Modeling Uncertainty in
Wireless Communication
Fabian Kuhn
Algorithms and Complexity Group
University of Freiburg
Germany
Fabian Kuhn
WRAWN 2013
Modeling Wireless Communication
Fabian Kuhn
WRAWN 2013
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Graph-Based Models
Network modeled as a graph 𝑮 = (𝑽, 𝑬)
• known as “radio network model” or “protocol model”
Variants:
• additional edges modeling only interference
• multiple communication channels
• with/without collision detection
• general graphs, geometric graphs, bounded growth, …
Fabian Kuhn
WRAWN 2013
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SINR Models
Signal-to-Noise-and-Interference Ratio:
• Many weak signals can together also prevent successful
communication
• Basic assumptions:
– 𝑃𝑥 : transmission power of node 𝑥
– 𝑑𝑥𝑦 : distance between nodes 𝑥 and 𝑦 (e.g., in ℝ2 )
• Node 𝑦 receives message from node 𝑥 if and only if
𝛼
𝑃𝑥 𝑑𝑥𝑦
𝛼 ≥𝛽
𝑁 + 𝑧≠𝑥,𝑦 𝑃𝑧 𝑑𝑧𝑦
• Also known as the physical model
• Variants: e.g., with/without carrier sensing
Fabian Kuhn
WRAWN 2013
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Wireless Models
Typical Current Wireless Models:
• E.g., graph-based, SINR, MIMO, …
Whether 𝒚 receives a message from 𝒙 is determined by a
deterministic rule
• Graph-based: no other neighbor transmits
• Physical model: SINR is above threshold 𝛽
Many existing algorithms make heavy use of this!
• Graph-based: message recv. ⟺ exactly one sender
• Use exact knowledge of parameters (e.g., position)
Fabian Kuhn
WRAWN 2013
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Simulating Collision Detection
Assumption: Single-hop network, no collision detection
Goal: Simulate collision detection
• Subset 𝑆 of nodes transmit
• Need to be able to distinguish 𝑆 = 0, 𝑆 = 1, 𝑆 > 1
Solution: Use 2 rounds to simulate 1 round
• Assume, we have a leader node ℓ
𝒗 ∈ 𝑺 transmit
𝒗 ∈ 𝑺 ∪ {ℓ} transmit
𝑆 =0
𝑆 = 1, 𝑆 ≠ {ℓ}
𝑆 = 1, 𝑆 = {ℓ}
𝑆 >1
Fabian Kuhn
WRAWN 2013
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Wireless Communication
Whether 𝒚 receives a message of 𝒙 depends on
• Distance of 𝑥 and 𝑦, transmission power of 𝑥
• Power, position of concurrent transmissions
But also on
• Phase difference of wireless transmissions
• Structure/geometry of the environment
– buildings, walls, doors, furniture, …
• Transmissions of unrelated wireless devices
– Devices of other networks, cell phones, …
• Any other electrical devices (e.g., microwave oven)
• Environmental factors
– Temperature, humidity, …
Fabian Kuhn
WRAWN 2013
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Dynamic Behavior
Environmental conditions change over time
• External interference changes over time
• Geometry, environmental factors can change
Network itself might be dynamic
• Devices can be mobile
(mobile ad hoc networks)
© James McLurkin, Rice U.
Fabian Kuhn
WRAWN 2013
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Uncertainty
Communication inherently unreliable / unpredictable
• Impossible to explicitly model every relevant factor
• Hard to exactly measure behavior of comm. channels
• Hard to exactly coordinate transmissions of different nodes
• Communication channels change over time
– Possibly frequently and significantly
• Network might even be dynamic
Fabian Kuhn
WRAWN 2013
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Modeling Uncertainty
Two basic approaches
1. Add random “noise” to the model
–
Depending on distance, trans. powers, etc., a message is
received with a certain probability
2. Add non-deterministic behavior to the model
–
•
Fabian Kuhn
In some cases, an adversary decides whether a message is
received
Leads to particularly robust algorithms
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Example: Dual Graph Model
Non-deterministic variant of graph-based model
[Kuhn,Lynch,Newport,Oshman,Richa; PODC ‘10]
Graphs 𝐺 (reliable edges), 𝐺 ′ ⊇ 𝐺 (𝐺 ′ ∖ 𝐺: unreliable edges)
𝑮
𝑮′ ∖ 𝑮
Fabian Kuhn
WRAWN 2013
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Single-Message Broadcast
Goal: Broadcast a single message from 𝑠 to everyone
Lower Bound: Ω(𝑛) rounds are needed even in
networks with a broadcast schedule of length 2.
