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Simultaneous routing and resource allocation via dual decomposition AUTHOR: Lin Xiao, Student Member, IEEE, Mikael Johansson, Member, IEEE, and Stephen P. Boyd, Fellow, IEEE http://www.s3.kth.se/~mikaelj/. Reporter D95725003 D95725006 IEEE Transactions on Communications, Vol. 52, No. 7, pages 1136-1144, July 2004 Slide 2 Contents Motivations System Variables System Model Design Objective and Performance Index Solution Motivation & Contribution Conclusion Slide 3 Motivations Technological motivation Wireless ad-hoc networks promising emerging technology Intellectual motivation Will ad-hoc networks deliver the required performance (capacity)? Compute the optimal parameters for a given network configuration Devise simple, distributed protocols that ensure efficient network operation Control-theoretic motivation Distributed resource allocation problems roots of distributed control theory New technological challenges/problems may inspire theoretical advances Pedagogical motivation To convey ideas and techniques from distributed convex optimization Slide 4 System Variables Controlled Input variables : communications variable (or r) including media access scheme, such as transmit power, bandwidth, time slot fraction and etc., weights on flow State variables : collection of source sink vector s, collection of flow vector x, Internal variables : none Internal parameters : total node number N, total destination number D, total link number L media access methods, coding and modulation scheme, network topology A nl, Measured output variable : average behavior of data transmission, total energy consumption, total Throughput, total utilization Controlled output variables : the same variable above Design Objective : Optimization network total utility, total power consumption System: Internal Variable Input variablesOutput Variable Slide 5 System model Assumption fixed topology fixed coding, modulation and optimize rates, routing & resource allocation Modeling multiple data flows influence of resource allocation on link capacities local & global resource limits Slide 6 System Model- Network topology Directed graph with nodes, links set of outgoing links at node, incoming links at Incidence matrix Link, L{1,..,m} node Slide 7 System Model- Network flow model Model average data rates, multiple source/destination pairs Identify flows by destination source flows flow from node to node link flows flow on link to node Flow conservation laws A is node-link incidence matrix in previous slide Slide 8 System Model- Multicommodity network flow Some traditional formulations: fixed, minimize total delay: fixed, maximize total utility: Total Traffic on link l U be a concave and strictly increasing utility function Capacity and source flow k link Slide 9 System Model- Communications model Capacities determined by resource (power, bandwidth) allocation Communications model(p.1138) Where is a vector of resources allocated to link, e.g., is concave and increasing resource limits local (power at node) or global (total bandwidth) Many (most?) channel models satisfy these assumptions! (transmit power, bandwidth) l = 1,..,L ; link l total traffic t c l Slide 10 System Model- Description Model for solving SRRA Problem Maximize weighted sum of capacities, subject to resource limits Convex problem Special methods for particular cases, e.g. water filling for variable powers, fixed bandwidth are concave and monotone increasing in r w, nonnegative scalar weight Slide 11 Simultaneous optimization of routing and resource allocation Solution to optimization problem We assume that are convex When communication resource allocation r is fixed, get convex multicommodity flow problem SRRA is a convex optimization problem, hence readily solved total traffic flows f(link, node, traffic)+ f(resource) Slide 12 Examples SRRA formulation is very general, includes Maximum utility routing (QoS) Minimum power routing as well as minimum bandwidth, minimax link utilization, etc. Slide 13 Solution - Solution Methods Slide 14 Solution - Optimal Routing in a Example 50 nodes, 340 links (transmitters) 5 nodes exchange data (i.e., 20 source-destination pairs) transmitters use FDMA, power limited in each node goal: maximize network utility Slide 15 Solution - Optimal Routing in a Example Slide 16 Slide 17 Optimal Routing Methods Slide 18 Solution - Dual Decomposition Method structure of SRRA problem objective separable in network flow and communications variables only capacity constraints couple x; s; t and dual decomposition (Lagrange relaxation) relax coupling capacity constraints by introducing Lagrange multipliers( http://episte.math.ntu.edu.tw/entries/en_lagrange_mul/index.html ) http://episte.math.ntu.edu.tw/entries/en_lagrange_mul/index.html decompose SRRA into two subproblems, both highly structured, efficient algorithms exist for each (dual decomposition again) Sub-problems coordinated by master dual problem Slide 19 Solution - Dual Decomposition Method Slide 20 Solution - SRRA Solution Hierarchical Dual Decomposition Multicommodity Network Flow Resource Allocation Problem Slide 21 Solution Subproblem Method Analysis ACCPM:Analytic center cutting-plane method, Goffin and Vial [GV93] Dual Function Slide 22 Conclusion Significance New Network model considering wireless environment Optimal operation of network using SRRA Method Problem Do not mention the important issues in wireless network such as QoS, Dynamic Routing, media access methods, coding and modulation scheme Cannot explain some situations packet loss, retransmissions, time varying fading, topology change Extension Improve algorithm Asynchronous distributed algorithm Dynamic routing and resource allocation Slide 23 Thanks for Listening Slide 24 Solution Introduction to Optimal Routing &resource Allocation Slide 25 System Model- Example: Gaussian broadcast with FDMA Communication variables: Shannon capacity: Total power, bandwidth constraint on outgoing links (Total Resource Limits) p.1138 Shannon entropy, H1 Probability of observing a particular symbol or event, pi, with in a given sequence and Slide 26 An example with FDMA Slide 27 Solution - Dual Decomposition Method Slide 28 Throughput, fairness, QoS End-to-end delay Gin index(fairness )