Post on 28-Mar-2015
Empirical tests Numerical evaluations Theoretical proofs
for stationary settings
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Relocation Analysis of
Stabilizing MAC
Algorithmsfor Large-Scale Mobile Ad Hoc
NetworksPierre Leone, Geneva (Switzerland)
Marina Papatriantafilou, Chalmers (Sweden)
Elad Michael Schiller, Chalmers (Sweden)
Challenges
• Collisions • When neighboring nodes simultaneously broadcast
Collision
Collision
Collision
CollisionCollision
Collision
Challenges
• Collisions
• Mobile nodes
• Random moves
Challenges
• Collisions
• Mobile nodes
• Random moves• Possibly adversarial
Challenges
• Collisions
• Mobile nodes
• Transient faults
Modeling the location of mobile nodes
• Arbitrary violations of the assumptions that model the locations
Short-lived malfunctions• hardware/software
Opportunities
• Synchronization
• Clock synchronization algorithms
and/or GPS
Opportunities
• Synchronization
• Wireless broadcast
A powerful primitive ensuring that:
• Nodes reach nearby nodes and receive the same messages
Our approach
• Stabilization • Assume that after the last transient fault the system state is random
• Steady state behaviors do not depend on that random state
• Natural self-stabilizing extensions
• arbitrary starting state
• guaranteed system recovery
• using periodic restarts
• Our negative results hold for self-stabilizing systems
Outline
3
Bounded Relocation
Rate
2
Negative Results
1
Relocation
Analysis
Relocation Analysis
• Random moves • Each mobile node randomly moves in the Euclidian plane
• Two mobile nodes can directly communicate if their distance is less than a threshold
In the short run
Gt , Gt+1 ∊ G
Many mobile nodes have similar neighborhoods in Gt and Gt+1
e.g., large communication radius
In the long run this similarity disappears
There are independent random relocations of the mobile nodes
e.g., Gt,Gt+x are independent when x → ∞
Relocation Analysis
• Random moves
• Evolving graphs
• Ferreira’04, Avin et al.’08
G=(G1, G2, …)
• In time t , graph Gt ∊ G models the communications and interferences
Relocation Analysis
• Between Gt, Gt+1 ∊ G, α|V| nodes
relocate to new neighborhoods
– α ∊in [0, 1] is the relocation rate
• Relocating nodes and their new neighborhoods are chosen randomly
• Random moves
• Evolving graphs
• Relocation rate
Relocation Analysis
• Between Gt, Gt+1 ∊ G, α|V| nodes
relocate to new neighborhoods
– α ∊in [0, 1] is the relocation rate
• Relocating nodes and their new neighborhoods are chosen randomly
• Random moves
• Evolving graphs
• Relocation rate
Our assumptions are different from:
• Random walks
do not consider short-term
(independent) random relocations
• Population protocols
do not consider long-term
neighborhood similarity
Relocation Analysis
• Between Gt, Gt+1 ∊ G, α|V| nodes
relocate to new neighborhoods
– α ∊in [0, 1] is the relocation rate
• Relocating nodes and their new neighborhoods are chosen randomly
• Random moves
• Evolving graphs
• Relocation rate
Relocation Analysis
• Between Gt, Gt+1 ∊ G, α|V| nodes
relocate to new neighborhoods
– α ∊in [0, 1] is the relocation rate
• Relocating nodes and their new neighborhoods are chosen randomly
• Random moves
• Evolving graphs
• Relocation rate
High rate=more collisions
Relocation Analysis
• Between Gt, Gt+1 ∊ G, α|V| nodes
relocate to new neighborhoods
– α ∊in [0, 1] is the relocation rate
• Relocating nodes and their new neighborhoods are chosen randomly
Useful in analyzing MAC algorithms
• Random relocation causes unexpected
interferences
• Expressiveness
• A single parameter defines the
rate of unexpected interferences
• Simpler proofs than Kinetic models
• Random moves
• Evolving graphs
• Relocation rate
Outline
3
Bounded Relocation
Rate
1
Relocation Analysis
2
Negative Results
Impossibility Result
Collision
Claim 1: There is no efficient and
deterministic MAC algorithm
Impossibility Result
Claim 1: For arbitrary relocation rate, there is no efficient and deterministic MAC algorithm
CollisionCollision
Focus on randomized MAC algorithms
Lower Bound
Claim 2: For arbitrary relocation rate, oblivious strategies are the best that you can hope for …
Oblivious strategies ignore
the broadcast history
•Consider random relocation of all
nodes after every algorithm step
•Learning the history is of no use
Focus on bounded relocation rate
Outline
1
Relocation Analysis
2
Negative Results
3
Bounded Relocatio
n Rate
Throughput Related Trade-off
• Oblivious • Ignores the history of broadcasts
Throughput Related Trade-off
• Oblivious
• Non-oblivious
P =
P =
P =
P =
•E.g., based on vertex-coloring– Luby '93
Throughput Related Trade-off
P =
P =
P =
P =
Can I use ?
