Network with Costs: Timing and Flow Decomposition

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