Throughput-Optimal Broadcast in
Dynamic Wireless Networks
Abhishek Sinha
Joint work with Prof. Leandros Tassiulas and Prof. Eytan Modiano
MIT, Yale University
Talked at: MobiHoc, 2016
Outline
•
Main problem
• Terminology
• Prior Work
• Optimal Algorithm
• Characterization of Broadcast-Capacity
• Numerical Simulation
• Conclusion
Main problem
Static wireless network model
• The network-topology is represented by a directed graph G(V; E).
• Time is slotted.
• Packet transmissions are point-to-point and subject to wireless interference constraints.
Main problem
Static wireless network model
• The network-topology is represented by a directed graph G(V; E).
• Time is slotted.
• Packet transmissions are point-to-point and subject to wireless interference constraints.
Main problem
Static wireless network model
• The network-topology is represented by a directed graph G(V; E).
• Time is slotted.
• Packet transmissions are point-to-point and subject to wireless interference constraints.
Main problem
Static wireless network model
• The network-topology is represented by a directed graph G(V; E).
• Time is slotted.
• Packet transmissions are point-to-point and subject to wireless interference constraints.
Main problem
Time Variation model
• channel states vary with time because of random fading, shadowing and node-mobility.
• a link could be either ON or OFF.
• All links are i.i.d.
e.g. network configurations:
Main problem
Time Variation model
• channel states vary with time because of random fading, shadowing and node-mobility.
• a link could be either ON or OFF.
• All links are i.i.d
e.g. network configurations:
Main problem
Time Variation model
• channel states vary with time because of random fading, shadowing and node-mobility.
• a link could be either ON or OFF.
• All links are i.i.d.
e.g. network configurations:
Main problem
Time Variation model
• channel states vary with time because of random fading, shadowing and node-mobility.
• a link could be either ON or OFF.
• All links are i.i.d
e.g. network configurations:
Main problem
Time Variation model
• channel states vary with time because of random fading, shadowing and node-mobility.
• a link could be either ON or OFF.
• All links are i.i.d
e.g. network configurations:
Main problem
Time Variation model
• channel states vary with time because of random fading, shadowing and node-mobility.
• a link could be either ON or OFF.
• All links are i.i.d
e.g. network configurations:
Main problem
How to efficiently disseminate packets, arriving at source node r, to
all other nodes in a multi-hop network?
Applications
live multimedia streaming
software updates
etc…
military communications
disaster management
Terminology
• Policy
A sequence of actions executed at every time slot.
Terminology
• Policy
A sequence of actions executed at every time slot.
•
Broadcast policy
A policy is called broadcast policy of rate λ if all nodes receive
distinct packets at rate λ.
Terminology
• Policy
A sequence of actions executed at every time slot.
•
Broadcast policy
A policy is called broadcast policy of rate λ if all nodes receive
distinct packets at rate λ.
•
Broadcast Capacity λ*
The supremum of all arrival rates λ
Terminology
• Policy
A sequence of actions executed at every time slot.
•
Broadcast policy
A policy is called broadcast policy of rate λ if all nodes receive
distinct packets at rate λ.
•
Broadcast Capacity λ*
The supremum of all arrival rates λ
Observation
An Upper bound on broadcast capacity λ*
(Min cut Max flow)
λ* ≤ Min Max-Flow(r t)
t ∈V\{r}
Prior Work
Edmond's Tree-Packing Theorem [1965]
Prior Work
Edmond's Tree-Packing Theorem [1965]
There exist λ* edge-disjoint spanning trees to achieve the
broadcast capacity.
Prior Work
Pre-computes the set of all spanning trees offline in wireline networks.
Prior Work
Pre-computes the set of all spanning trees offline in wireline networks.
Prior Work
Pre-computes the set of all spanning trees offline in wireline networks.
• Impractical for large and time-varying networks
• Wireless case is studied with a fixed activation schedule only
Optimal Algorithm
The Policy
A feasible broadcast policy executes following two actions at every slot t :
• Link Activation : Activate a subset of links (e.g., a matching) subject to the
underlying interference constraints.
• Packet Scheduling π(S) : Transmit packets over the set of activated links.
Optimal Algorithm
state-space П
The number of packets present at each subset of nodes
Optimal Algorithm
state-space П
The number of packets present at each subset of nodes
Optimal Algorithm
state-space П
The number of packets present at each subset of nodes
An arbitrary packet Scheduling π(S) is hard to
describe
State space grows exponentially.
Optimal Algorithm
Question
What can we do to simplify the Packet Scheduling π(S)?
Optimal Algorithm
Question
What can we do to simplify the Packet Scheduling π(S)?
Answer
To simplify π(S), we consider the sub-space Пin-orderс П in which all packets
are delivered to every node in-order.
in other words: if a node receive packet with index j it will accept it only if it has
all the lower index packets [1, j-1].
Optimal Algorithm
П* C Пin-order
For all П* C Пin-order, a packet p is
eligible for transmission to node d
All in-neighbors of node n contain the
packet p.
Optimal Algorithm
П* C Пin-order - Policy hierarchies
П: all policies that perform arbitrary packet-forwarding.
Пin-order : policies that enforce in-order packet-forwarding.
П*: policies that allow reception only if all in-neighbors have
received the specific packet.
Optimal Algorithm
П* C Пin-order - Policy hierarchies
П: all policies that perform arbitrary packet-forwarding.
Пin-order : policies that enforce in-order packet-forwarding.
П*: policies that allow reception only if all in-neighbors have
received the specific packet.
Punchline : There exists a policy in П* , which is optimal for DAG!
Optimal Algorithm
П* C Пin-order
Redefinition of network state:
S(t) ={R1(t), R2(t),…,Rn(t)}
Where Ri(t) is the total number of packets received by node i up
to time t.