Can I use ?
Can I use ?
Can I use ?
Great! I will keep
Great! I will keep
I will pick another one
CollisionI will pick
another one
• Oblivious
• Non-oblivious
•E.g., based on vertex-coloring– Luby '93– Color uniqueness:
•A node has a color different than its neighbors, i.e., no collisions
Throughput Related Trade-off
•E.g., based on vertex-coloring– Luby '93– Color uniqueness:
•A node has a color different than its neighbors, i.e., no collisions
P =
Next round
• Oblivious
• Non-oblivious
Throughput Related Trade-off
• Oblivious
• Non-oblivious
• Trade-off
Critical-threshold relocation
rate
• Above which oblivious is better
• Below which non-oblivious is better
Simplifying assumptions
1. No dependencies among neighbors
2. All relocations occur once in every round
3. Collision detection is easy
Broadcasts can inform about color choices
Later today: Remove these assumptions
How many nodes no longer have
unique colors after a round?
Collision
Collision Collision
Uniform distribution of colors
• Assume uniformity in the starting
configuration
• Show uniformity in every
configuration that follows
At most ~α|V| nodes are expected to no longer have unique colors after a round
From Stationary to Non-stationarySimplifying assumptions
1. No dependencies among neighbors
2. All relocations occur once in every round
3. Collision detection is easy
Broadcasts can inform about color choices
Later today: Remove these assumptions
How many nodes start having
unique colors after a round?
From Stationary to Non-stationary
Using the vertex-coloring algorithm
(1- β) ≅1/e• Let y be the number of nodes with
conflicting colors
• Let (1- β)y be the expected
number of nodes whose colors
become unique within a
broadcasting round
Conflicting Unique
From Stationary to Non-stationary
• When the recovery is slower than the relocation (1 - β) < α • Convergence
Recovery
Relocation
Conflicting Unique
From Stationary to Non-stationary
• When the recovery is slower than the relocation (1 - β) < α
• When the relocation is slower than the recovery (1 - β) > α
• Convergence
Recovery
Relocation
Throughpu
t
80%
60%
40%
20%
0.2 0.4 0.6 0.8
Oblivious strategy
non-oblivious strategy
0.2 0.4 0.6 0.8
24.0c
Too good to be true!We discover a critical-threshold relocation rate
Guaranteed throughput of non-oblivious strategies
Eventual Throughput Simplifying assumptions
1. No dependencies among neighbors
2. All relocations occur once in every round
3. Collision detection is easy
Broadcasts can inform about color choices
Later today: Remove these assumptions
We remove the simplifying assumptions and bound the recovery ratio
(1- β) ∊ [σ(1- α) ,1/e], where σ=(5+3/e)/32
Relocation Rate α
• CSMA/CA
Existing Approaches
• Such as Herman Tixeuil ‘04• Divided ratio time
– Overhead– TDMA time slots
• CSMA/CA in overhead part for
• Local leader election
• Vertex coloring
• When nodes relocate in every broadcasting round
• No guarantees for leader election
Overh
ea
dOverh
ea
d
RTS
Existing Approaches
• CSMA/CA
• RTS/CTS dialog
• Request to Send / Clear to Send• Facilitates short exposure time
– period during which a transmitted packet might be intercepted
– shorter than a time slot
Defer
CTS
Existing Approaches
• CSMA/CA
• RTS/CTS dialog
• Request to Send / Clear to Send• Facilitates short exposure time
– period during which a transmitted packet might be intercepted
– shorter than a time slot
RTS
DeferDefer
ACK
Data
Existing Approaches
• CSMA/CA
• RTS/CTS dialog
