1. 1 TCOM 541 Session 4

2. 2 Web Page http://teal.gmu.edu/ececourses/tcom540/TC OM540541.htm

3. 3 Backbone Reliability What do we mean by “reliability”? –Probability that the working nodes are connected? –What if connectivity is maintained but capacity is reduced? Can define an “outage” as occurring “whenever any interface, SDP-SDP service, or network performance parameter is not within specified performance limits” (FAA)

4. 4 Backbone Reliability (2) Then the desired reliability can be specified in terms of –Mean time between outages –Restoral time –Availability = (Total_Time_Outages)/Total_Time Notice this definition does not address network-wide reliability We will address network reliability in terms of 2- connectedness

5. 5 2-Connected Backbones - Recap A vertex v of a connected graph G = (V, E) is an articulation point if removing the vertex and all attached edges disconnects the graph If a connected graph has no articulation points, it is said to be 2-connected

6. 6 Augmenting the Backbone Suppose 1.We have completed an initial backbone design 2.We have further identified a subset of backbone nodes that require 2-connectivity How do we add links to the backbone to satisfy (2)? –Discuss two algorithms, AMENTOR and MENTour

7. 7 AMENTOR Add minimal set of links to backbone to ensure 2-connectivity –At minimum increase in cost Cannot do this by enumeration for large networks– need to develop a heuristic approach

8. 8 AMENTOR Algorithm Find articulation points a 1 …a k and 2-connected components C 1 … C m –If there are none, the network is 2-connected already Build an auxiliary graph G –Nodes of G correspond to a i and C j Thus there are k + m nodes in G If a r is in C s than there is an edge in G between a r and C s –G is a tree – if there were a cycle in G, all components in the cycle would collapse into one 2-connected component

9. 9 AMENTOR Algorithm (2) Give each node a i a weight of 0 and each node C j a weight of 1 –All edges have a weight of 0 – because articulation points lie between 2-connected components Compute shortest paths in the graph G –Several algorithms available, some we’ve discussed –Distance from C j to C h is the number of 2-connected components traversed in going from a node in C j to a node in C h Adding an edge will collapse all these components into a single 2-connected component

10. 10 AMENTOR Algorithm (3) Use G to provide figure of merit to possible edges in the original network N –Consider all node pairs (n1, n2) in N –Reject if either node is an articulation point Then each node belongs to a unique 2-connected component – call them C1 and C2 Merit = cost N (n1,n2)/dist G (C1,C2) Add link between pair with lowest merit Return to beginning of algorithm

11. 11 MENTour Algorithm Rather than adding links one at a time, MENTour builds tours from the beginning Same steps as MENTOR-II except that instead of building a hybrid Prim-Dijkstra tree, we build a TSP tour on the backbone sites Recall that there is no method to develop an optimal TSP tour …

12. 12 MENTOR Summary Backbone selection –Threshold clustering –K-means clustering –Automatic clustering Initial topology –Prim-Dijkstra tree –TSP tour Link addition –Home-based routing –ISP-based routing Access topology –Star –Esau-Williams –MSLA

13. 13 Constraints on Designs Cheapest design may not be acceptable for reasons of reliability, performance, … or organizational or political reasons May have to modify algorithms or write problem-specific code to produce an acceptable design

14. 14 Types of Constraints Hop constraints –Worst case –Average –Node-pair Equipment constraints –Degree –Throughput Link constraints –Required or forbidden Performance constraints –Worst case –Average –Node-pair Reliability constraints –Entire network –Backbone –Node-pair Miscellaneous ….

15. 15 Over-Constrained Networks It is possible – even easy – to specify so many or so severe constraints that there is no feasible network design –E.g., 7 nodes, each must be of degree three –E.g., suppose each site has equipment that can only terminate a LAN and two leased lines Can only build a ring As network grows, link capacity will be overloaded at some point

16. 16 Approaches Culling –Generate a lot of designs with varying parameters –Throw away all that do not meet hop constraints –Select best remaining design –Brute force method may be quite acceptable if it avoids writing problem-specific code –May not work if constraints are too severe – I.e., frequency of feasible designs may be too low to be practicable

17. 17 Max Hop Constraints Culling may work if hop constraints not too severe If not, two more possibilities –Reduce diameter of backbone –Reduce depth of access trees E.g., replace trees with stars –Relative attractiveness depends on cost split between access and backbone Another approach – augment the network

18. 18 Average Hop Constraints Easier to design to average hops than max hops Three approaches –Increase a Builds more star-like networks –Increase slack Introduces more links –Use lower-speed links Introduces more links

