What is a Flow

1
What is a Flow?

• Given a graph G(V, E, c), a source s and a
  target t,
• A s-t flow is a nonnegative function f on edge
  such that
• (conservation property)
• and the value of the flow is

2
What is the Max of Flows?

• Subgraph G(V, E, c)
• V, E are subsets of V and E respectively
• Suppose a s-t flow is ft , the max of flows is a
  subgraph

3
Sum of Max of Flows

• Superimposing communication is a suboptimal
• Need cross session network coding to achieve the
  optimality
• However, superimposing communication is still a
  viable way to the multisession multicast
  communication
• Moreover, superimposing communication problem can
  be easily transformed into a linear programming

4
Sum of Max of Flows(2)

• Capacity graph G(V, E, c)
• Multicast sessions
• By sharing bit-rate resources to sessions, we can
  form following equalities and inequalites

5
Cross-layer Optimization

• Capacity graph is spanned by a set of ECGs
• Different ECGs can support different capacity
  graph
• Practically, we can not find all the ECGs due to
  the large number of nodes in the network
• We can only choose a set of ECGs to span a
  reasonable good capacity graph

6
Cross-layer Optimization(2)

• Objective
• Given a set of traffic demands
  we pick up a set of ECGs and
  corresponding timesharing factor , therefore,
  we get the capacity graph such
  that it supplies U and minimizes a desired cost
  function

7
Cost function-Bandwidth

• Given a set of ECGs B
• V physical nodes virtual nodes
• E physical links links between physical
  nodes and virtual nodes
• timesharing factors
• Fixed B, max rate scaling factor alpha w.r.t. the
  variables timesharing factor and flows

8
Cost function-Bandwidth

• The linear programming is formulated as

9
Cost function-Bandwidth

• We can also rewrite the linear programming as

10
Cost function-Bandwidth

• Note that there is no constraint that the sum of
  the timesharing factor is no more than 1 in the
  rewritten linear programming
• With increasing, the traffic load will also
  increase. It is possible that it comes to a point
  that congestion occurs.
• Define aggregate congestion measure

11
Cost function-Energy

• If the total power consumption for ECG ,
  then the linear programming for minimizing the
  total power can be formulated as