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Stochastic Multicast with Network Coding

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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

- Capacity planning at multicast service provider
- Solution 1 – Heuristic
- Usually but not always good solutions

- Solution 2 – Sampling
- Provable performance bound

- Simulations
- Conclusion

Network Service Provider

Content Provider

SLA

negotiate

negotiate

Usage beyond SLA incurs penalties!

Network

P(t)

Potential Customers

- Content provider’s goal:
- Minimize expectedcost
- 2-stage stochastic optimization

- Minimize expectedcost

- 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

- 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

- Set of customers is non-deterministic

- Exploits replicable property of information
- Reduce redundant transmissions
- Efficient bandwidth usage => cost savings!

Traditional multicast

- Finding and packing Steiner trees – NP-Hard!
Network coding

- Exploit encodable property of information
- Polynomial time solvable
- linear programming formulation

- 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”

- Directed graph
- Edge has cost and capacity
- Receiver has set of paths to the source

- 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 -

- Optimal
- But intractable!
- Exponentially sized
- #P-Hard in general

- Can we approximate the optimal solution?

- 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

- Simulations show excellent performance in most cases
- No provable performance bound
- In fact, it is unbounded

- 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

- 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)

- 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

- 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!

- 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

- Heuristic