Routing. Effect of Routing in Flow Control. Relevant Graph Terms. Effect of Routing Path on Flow Control. Effect of Routing Path on Flow Control

Transcription

1 Routing Third Topic of the course Read chapter of the text Read chapter of the reference Main functions of routing system Selection of routes between the origin/source-destination pairs nsure that the data packets are delivered using the selected routes (use route tables) Routing system requires coordination among all the nodes of the network Routing system should be able to ope with node/link failures Reroute traffic based on cost metrics in case of failures or congestions Routing Performance Measures affected by routing Throughput (quantity of service) verage packet delay (quality of service) ffect of Routing in low ontrol ffect of Routing Path on low ontrol If the offered load is light Offered load Throughput = offered load If the offered load is excessive Throughput = offered load low ontrol - rejected load Throughput Lower Routing delay Rejected load allows more traffic to flow Routing across the network elay Link capacity: 0 units/s Origins (O) :, estination (): Routing paths (--); (--) have flow below link capacity (good throughput) Routing path (--); (--) will equal capacity ( leads to large delays) low: units/s low: units/s estination ffect of Routing Path on low ontrol Node has offered load of units Routing paths (-- ); (--) will lead to flow rejection in (--) qually splitting into (- -); (--) leads to fully accepted flow low: units/s low: units/s Relevant Graph Terms ef: graph G = (N, ) is a finite nonempty set of nodes N and a collection of distinct pairs of nodes from N. ach pair in is called arc (link or edge) Walk: sequence of nodes (n, n, n,,n k ) with links (n i,n i+ ) ycle: walk with n =n k Path: walk with no cycle Node Link/rc irected Link/rc

2 Relevant Graph Terms onnected Graph: or any given node i N, there is a path to each node j N Subgraph S of G is any nonempty strict subset of nodes N and arcs Tree: connected graph that contains no cycle Spanning tree: subgraph of G that is a tree and contains all the nodes from G igure for xamples Source node 7 8 onstructing a Spanning Tree Problem: Given a connected graph G=(N,) construct a spanning tree.. Let n N. Let S = {n} and ={}. If S=N stop; Return (S, ); else go to. Let (i, j) be a link with i S and j S Set S := S U{j}; := U {(i, j)} Go to step xample 9 (Min Weight) Spanning Trees Problem: Given the link weights, d ik between nodes i and k, compute a spanning tree that leads to minimum sum of link weights. Kruskal s lgorithm Sort the link weights m=; While (m <N){ hoose the min weight link break ties arbitrarily heck for cycles (if cycle, discard the link) If no cycle (m++) } 0 (Min Weight) Spanning Trees Prim s lgorithm Start from any node i hoose the min weight link (d ij ) of node i onsider super node ij and choose the min weight link from ij (added node is k) heck for cycles and discard any link that forms cycle. (Need special data structure for cycle detection) ontinue till (N-) links are added. Types of Routing lgorithms entralized--route choices are made at a central node istributed--omputation of route choices is shared among participating nodes Static--Routing table remains fixed during the session (unless there is link/node failure) daptive-- Routes are updated based on network parameters such as congestion, node battery lifetime

3 Least ost Path Routing very Link (i,j) has an associated cost (eg., bandwidth) Problem: Given source and destination node(s) as well as the link costs, find the least total cost path from the source(s) to destination(s) It is an optimization problem If all the link costs are identical, the least-total cost path is also the shortest path entralized Least-ost Path Routing: ue to ijkstra (Link State lgorithm) ssumes: link costs are nonnegative dij : ost of the link between nodes i and j (dij = if there is no link between i and j) i : stimate of the least-cost of node i from the source P : Set of nodes for which the least-cost path from the source is determined lgorithm Initialization: P={Source node; say #}; =0; i = di ; Loop{ Step: (ind next least-cost Node) ind node i not in P such that i = min j Step: (augment the set P) P = P U {i} Step: (update the least-cost for all nodes that are not in P) j := min [ j, {dij +i}] } n xample n xample Node is source node. onstruct the least-cost routes from to all other nodes Iteration Set P 0 {} (Src) {,} {,,} {,,,} {,,,, } {,,,, } n xample nalysis of ijstra s lgorithm Straight forward implementation requires (n-) + (n-) + = n(n-)/ computational steps for completion or nodes i P, j P; i j Node is source node. onstruct the least-cost routes from to all other nodes 7 8

4 Routing Instability: Oscillations If the link cost is proportional to the traffic flow (including the direction), the routing may lead to unwanted oscillations 0 0 e e + e 9 ellman-ord lgorithm for Shortest Path Given G=(N,) and the link costs d ik Problem: ompute shortest path from a source node (denote by #) to all other nodes efine: h i = Shortest walk from node i to node with at most h links; Set h = 0 for all h ellman-ford lgorithm 0 i = h+ i = min [ dij + j h ] for all i h h- t termination i = i for all i Iterative, (can be) asynchronous and distributed (suitable features for Internet Routing) 0 Source node n xample Problem: ind the shortest path from Node to all other nodes Update Table for ellman-ord with node as source Node link link link (due to ) (ue to ) (due to ) 0(due to ) (ue to ) Source node inal Result Practical issues with the distributed algorithm Link cost change can lead to problems Link cost decrease spreads fast (Good news travel s fast) (rom Kurose & Ross, chapter, pp. 89-9) 0 ompare with ijkstra s from last class

5 Practical issues with the distributed algorithm Link cost change can lead to problems Link cost increase spreads slowly (count to infinity problem) (rom Kurose & Ross, chapter, pp. 89-9) 0 0 Practical issues with the distributed algorithm Limited solution (alled Poisoned Reverse) does not work for more than two nodes case) to count to infinity If node has shortest route to via, it informs that h = nalysis of ellman-ord Let i 0 = for all i in. laims: i h are the shortest walk ( h) length from node i to node. The algorithm terminates after finite steps iff all cycles not containing node have nonnegative length. If the algorithm terminates in h steps, then h N. Worst case computations for convergence # of iterations =N; ach time it uses N- nodes and each node compares N- values. Leading to O(N ) 7