Congestion Control for Multicast Flows with Network Coding

1
Congestion Control for Multicast Flows with
Network Coding

• Lijun Cheny, Tracey Hoy, Mung Chiangz, Steven H.
  Lowy, and John C. Doyley
• Presented by
• Subhash Lakshminarayana

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Introduction

• Flow control at end-systems for network coding
  based multicast flows with elastic rate demand.
• Optimization based models for network resource
  allocation.
• Decentralized controllers at sources and
  links/nodes for congestion control in wired
  networks.
• Two scenarios
• Use coding subgraphs that are un-capacitated
• Do not explicitly find coding subgraphs
• dynamic routing and coding decisions based on
  queue length gradients
• Work at the transport layer - adjust the source
  rates
• Work at the network layer - carry out network
  coding
• Random linear Network coding approach used
• Partially-primal and dual gradient algorithm

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Network Coding-1

• Extends the functionality of network nodes from
  storing/forwarding packets to performing
  algebraic operations on received data
• Bandwidth Gains
• Maximize multicast throughput
• Robustness to link/node failures
• Robustness to packet losses

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Network Coding-2
Graph with set N of nodes and a set L of directed
links
Set of multicast sessions
Source of session m
Destination set of session m
Session rate

• Information must flow at rate to
  each destination
• With network coding the actual physical flow on
  each link need only be the maximum of the
  individual destinations flows

Information flow for destination d of session m
Amount of link capacity allocated to session m
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Network Coding-3

• Only intra-session Network coding considered

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Network Coding-4

• 1Theorem 1
• The rate vector g satisfies the above
  constraints if and only if there exists a network
  code that sets up a multicast connection at rate
  arbitrarily close to from source to
  destinations in set and that injects packets
  at rate arbitrarily close to on each link

• 2Random Linear Network Coding
• Network nodes form output packets by taking
  random linear combinations of corresponding
  blocks of bits in input packets
• Linear combination specified by the coefficient
  vector in the packet header
• Each sink receives with high probability a set of
  packets with linearly independent coefficient
  vectors, allowing it to decode

1. D. S. Lun, N. Ratnakar, M. Medard, R. Koetter,
D. R. Karger, T. Ho and E. Ahmed, Minimum-cost
multicast over coded packet networks, IEEE Trans.
Inform. Theory, 2006. 2. Tracey Ho, Muriel
Médard, Ralf Koetter, David R. Karger, Michelle
Effros, Jun Shi, and Ben Leong , A Random Linear
Network Coding Approach to Multicast, IEEE Trans.
Inform. Theory, 2006
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Multicast with Given Coding Subgraphs-1

• Coding is done on overlapping segments of
  different trees of a session that have disjoint
  sets of downstream destinations.
• Coding subgraphs chosen in a variety of ways
• Delay
• Resource Usage
• Commercial relationship amongst network
  providers.

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Multicast with Given Coding Subgraphs-2
Coding subgraph for multicast session m
Set consisting of union of links of possibly
overlapping multicast trees connection source to
all destinations
Set of all destinations of session m
Information flow rate for session m
Multicast Matrix with of dimension
Amount of link capacity associated with session m

• Network Coding Constraint

There exists a corresponding multicast network
code of rate arbitrarily close to
from source to destinations
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Utility Maximization problem
Multicast with Given Coding Subgraphs-3
Continuously differentiable, increasing and
strictly concave
Objective Decentralized controllers at source
and links/nodes to achieve optimum

• Linear constraints form a convex set
• Convex optimization problem!
• Strong duality holds
• Decentralized solution using Lagrange dual methods

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Multicast with Given Coding Subgraphs-4
Traffic split variable for each multicast tree
Rate supported by each tree
Two-Level Optimization Problem

• Source rate, session allocation problem fast
  timescale
• Traffic splitting problem slow traffic
  engineering timescale

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Multicast with Given Coding Subgraphs-5
Congestion Price at link l for multicast tree

• Congestion Control

Aggregate congestion price over multicast tree

• Similar to TCP congestion control algorithm
• End to End congestion control mechanism

