1. 1 Network Information Flow in Network of Queues Phillipa Gill, Zongpeng Li, Anirban Mahanti, Jingxiang Luo, and Carey Williamson Department of Computer Science University of Calgary Now at University of Toronto. Now at IIT Delhi, India. 1 2 2 1 IEEE/ACM MASCOTS 2008

2. 2 The Story: Network Modeling Queueing networks Well-established modeling methodology Network information flow Another well-established approach These two different approaches have different strengths and weaknesses Q: Can we blend the two together? A: We think so.

3. 3 Queueing Networks (1 of 3) Single Queue: Server Queue (finite or infinite) Arrival Process Departure Process  Literature: ACM SIGMETRICS, IFIP Performance, IEEE/ACM MASCOTS, Performance Evaluation, Queueing Systems, … M/M/1: q = / (1 – ) where = /    

4. 4  Queueing Networks (2 of 3) Chain of Queues: 

5. 5  Queueing Networks (3 of 3) Networks of Queues:  ...

6. 6 Queueing Networks: Summary Good: Models finite node capacity (rate, storage) Realistic models of stochastic traffic Realistic models of nodal delay and loss Bad: Naïve and unrealistic network topology Ignores multi-hop flow routing concept Hop-by-hop (atomic) view, not end-to-end

7. 7 Network Flow (1 of 3) Maximize unicast flow from S to T ST A B CD Capacity Cost 3 5 8 10 3 4 5 6 8 9 5 3 8 5 5 5 Literature: IEEE INFOCOM, CORS/ORSA, STOC, IEEE JSAC, Trans. on Information Theory, …

8. 8 Network Flow (2 of 3) Minimize cost of unit flow from S to T ST A B CD 3 5 8 10 3 4 5 6 8 9 5 3 8 5 5 5 Capacity Cost

9. 9 Network Flow (3 of 3) Multicast flow from S to R1, R2, and R3 Assumptions: - Multicast flow has unit capacity (i.e., 1). - All edges have the same unit capacity, and the same cost. - Information flows are replicable and encodable. S R R R 1 2 3 Multicast approach has server cost 3, network cost 6

10. 10 Network Flow (3 of 3) Multicast flow from S to R1, R2, and R3 Assumptions: - Multicast flow has unit capacity (i.e., 1). - All edges have the same unit capacity, and the same cost. - Information flows are replicable and encodable. S R R R 1 2 3 Better approach has server cost 2, network cost 5

11. 11 Network Flow (3 of 3) Multicast flow from S to R1, R2, and R3 Assumptions: - Multicast flow has unit capacity (i.e., 1). - All edges have the same unit capacity, and the same cost. - Information flows are replicable and encodable. S R R R 1 2 3 Network coding approach has server cost 1.5, network cost 4.5 a a a b b b abab abababab

12. 12 Network Flow: Summary Good: Properly reflects network topology Captures the multi-hop flow routing aspect Can exploit benefits of network coding Bad: Implicitly assumes nodes are very powerful Ignores nodal processing delay Ignores queueing delay and loss

13. 13 Research Questions Can we combine the two approaches, so that we get the best of both worlds? Non-trivial network topologies Multi-hop routing Stochastic traffic Nodal processing, queuing delay, loss... Does such an approach lead to new, interesting, and different insights, compared to classic network information flow or queueing network models?

14. 14 Methodology Mathematical modeling as a convex optimization problem (optimal routing) Deterministic, non-trivial multi-hop flows Stochastic traffic, nodal queueing delays For each routing scenario: Construct the mathematical program Prove that objective function and the feasibility region are convex (solvable) Perform simulation for numerical results

15. 15 Example: Single Unicast Minimize: Subject to:    f (u) (u) u V in  f (u) < in u V  f (u) = f (u) outin f(ts) = throughput  f(uv) C(uv)  (u) = 1 - f (u) in  uv E  (u) 0   f(uv) 0  Queueing Delay Weighted Delay (“cost”) Stability Target Volume Flow Conservation Capacity Constraint Non-negative values u u

16. 16 Evaluation Methodology Network topology generation: BRITE Convex optimization: MATLAB and cvx Numerical results interpreted Model correctness verified Results with the new model can behave differently than (for example) network models based on (linear) link costs

17. 17 Example: Numerical Results

18. 18 Modeling Issues Analysis of queueing network models often relies on memoryless property (i.e., Poisson arrivals and departures) Network coding implies some sort of “synchronization” between two streams Q: Is our approach doomed? A: No. Poisson property is preserved! (see proof in paper via Markov chains)

19. 19 Summary and Conclusions We proposed a new approach to network modeling, which combines classic network information flow with queueing networks Preliminary results with this model look very promising Memoryless property preserved New insights on optimal routing Ending of the story is yet to be written!