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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 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 • Solution for Point-to-point link • Extending to general network • Conclusion WNCG, UT AUSTIN
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
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
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
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
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
Problem set up (contd.) No Interference No BC Constraint A series-and-parallel network WNCG, UT AUSTIN
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
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
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
Point-to-point Link (contd.) For large packet sizes, the rate-cost curve will look like WNCG, UT AUSTIN
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
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
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
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
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
General S-P Network (contd.) Box # 2 Box # 1 These “boxes” are connected in parallel, to give… WNCG, UT AUSTIN
General S-P Network (contd.) … a black-box with this rate-cost curve. WNCG, UT AUSTIN
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
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