1. 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

2. Outline Main problem Terminology Prior Work Optimal Algorithm
Characterization of Broadcast-Capacity
Numerical Simulation
Conclusion

3. 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.

4. 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.

5. 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.

6. 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.

7. 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:

8. 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:

9. 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:

10. 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:

11. 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:

12. 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:

13. 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 military communications software updates disaster management
etc…

14. Policy A sequence of actions executed at every time slot.
Terminology
Policy
A sequence of actions executed at every time slot.

15. 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.

16. 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

17. 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* λ* ≤ Min Max-Flow(r t)
(Min cut Max flow)
t ∈V\{r}

18. Edmond's Tree-Packing Theorem [1965]
Prior Work
Edmond's Tree-Packing Theorem [1965]

19. 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.

20. Prior Work
Pre-computes the set of all spanning trees offline in wireline networks.

21. Prior Work
Pre-computes the set of all spanning trees offline in wireline networks.

22. 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

23. 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.

24. 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

25. 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

26. 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.

27. 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)?

28. 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].

29. 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.

30. 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.

31. 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!

32. 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.

33. Optimal Algorithm П* Definition of state Variables:
For each node j∈ V\{r} define:
𝑿 𝒋 𝒕 = Min 𝑖:(𝑖,𝑗)∈𝐸 ( 𝑹 𝒊 𝒕 − 𝑹 𝒋 (𝒕)) - amount of message that j can get at time t
𝒊 𝒋 ∗ 𝒕 =𝒂𝒓𝒈 Min 𝑖:(𝑖,𝑗)∈𝐸 ( 𝑹 𝒊 𝒕 − 𝑹 𝒋 (𝒕)) - the in-neighbor with the potential to deliver message

34. Optimal Algorithm П* Definition of state Variables:
For each node j∈ V\{r} define:
𝑿 𝒋 𝒕 = Min 𝑖:(𝑖,𝑗)∈𝐸 ( 𝑹 𝒊 𝒕 − 𝑹 𝒋 (𝒕)) - amount of message that j can get at time t
𝒊 𝒋 ∗ 𝒕 =𝒂𝒓𝒈 Min 𝑖:(𝑖,𝑗)∈𝐸 ( 𝑹 𝒊 𝒕 − 𝑹 𝒋 (𝒕)) - the in-neighbor with the potential to deliver message
Lemma:
Under П*, any algorithm stabilizing X(t) is a broadcast policy in a DAG.

35. Optimal Algorithm П* Reminder:
Broadcast policy - A policy is called broadcast policy of rate λ if all nodes receive distinct packets at rate λ.
Optimal Algorithm П*
Definition of state Variables:
For each node j∈ V\{r} define:
𝑿 𝒋 𝒕 = Min 𝑖:(𝑖,𝑗)∈𝐸 ( 𝑹 𝒊 𝒕 − 𝑹 𝒋 (𝒕)) - amount of message that j can get at time t
𝒊 𝒋 ∗ 𝒕 =𝒂𝒓𝒈 Min 𝑖:(𝑖,𝑗)∈𝐸 ( 𝑹 𝒊 𝒕 − 𝑹 𝒋 (𝒕)) - the in-neighbor with the potential to deliver message
Lemma:
Under П*, any algorithm stabilizing X(t) is a broadcast policy in a DAG.

36. Optimal Algorithm П* Reminder:
Broadcast policy - A policy is called broadcast policy of rate λ if all nodes receive distinct packets at rate λ.
Optimal Algorithm П*
Definition of state Variables:
For each node j∈ V\{r} define:
𝑿 𝒋 𝒕 = Min 𝑖:(𝑖,𝑗)∈𝐸 ( 𝑹 𝒊 𝒕 − 𝑹 𝒋 (𝒕)) - amount of message that j can get at time t
𝒊 𝒋 ∗ 𝒕 =𝒂𝒓𝒈 Min 𝑖:(𝑖,𝑗)∈𝐸 ( 𝑹 𝒊 𝒕 − 𝑹 𝒋 (𝒕)) - the in-neighbor with the potential to deliver message
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.

37. 𝑾 𝒊𝒋 𝒕 = 𝑿 𝒋 𝒕 − 𝒌:𝒋= 𝒊∗ 𝒕 (𝒌) 𝑿(𝒌) (under the restrictions)
Reminder:
𝑿 𝒋 𝒕 = Min 𝑖:(𝑖,𝑗)∈𝐸 ( 𝑹 𝒊 𝒕 − 𝑹 𝒋 (𝒕))
𝒊 𝒋 ∗ 𝒕 =𝒂𝒓𝒈 Min 𝑖:(𝑖,𝑗)∈𝐸 ( 𝑹 𝒊 𝒕 − 𝑹 𝒋 (𝒕))
Optimal Algorithm
Algorithm:
𝟏. 𝑻𝒐 𝒆𝒂𝒄𝒉 𝒆𝒅𝒈𝒆 𝒊, 𝒋 ∈𝑬, 𝒂𝒔𝒔𝒊𝒈𝒏 𝒂 𝒘𝒆𝒊𝒈𝒉𝒕 𝑾 𝒊𝒋 𝒕 , 𝒘𝒉𝒆𝒓𝒆
𝑾 𝒊𝒋 𝒕 = 𝑿 𝒋 𝒕 − 𝒌:𝒋= 𝒊∗ 𝒕 (𝒌) 𝑿(𝒌) (under the restrictions)
𝟐. 𝑪𝒉𝒐𝒐𝒔𝒆 𝒂 𝒎𝒂𝒙 𝒘𝒆𝒊𝒈𝒉𝒕 𝒂𝒄𝒕𝒊𝒗𝒂𝒕𝒊𝒐𝒏 𝒘𝒊𝒕𝒉 𝒘𝒆𝒊𝒈𝒉𝒕𝒔 𝑾(𝒕)

38. Optimal Algorithm
The П* optimality is proven for DAG using queueing theory:
lim 𝒕→∞ 𝑹 𝒊 𝝅∗ 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 λ 𝑫𝑨𝑮 ∗ .

39. 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.

40. Numerical Simulation
Wireless network : 3X3 grid

41. 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).

42. Numerical Simulation static case

43. 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.