Stochastic Multicast with Network Coding Ajay Gopinathan, Zongpeng Li Department of Computer Science University of Calgary ICDCS 2009, June , Montreal

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 SLA Content Provider Network Service Provider Potential Customers Usage beyond SLA incurs penalties! negotiat e P(t)

The Content Provider’s Dilemma Content provider’s goal: – Minimize expected cost 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

Multicast Routing LP

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 A and 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

Multicast LP dual formulation

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!

Simulations

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

Steiner Trees