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Network with Costs: Timing and Flow Decomposition

Network with Costs: Timing and Flow Decomposition. Shreeshankar Bodas, Jared Grubb, Sriram Sridharan, Tracey Ho, Sriram Vishwanath The University of Texas at Austin California Institute of Technology. Outline. Introduction Previous work Results and problem setup Role of timing information

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Network with Costs: Timing and Flow Decomposition

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  1. Network with Costs: Timing and Flow Decomposition Shreeshankar Bodas, Jared Grubb, Sriram Sridharan,Tracey Ho,Sriram Vishwanath The University of Texas at Austin California Institute of Technology

  2. Outline • Introduction • Previous work • Results and problem setup • Role of timing information • Solution for Point-to-point link • Extending to general network • Conclusion WNCG, UT AUSTIN

  3. Introduction • Networks with costs arise in many wire-line networks. • Costs for using existing infrastructure. • Characterize data rate vs. incurred cost trade-off. WNCG, UT AUSTIN

  4. Previous Work • D. Lun, M. Medard, T. Ho and R. Koetter, “Network coding with a cost criterion”, Oct. 2004. • R. Koetter, “Flow Decomposition of Capacitated Networks”, Nov. 2006. • J. Giles and B. Hajek, “An information-theoretic and game-theoretic study of timing channels”, Sep. 1988. • V. Anantharam and S. Verdú, “Bits through queues”, Jan. 1996. WNCG, UT AUSTIN

  5. Main Results • “Series-and-parallel” network can be thought of as capacitated network for large packet sizes over the links. • Step-by-step algorithm for constructing rate-cost trade-off curve of such networks. • Contribution of timing information is negligible for large packet sizes. WNCG, UT AUSTIN

  6. Problem Setup • “Series-and-parallel” network • Single source, single destination • No interference / broadcast constraints • Link has three parameters: Capacity (C), Cost of usage (S), Packet size (m) WNCG, UT AUSTIN

  7. Problem Setup (contd.) • Channel can be in idle state. Keeping it idle costs nothing. Sending packets alone incurs cost. • Questions: • How much information can we send from source to destination, given capacity and cost constraints? • Transmission strategy for every channel? WNCG, UT AUSTIN

  8. Problem set up (contd.) No Interference No BC Constraint A series-and-parallel network WNCG, UT AUSTIN

  9. Timing Information • Point-to-point link • 3 packets transmitted in 4 time-slots… Possible schemes: Clever sequencing of packets and silences gives “extra” data rate. Cost incurred = 3 units, Data transferred > 3 packets ! WNCG, UT AUSTIN

  10. Timing Information (contd.) • Timing information := Total information conveyed - Information conveyed by packets • Expected to be “small”. Indeed so, for large packets. • Theorem: For a point-to-point link, if packet size = m, then timing capacity is no larger than 1/m. Idea behind the proof: Separating two events (packet transmission and idle slot), and using Fano’s inequality for upper bound. WNCG, UT AUSTIN

  11. Point-to-point Link • Point-to-point link: • Capacity = C, • Usage cost (per time slot) = S, • Packet size = m, • Average cost constraint = S0 (per time slot), then we prove that (C - 1/m) min(1, S0/S) ≤ Cpp ≤ min(C, CS0/S + 1/m) • Bounds match as m → ∞. WNCG, UT AUSTIN

  12. Point-to-point Link (contd.) For large packet sizes, the rate-cost curve will look like WNCG, UT AUSTIN

  13. Pure Series/Parallel Links • Network with k series links. Derive upper and lower bounds on capacity under average cost constraint. • Upper and lower bounds match as m → ∞. • Proof technique: Assume that ith time-slot carries a packet with probability γi and use Fano’s inequality… • Repeat for network with k parallel links. WNCG, UT AUSTIN

  14. Pure Series/Parallel Links (contd.) • The typical rate-cost curves for the pure-series and pure-parallel assemblies of 2 channels are here: Series Parallel WNCG, UT AUSTIN

  15. General S-P Network • Recall: “Series-and-parallel” network. • Large packet-sizes over all links. Then, the achievable rate over the network is a: • Concave function of the allowed average cost, • Piecewise linear function. • Black-box interpretation: Network is characterized by rate-cost curve. Internal details hidden. WNCG, UT AUSTIN

  16. General S-P Network (contd.) A series combination of two components, or a parallel combination, can be thought of as a single black-box. WNCG, UT AUSTIN

  17. General S-P Network (contd.) • Series assembly of two black-boxes: • Each individual box must operate at a rate R • Incur a total cost = Σ(costs of operating ith box at a rate = R) • The rate-cost curves are “added” along the cost axis. • Parallel assembly of two black-boxes: • Each segment in rate-cost curve represents a channel inside the black-box. • Use channels in decreasing rate/cost returns. WNCG, UT AUSTIN

  18. General S-P Network (contd.) Box # 2 Box # 1 These “boxes” are connected in parallel, to give… WNCG, UT AUSTIN

  19. General S-P Network (contd.) … a black-box with this rate-cost curve. WNCG, UT AUSTIN

  20. General S-P Network (contd.) • For a general network: • Successively break down into series and parallel assemblies of two black-boxes • Apply the previous construction to get rate-cost curve. • Thus get the rate-cost trade-off for entire network. WNCG, UT AUSTIN

  21. Conclusion • The network can be thought of as a capacitated network. • A step-by-step algorithm for constructing the rate-cost trade-off curve of a series-and-parallel network. • The contribution of timing information is negligible for large packet sizes. WNCG, UT AUSTIN

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