• Request to Send / Clear to Send• Facilitates short exposure time
– period during which a transmitted packet might be intercepted
– shorter than a time slot
DeferDefer
Our Approach
• Divided time slots
• Competition part – MaxRnd rounds
• DATA part
Competition rounds
DATA packetDATA packet
Competition rounds
DATA packet
Competition rounds
DATA packet DATA packet
Competition rounds
slot 1 slot 2 slot 3 slot 4
round 1
round 2
round 3
round 4
Max competition rounds, e.g., MaxRnd = 4
Our Approach
• Divided time slots
• Round based competition
Timeslot
Data partCompetition partRecovery is facilitated because of:
1. Simple winner and losers
2. “Unlucky winners” and “lucky losers”
Neighbors may choose the same slot
On the k competition round,
• competitors send RTS with
probability 2 (k-MaxRnd )
Our Approach
• Divided time slots
• Round based competition
Neighbors may choose the same slot
On the k competition round,
• competitors send RTS with
probability 2 (k-MaxRnd )
Timeslot
Data partCompetition part
RTC
CTS
DATA
Simple winner and losers
The simple winner chooses the slot as its “permanent” one
The simple losers are aware of the winner’s broadcast and continue to look for other broadcasting slots
Neighbors may choose the same slot
On the k competition round,
• competitors send RTS with
probability 2 (k-MaxRnd )
Our Approach
• Divided time slots
• Round based competition
Timeslot
Data partCompetition part
CTS
“Unlucky winners” and “lucky losers”
RTC
RTCDATA
DATA
Collision
“Unlucky winners” are not aware of their coalitions and continue to compete for this slot on the next round
“Lucky losers” are aware of the winners’ collisions and stop competing for this slot on the next round
Neighbors may choose the same slot
On the k competition round,
• competitors send RTS with
probability 2 (k-MaxRnd )
Our Approach
• Divided time slots
• Round based competition
Timeslot
Data partCompetition part
Simple winner and losers +
“Unlucky winners” + “Lucky losers” =
Recovery rate of (5+3/e)/8
Simplifying assumptions
1. No dependencies among neighbors
2. All relocations occur once in every round
3. Collision detection is easy
Broadcasts can inform about color choices
Later today: Remove these assumptions
Recovery rate of (5+3/e)/8
Recovery rate of (1- α) (5+3/e)/8
Recovery rate of (1- α) (5+3/e)/32
Confused?!
μ-M
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AI-LM
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BitM
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PS A
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C HM
AC LM
AC LPL
MF
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AC M
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Let us help you!
Consider your favorite decentralized vertex-coloring algorithm
The good news: the algorithm possibly can migrate from stationary settings to non-stationary ones
Conclusions
• Novel throughput-related trade-off– between oblivious and non-oblivious strategies
– depends on the relocation rate of mobile nodes
• Circumventing the difficulties of– collision detection
– modeling the locations of mobile nodes
• A study of a fault-tolerant and “stateful” algorithm – Extendable to consider self-stabilization
Algosensors’09: 5th International Workshop onAlgorithmic Aspects of Wireless Sensor Networks
July 10-11, 2009
Thank you for your attention
Contact info. elad@chalmers.se
TR-2008:23, Department of Computer Science and Engineering, Chalmers University of Technology