Optimal Algorithm
П*
Definition of state Variables:
For each node j∈ V\{r} define:
•
𝑿𝒋 𝒕 = Min (𝑹𝒊 𝒕 − 𝑹𝒋 (𝒕)) - amount of message that j can get at time t
•
𝒊𝒋 ∗ 𝒕 = 𝒂𝒓𝒈 Min (𝑹𝒊 𝒕 − 𝑹𝒋 (𝒕)) - the in-neighbor with the potential to deliver message
𝑖:(𝑖,𝑗)∈𝐸
𝑖:(𝑖,𝑗)∈𝐸
Optimal Algorithm
П*
Definition of state Variables:
For each node j∈ V\{r} define:
•
𝑿𝒋 𝒕 = Min (𝑹𝒊 𝒕 − 𝑹𝒋 (𝒕)) - amount of message that j can get at time t
•
𝒊𝒋 ∗ 𝒕 = 𝒂𝒓𝒈 Min (𝑹𝒊 𝒕 − 𝑹𝒋 (𝒕)) - the in-neighbor with the potential to deliver message
𝑖:(𝑖,𝑗)∈𝐸
𝑖:(𝑖,𝑗)∈𝐸
Lemma:
Under П*, any algorithm stabilizing X(t) is a broadcast policy in a DAG.
Optimal Algorithm
П*
Reminder:
Broadcast policy - A policy is called
broadcast policy of rate λ if all nodes
receive distinct packets at rate λ.
Definition of state Variables:
For each node j∈ V\{r} define:
•
𝑿𝒋 𝒕 = Min (𝑹𝒊 𝒕 − 𝑹𝒋 (𝒕)) - amount of message that j can get at time t
•
𝒊𝒋 ∗ 𝒕 = 𝒂𝒓𝒈 Min (𝑹𝒊 𝒕 − 𝑹𝒋 (𝒕)) - the in-neighbor with the potential to deliver message
𝑖:(𝑖,𝑗)∈𝐸
𝑖:(𝑖,𝑗)∈𝐸
Lemma:
Under П*, any algorithm stabilizing X(t) is a broadcast policy in a DAG.
Optimal Algorithm
П*
Reminder:
Broadcast policy - A policy is called
broadcast policy of rate λ if all nodes
receive distinct packets at rate λ.
Definition of state Variables:
For each node j∈ V\{r} define:
•
𝑿𝒋 𝒕 = Min (𝑹𝒊 𝒕 − 𝑹𝒋 (𝒕)) - amount of message that j can get at time t
•
𝒊𝒋 ∗ 𝒕 = 𝒂𝒓𝒈 Min (𝑹𝒊 𝒕 − 𝑹𝒋 (𝒕)) - the in-neighbor with the potential to deliver message
𝑖:(𝑖,𝑗)∈𝐸
𝑖:(𝑖,𝑗)∈𝐸
Lemma:
Under П*, any algorithm stabilizing X(t) is a broadcast policy in a DAG.
Intuition : The state-vector X(t) mathematically corresponds to “queue-sizes" in the
traditional queuing network.
Reminder:
Optimal Algorithm
•
𝑿𝒋 𝒕 = Min (𝑹𝒊 𝒕 − 𝑹𝒋 (𝒕))
•
𝒊𝒋 ∗ 𝒕 = 𝒂𝒓𝒈 Min (𝑹𝒊 𝒕 − 𝑹𝒋 (𝒕))
𝑖:(𝑖,𝑗)∈𝐸
Algorithm:
𝟏. 𝑻𝒐 𝒆𝒂𝒄𝒉 𝒆𝒅𝒈𝒆 𝒊, 𝒋 ∈ 𝑬, 𝒂𝒔𝒔𝒊𝒈𝒏 𝒂 𝒘𝒆𝒊𝒈𝒉𝒕 𝑾𝒊𝒋 𝒕 , 𝒘𝒉𝒆𝒓𝒆
𝑾𝒊𝒋 𝒕 = 𝑿𝒋 𝒕 −
𝒌:𝒋=𝒊∗𝒕 (𝒌) 𝑿(𝒌)
(under the restrictions)
𝟐. 𝑪𝒉𝒐𝒐𝒔𝒆 𝒂 𝒎𝒂𝒙 𝒘𝒆𝒊𝒈𝒉𝒕 𝒂𝒄𝒕𝒊𝒗𝒂𝒕𝒊𝒐𝒏 𝒘𝒊𝒕𝒉 𝒘𝒆𝒊𝒈𝒉𝒕𝒔 𝑾(𝒕)
𝑖:(𝑖,𝑗)∈𝐸
Optimal Algorithm
The П* optimality is proven for DAG using queueing theory:
𝑹𝝅∗
𝒕
lim 𝒊 𝒕
𝒕→∞
= λ, ∀𝒊 ∈𝑽
Observation from algorithm:
broadcast capacity of DAGs is limited by the minimum in-degree of the time∗
average graph – will note as λ𝑫𝑨𝑮
.
Characterization of Broadcast-Capacity
Since the only thing that determine the broadcast capacity is the minimum in degree of a
node, the broadcast capacity of every wireless DAG - Even when there is a corollary
between them can be computed in polynomial time.
Numerical Simulation
Wireless network : 3X3 grid
Numerical Simulation
Average broadcast-delay as function
of the packets arrival rates
Wireless network : 3X3 grid
Each link is ON with probability p at
every slot (i.i.d).
Numerical Simulation
static case
Conclusions
• Classical algorithms require online computation of spanning trees, which is
practically infeasible for large dynamic networks.
• The Authors derived the first online, provably optimal broadcast algorithm for
wireless DAGs with dynamic topologies.
• Broadcast Capacities of Wireless DAGs have been characterized mathematically
and algorithmically.