• Session Allocation

where

• Session with higher link price is allocated more
  capacity

Note
Net flow through a link remains constant
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Multicast with Given Coding Subgraphs-6

• Congestion price update

• Tries to match supply and demand
• Distributed and can be implemented at each link
  with local information

• Tree adaptation Algorithm

Much slower minimization of
over p can be seen instantaneous

• Traffic split vector updates

• Discourages more congested trees
• Slow algorithm

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Convergence Analysis-1

• Optimality conditions for convex program

Theorem 2 Under congestion control and session
allocation defined above, the system converges to
the optimum of the problem P1a.

• Lyapunov function

• Negative Lyapunov drift

• System converges to an invariant set specified
  by

only if p,y,x satisfy optimality conditions
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Convergence Analysis-2
Theorem 3 The tree adaptation algorithm (9)
converges to the optimum of the system problem P1.
Prove that
Tree adaptation algorithm will converge to
equilibrium
such that
and
solves the system problem

• Implementation of price feedback

• Each link keeps a separate virtual queue
  for each multicast tree
• Packet header contains indices of trees whose
  information it contains
• Each node i will pass aggregate price along the
  links from the receivers till itself to the
  upstream node j.

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Networks without Given Coding Subgraphs-1
Information flow for destination d of session m
Amount of link capacity allocated to session m
Capacity of link (i,j).

• Network coding constraint

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Networks without Given Coding Subgraphs-2

• Congestion Control

• Local congestion price at the source node
• Congestion in the network is propagated to source
  node through backpressure.

• Session Allocation

• Session weights along each link

• Capacity gradient

Sessions having higher weight gets more higher
capacity allocation
No change in the net capacity allocated
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Networks without Given Coding Subgraphs-3

• Over link (i,j), transmit a random linear
  combination of data of multicast session m to all
  destinations having positive price gradient.

• Congestion Price Update

• Each node adjusts its sending rate according to
  local congestion price.

• Implementation of price feedback

• Each packet need to carry a vector of destination
  identities in the packet header
• Each nodes i keep a separate virtual queue
  as congestion price for each multicast session
  m and

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Convergence Analysis
Networks without Given Coding Subgraphs-4

• Theorem 4
• Under congestion control and session allocation
  algorithm defined above, the system converges to
  the optimum of the problem P2.

• Prove that Lyapunov drift is negative

• System converges to an invariant set specified by

only if p,f,g,x satisfy the optimality conditions
for convex problem
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Extension to Wireless Networks-1
Hyperarcs transmitting node
receiving neighbors
Hyperarc flow vector
static topology finite fixed broadcast capacity
capacity region at the link layer
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Extension to Wireless Networks-2

• Congestion Control

• Scheduling

• Choosing Multicast Session

• Session scheduling within the hyperarc

• Price Differential

• Random linear combination of data of chosen
  multicast session at a rate

such that
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Extension to Wireless Networks-3

• Hyperarc scheduling is centralized and is
  NP-complete
• Yet to design distributed algorithms

• Congestion price update

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Numerical Examples-1

• With Coding subgraphs

• 2 Sessions routed
• Session1 source s and destinations x and y
• Session2 source t and destinations u and z

Session Utilities
Period of update of cong control/session
allocation algorithm
Period of update of cong tree adaptation
algorithm
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Numerical Examples-2

• Comments
• Source rate reach 5 of optimal within 5
  iterations.
• Traffic split vectors reach 5 of optimal within
  10 iterations.
• Inner loop runs for 500 iterations
• End users can dynamically control number of
  iterations by monitoring the congestion prices
  over different trees
• Dynamic step sizes

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Without Coding Subgraphs
Numerical Examples-3

• Comments
• Smaller step size, slower the convergence and the
  closer to the optimal
• Throughput achieved for the case without given
  subgraphs is larger than that for the case with
  given coding subgraphs.
• Capacity region for the case with given coding
  subgraph is a subset of the capacity region with
  the coding subgraphs unspecified.