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ICDCS 2009, June 24 2009, Montreal. Stochastic Multicast with Network Coding. Ajay Gopinathan, Zongpeng Li Department of Computer Science University of Calgary. Outline. Capacity planning at multicast service provider Solution 1 – Heuristic Usually but not always good solutions

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stochastic multicast with network coding

ICDCS 2009, June 24 2009, Montreal

Stochastic Multicast with Network Coding

Ajay Gopinathan, Zongpeng Li

Department of Computer Science

University of Calgary

outline
Outline
  • Capacity planning at multicast service provider
  • Solution 1 – Heuristic
    • Usually but not always good solutions
  • Solution 2 – Sampling
    • Provable performance bound
  • Simulations
  • Conclusion
problem statement
Problem Statement

Network Service Provider

Content Provider

SLA

negotiate

negotiate

Usage beyond SLA incurs penalties!

Network

P(t)

Potential Customers

the content provider s dilemma
The Content Provider’s Dilemma
  • Content provider’s goal:
    • Minimize expectedcost
      • 2-stage stochastic optimization
two stage stochastic optimization
Two-stage stochastic optimization
  • Stage 1:
    • Estimate capacity needed
    • Purchase capacity at fixed initial pricing scheme
  • Stage 2:
    • Set of multicast receivers revealed
    • Bandwidth price increases by factor
    • Augment stage 1 capacity, for sufficient capacity to serve everyone
  • Stage 1 purchasing decision should minimize cost of both stages in expectation
the content provider s dilemma1
The Content Provider’s Dilemma
  • Content provider’s goal:
    • Minimize expected cost
  • Obstacles
    • Set of customers is non-deterministic
      • Assume probability of subscription
      • Based on market analysis/historical usage patterns
    • Employ the cheapest method for data delivery
      • Multicast
why multicast
Why multicast?
  • Exploits replicable property of information
    • Reduce redundant transmissions
    • Efficient bandwidth usage => cost savings!
content provider s routing solution
Content Provider’s Routing Solution

Traditional multicast

  • Finding and packing Steiner trees – NP-Hard!

Network coding

  • Exploit encodable property of information
  • Polynomial time solvable
  • linear programming formulation
multicast with network coding
Multicast with network coding
  • Take home message
    • Compute multicast as union of unicast flows
    • Union of flows do not compete for bandwidth
      • Conceptual flows

“A multicast rate of d is achievable if and only if d is a feasible unicast rate to each multicast receiver separately”

network model
Network Model
  • Directed graph
  • Edge has cost and capacity
  • Receiver has set of paths to the source
how to minimize expected cost
How to minimize expected cost?
  • First stage, buy capacity at unit cost
  • Second stage, cost increases by
    • Unit capacity cost
  • For every let be probability that set is the customer set in second stage
  • Capacity bought in first stage –
  • Capacity bought in second stage -
two stage optimization1
Two-stage optimization
  • Optimal
  • But intractable!
    • Exponentially sized
    • #P-Hard in general
  • Can we approximate the optimal solution?
solution 1 heuristic
Solution 1 - Heuristic
  • Idea – Future is more expensive by
    • Buy units of capacity in stage one if probability of requiring is
  • Algorithm overview
    • Compute optimal flow to all receivers
    • Compute probability of requiring amounts of capacity on each edge
    • Buy on if above condition is met
solution 1 heuristic1
Solution 1 - Heuristic
  • Simulations show excellent performance in most cases
  • No provable performance bound
    • In fact, it is unbounded
solution 2 sampling
Solution 2 - Sampling
  • Basic idea – sample from probability distribution to get estimate of customer set
  • Is sampling once enough?
    • Need to factor in inflation parameter
  • Theorem [Gupta et al., ACM STOC 2004]
    • Optimal – sample times
    • Possible to prove bound on solution
cost sharing schemes
Cost sharing schemes
  • Method for allocating cost of solution to the service set (multicast receivers)
  • Denote as the cost share of in A
  • A -strict cost sharing scheme for any two disjoint sets Aand B:

1)

2)

3)

cost sharing schemes1
Cost sharing schemes
  • Theorem [Gupta et al., ACM STOC 2004]

If there exists a -strict cost sharing scheme, then sampling provides a (1 + )-approximate solution

  • Does network coded multicast have such a scheme?
    • Yes! Use dual variables of primal multicast linear program
a 2 strict cost sharing scheme
A 2-strict cost sharing scheme
  • Theorem

The variables in the dual linear program for multicast constitute a 2-strict cost sharing scheme

  • Proof using LP duality and sub-additivity
  • Sampling guarantees a 3-approximate solution!
conclusion
Conclusion
  • Problem – minimize expected cost for content provider when set of customers are stochastic
  • Two solutions
    • Heuristic
      • Performs well in most cases
      • No performance bound
    • Sampling
      • Performs less well than heuristic in simulations
      • Guaranteed performance bound
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