19. 19 Node-Pair Constraints Traffic between certain node pairs must meet a hop constraint Two algorithms –Simply add cheapest link that meets constraints However this will also attract traffic from other nodes –Modify and use ISP algorithm in MENTOR Re-optimize network by setting new link length to minimize cost increase

20. 20 Equipment Constraints Equipment limitations may have significant effect on network design –Degree constraints –Processing constraints

21. 21 Degree Constraints E.g., routers with limitations on number of lines that can be attached –Say each site sends and receives 500 kbps and the routers can only handle four T1s plus a LAN card –Clearly, each backbone node can have at most one edge site attached –Traffic loading algorithm can have a major effect on feasibility of any given design

22. 22 Processing Constraints E.g., limitation on number of packets that can be processed per second, say pmax –Problem lends itself to use of a drop algorithm Build a complete graph Do initial loading – node terminates more than pmax/2, problem is infeasible Order links by merit = 100*u + (1-cost/max_cost) Choose link with lowest merit, compute alternate path If alternate path shows feasible loads, drop the link, else set merit = infinity

23. 23 Processing Constraints (2) Alternatively, build composite nodes –E.g., use two processors at a node Link by high speed cable Terminate half of the links on each box Cap = 1000 167 333

24. 24 Link Constraints Generally involve either forbidden of required links –Want to use existing link capacity –Diversity –Backup (disaster recovery) –Unavailability (within reasonable timeframe) –Inaccurate tariff data –Unsuitable media (e.g., satellite) –Lack of confidence in carrier

25. 25 Modifying MENTOR for Link Constraints During tree or tour-building phase, to forbid a link, simply assign a high-enough price that the algorithm would never choose it However, direct link addition phase will add a link if u > umin –This has to be changed to add a link if u > umin and cost < “high-enough price ”

26. 26 Modifying MENTOR for Link Constraints (2) Required links may be included –During tree/tour building E.g., by assigning a low cost to the link, and culling the designs But remember the real cost is not the artificially low cost! –During direct-link addition Can add a table specifying links to be added directly –During post-processing Easy, but may introduce significant extra cost since link is not taken account of during design phase

27. 27 Performance Constraints Evaluating performance not easy for large network designs –Blocking for voice networks –Delay for packet-switched networks Analysis or simulation –Analysis relies heavily on queuing theory

28. 28 Capacity Assignment Algorithm Aims at improving performance of an existing network Keep topology fixed, add capacity to existing links to reduce delays List node and link options with associated cost –Compute contribution to total average end-to-end delay –Compute cost per ms delay reduction compared to current network Add capacity starting with lowest cost per delay reduction

29. 29 Capacity Assignment Algorithm (2) Algorithm may overkill on last step and add much more capacity than needed Problem is an example of the knapsack problem –Given a set of integers N = {N1, N2, …, Nk}, find a subset that add to exactly M –2 k possible combinations …

30. 30 Reliability Constraints Evaluating reliability not easy for large network designs –Must define failure Loss of connectivity vs. degradation of performance –Here we will discuss simple failure of connectivity –Simplest case is a tree – all nodes are connected if and only if all links and nodes are working

31. 31 Reliability Constraints (2) Let tree have nodes N 1, N 2, …, N n and links L 1, L 2, …Ln-1 Let failure probabilities be p i and p j * respectively Probability network is working is then  (1-p i )  (1-p j *) = (1-p) n (1-p*) n-1 if probabilities are uniform Obviously tends to 0 when n is large

32. 32 Reliability Constraints (2) For graphs more complicated than trees, calculations can be complex Try to reduce to simpler networks –Series reduction Replace node and two edges with a single edge, probability of working = p n p e1 p e2 –Parallel reduction Replace two parallel edges with single edge, probability of working = 1- (1-p e1 )(1-p e2 ) = p e1 + p e2 - p e1 p e2

33. 33 Reducing a Graph ABC FED 0.8 0.9 Assume nodes have reliability = 1.0

34. 34 Reducing a Graph BC FED 0.72 0.8 0.9 Apply series reduction to A

35. 35 Reducing a Graph B FED 0.72 0.8 0.9 Next, apply series reduction to C

36. 36 Reducing a Graph B E 0.576 0.9 Next, apply series reduction to D and F

37. 37 Reducing a Graph B E 0.982 Next, apply parallel reduction

38. 38 Assignment Cahn 10.13 Read Cahn Chapter 11