1. 2/14/2016  A. Orda, A. Segall, 1 Queueing Networks M nodes external arrival rate (Poisson) service rate in each node (exponential) upon service completion in node i, –user goes to node j with probability –and exits the system with probability if is arrival rate ( not necessarily Poisson) to node i, then system state, where is number of users at node i Jackson’s Theorem where, namely each queue behaves as an independent M/M/1 queue

2. 2/14/2016  A. Orda, A. Segall, 2

3. 2/14/2016  A. Orda, A. Segall, 3 Routing Definition: selection of continuous path(s) between source and destination over which data will travel from source to destination Routing process is complicated because: –requires cooperation between a large number of nodes ( not only neighbors ) –needs to cope with a large number of situations, sometimes unpredictable –performance requirements may be strict because it is a basic part of the operation of network Two aspects: –path selection –transfer of data over the path Normally second part is simpler than the first. Second operation may be performed using tables in the nodes or description of the path in the header of the message.

4. 2/14/2016  A. Orda, A. Segall, 4 Routing Protocol Requirements –Performance: delay, throughtput, etc. –Correctness: messages should reach destination –Reliability –Stability and Convergence –Fairness: in the enclosed example, maximum traffic will not allow traffic from X to X’ Performance Criteria –Throughput, i.e. amount of traffic –Low Delay Network point of view User point of view –Cost –Other network-specific criteria

5. 2/14/2016  A. Orda, A. Segall, 5 How does Routing Protocol affect Network Performance ? The network should apply procedures to limit the amount of traffic that enters the network, thus better routing will allow more traffic to traverse the network On the other hand, given the throughput, better routing will provide lower delay. Example: all links have 10 units capacity Low load: 5 units from 1 to 6 and from 2 to 6 –via 3 and 5 respectively: good routing –both via 4 : bad routing High load: 5 units from 1 to 6, 15 units from 2 to 6 –if NO flow splitting is allowed, ( at least ) 5 units are rejected –if splitting is allowed, we can handle all requirements, e.g. by sending the traffic from 1 via 3, half of the traffic from 2 via 4 and the other half via 5 Maximum Load: –both flows via 4, max. flow = 10, bad choice –no splitting, max. =20, better routing –with splitting, max = 30

6. 2/14/2016  A. Orda, A. Segall, 6 Static, non-traffic-splitting, routing Nodes maintain routing tables Table at a node holds the next hop routing node for every destination Tables differ from node to node Building the tables –select a set of paths –use the set of paths to deduct the corresponding neighbor –the set of paths should be selected such that there are no loops, for example if the path from A to G goes via E, then the path from E to G ( as reflected in the table of E) should not go through A

7. 2/14/2016  A. Orda, A. Segall, 7 Static Routing, splitting traffic Table contains the fractions of traffic a node forwards to each neighbor for each destination The decision for each packet is made by using a random number Multi-route static table at node J

8. 2/14/2016  A. Orda, A. Segall, 8 Routing in Local Area Networks Data is broadcast in each LAN All bridges on each LAN receive each packet The routing problem is to decide which bridge forwards packets for a given destination and which bridge discards them.

9. 2/14/2016  A. Orda, A. Segall, 9 Graphs - definitions Graph (N,A) –A - collection of links –N - collection of nodes link is denoted by (n 1, n 2 ) or by a i Walk - Arbitrary collection of successive links Path - Loop-less walk ( no node appears more than once ) Cycle - walk that starts and ends at the same node Connected graph - there is a path between every pair of nodes Non-connected graph Sub-graph - Part of a given graph Tree - Connected loop-less graph Spanning Tree for graph G - A tree that is a sub-graph of G and contains all nodes of G

10. 2/14/2016  A. Orda, A. Segall, 10 Spanning Tree Algorithm Given a connected graph G = ( N,A) 1. Select an arbitrary node V, N’={V}, A’={} 2. If N=N’, stop. A’ is the Spanning Tree. 3. Select a link (i,j), such that. Update 4. Go to 2 Proof: –there always exists a link in 3. ( since G is connected ) –A’ is always a tree –the final A’ is spanning Properties –Any connected graph G = (N,A) has a spanning tree –in any graph G holds –holds

11. 2/14/2016  A. Orda, A. Segall, 11 Routing in Networks of LAN’s Here the LAN’s are the “Network Nodes” and the bridges are the “Network Links” The routing is done on a Spanning Tree, because: –Need to be able to reach each LAN from each LAN –difficult to split traffic ( each LAN is a broadcast medium )

12. 2/14/2016  A. Orda, A. Segall, 12 Minimum Spanning Tree (MST) If Network links are not the same, different Spanning Trees have different properties, we want to select the Spanning Tree with the best performance Assumption: The “quality” of each link can be measured and expressed as a strictly positive number ( e.g. delay, load, etc.) Definition: “tree weight” = sum of tree link weights. Problem: Given a graph G=(N,A), with link weights {w ij }, select an MST, i.e. the Spanning Tree with minimum weight among all Spanning Trees Definitions: –segment = sub-graph of an MST –outgoing link = link with one end in the segment and the other outside the segment

13. 2/14/2016  A. Orda, A. Segall, 13 1 6 3 1 2 4 2 53 4 25