[Kuhn,Lynch,Newport,Oshman,Richa; PODC ‘10]
𝒔
Fabian Kuhn
𝒖
𝑲𝒏−𝟏
WRAWN 2013
𝒗
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Single-Message Broadcast
Upper Bound: For every connected 𝐺 and every 𝐺′,
broadcast can be done in 𝑂(𝑛 log 2 𝑛) rounds.
[Kuhn,Lynch,Newport,Oshman,Richa; PODC ‘10]
Algorithm:
• Upon receiving message, broadcast it with probability
1
1 ,1 ,…,1 ,1 ,…,1 ,1 ,…,1 ,…
,
…
,
4 5
2
2 3
3 4
5
Θ(log 𝑛)
Θ(log 𝑛)
Θ(log 𝑛)
Θ(log 𝑛)
• Guarantees that each node transmits alone once
Fabian Kuhn
WRAWN 2013
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Weaker Adversaries
Cost of broadcast depends on adversary
(Strongly) adaptive adversary
• Chooses edges dependent on which nodes transmit
• Ω(𝑛) lower bound applies
Oblivious adversary
• Has to commit to all edge sets at the beginning
• Broadcast possible in time 𝑂 𝐷 + log 𝑛 log 𝑛
[Ghaffari,Lynch,Newport; PODC ‘13]
Online adaptive adversary
• Does not know randomness of current round
• Ω 𝑛 log 𝑛 lower bound
[Ghaffari,Lynch,Newport; PODC ‘13]
Fabian Kuhn
WRAWN 2013
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Modeling Uncertainty
Same ideas can be applied to any wireless model
• Take your favorite wireless model and add a nondeterministic component
Example SINR model
• Define two thresholds 𝛽1 < 𝛽2 :
𝐒𝐈𝐍𝐑 ≥ 𝜷𝟐 :
message received
𝐒𝐈𝐍𝐑 < 𝜷𝟏 :
message not received
𝐒𝐈𝐍𝐑 ∈ [𝜷𝟏 , 𝜷𝟐 ): adversary decides
Dual graph model
• Generalizes every model for 𝐺 = (𝑉, ∅), 𝐺′ = 𝐾𝑛
– If nodes can receive at most one message per round
• Models an arbitrary dynamic graph
Fabian Kuhn
WRAWN 2013
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Dynamic Networks
First step towards understanding dynamic networks
• Non-deterministic behavior  partially dynamic network
• E.g., if 𝐺 = 𝑉, ∅ , 𝐺 ′ = 𝐾𝑛 ,
– dual graph model gives an arbitrary dynamic network
Dynamic networks to study wireless communication?
• Set of successful transmissions define dynamic network
• Design transmissions to satisfy
– reliable links are scheduled regularly
– connectivity over time
– …
• Use dynamic network algorithm on top
Fabian Kuhn
WRAWN 2013
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Some General Questions
How much non-determinism can we tolerate to still use
techniques of classic models?
• Or at least to still do things (almost) as efficiently?
What are the inherent costs of dealing with nondeterministic behavior?
• Which things are not possible or much more expensive?
– Depending on the amount of non-determinism allowed
Go beyond broadcast / inf. dissemination?
• How should we define local structures in a dynamic graph?
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WRAWN 2013
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Summary
• Wireless communication behaves unreliably
• Needs to be reflected in the models
• Leads to particularly robust algorithms
• And a better general understanding of the problems
– E.g., what’s the limit of each technique?
Fabian Kuhn
WRAWN 2013
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Thanks!
Fabian Kuhn
WRAWN 2013
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