- By
**belle** - Follow User

- 79 Views
- Uploaded on

Download Presentation
## PowerPoint Slideshow about ' Stochastic Multicast with Network Coding' - belle

**An Image/Link below is provided (as is) to download presentation**

Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author.While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server.

- - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - -

Presentation Transcript

### Stochastic Multicast with Network Coding

ICDCS 2009, June 24 2009, Montreal

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
- Solution 2 – Sampling
- Provable performance bound
- Simulations
- Conclusion

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

- Content provider’s goal:
- Minimize expectedcost
- 2-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 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?

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

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

- 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

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

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 optimization

- Optimal
- But intractable!
- Exponentially sized
- #P-Hard in general
- Can we approximate the optimal solution?

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 - Heuristic

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

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

- 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 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

- 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

- 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

Download Presentation

Connecting to Server..