose, M., Petit, J., and Serna, M.: Approximation Layout Problems on

Random Geometric Graphs. J. of Algorithms 39 (2001)

- Karonski, M., Scheinerman, E.R., and Singer-Cohen, K.B.: On Random Intersection

Graphs: The Subgraph Problem. Combinatorics, Probability and Computing Journal

8 (1999).

- S. Nikoletseas, C. Raptopoulos, and P. Spirakis, “The Existence and Efficient Construc-

tion of Large Independent Sets in General Random Intersection Graphs”, in the 31st

ICALP, 2004. Also in the TCS Journal, 2006.

- P. Leone and J. Rolim, “Towards a Dynamical Model for Wireless Sensor Networks”,

ALGOSENSORS 2004.

- J. Wiederman and J. van Leuwen, “The Emergent Computational Potential of Evolving

Artificial Living Systems”, AI Communications, IOS Press, 2002.

83

More problems / other important issues

• power management schemes

• heterogeneity

• incremental deployment of sensors

• collision avoidance

• obstacle avoidance

• application development environents/overlays

84

Additional References

- I. Chatzigiannakis, A. Kinalis and S. Nikoletseas, “An Adaptive PowerConservation Scheme for Heterogeneous Wireless Sensors”, in the Pro-ceedings of the 17th ACM Symposium on Parallel Algorithms and Ar-chitectures (SPAA 2005), 2005.

- A. Boukerche, I. Chatzigiannakis and S. Nikoletseas, “A New Energy Ef-ficient and Fault-tolerant Protocol for Data Propagation in Smart DustNetworks using Varying Transmission Range”, in the Computer Com-munications Journal, Elsevier, 2005.

- I. Chatzigiannakis, G. Mylonas and S. Nikoletseas, “Modeling and Eval-uation of the Effect of Obstacles on the Performance of Wireless SensorNetworks”, accepted, to appear in the Proc. 39th Annual ACM/IEEESimulation Symposium (ANSS 06), IEEE Computer Society Press, 2006.

85

- P. Leone, J. Rolim and S. Nikoletseas, “An Adaptive Blind Algorithm for Energy

Balanced Data Propagation in Wireless Sensor Networks”, in the Proc. of the IEEE

Conference on Distributed Computing in Sensor Networks (DCOSS), Lecture Notes in

Computer Science (LNCS), Springer Verlag, Vol. 3267, pp. 35-48, 2005.

- I. Chatzigiannakis, G. Mylonas and S. Nikoletseas, “jWebDust: A Java-based Generic

Application Environment for Wireless Sensor Networks”, in the Proceedings of the IEEE

International Conference on Distributed Computing in Sensor Networks (DCOSS), Lec-

ture Notes in Computer Science (LNCS), Springer Verlag, Volume 3267, pp. 376-386,

2005.

- I. Chatzigiannakis, T. Kinalis and S. Nikoletseas, “Wireless Sensor Networks Protocols

for Efficient Collision Avoidance in Multi-path Data Propagation”, in the Proceedings

of the ACM Workshop on Performance Evaluation of Wireless Ad Hoc, Sensor, and

Ubiquitous Networks (PE-WASUN R 2004), pp. 8-16, held in conjunction with ACM

MSWiM R 2004, 2004.

86

State of the art Surveys

- Akyildiz, I.F., Su, W., Sankarasubramaniam, Y., and Cayirci, E.: Wire-less Sensor Networks: a Survey. In the Journal of Computer Networks38 (2002) 393-422.

- Estrin, D., Govindan, R., Heidemann, J., and Kumar, S.: Next CenturyChallenges: Scalable Coordination in Sensor Networks. In Proc. 5thACM/IEEE International Conference on Mobile Computing V MOBI-COM R1999.

- Kahn, J.M., Katz, R.H., and Pister, K.S.J.: Next Century Challenges:Mobile Networking for Smart Dust. In Proc. 5th ACM/IEEE Interna-tional Conference on Mobile Computing (September 1999) 271-278.

87

Thank you!

88