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TCOM 541

Session 4

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Web Page

• http://teal.gmu.edu/ececourses/tcom540/TCOM540541.htm

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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)

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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

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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

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Augmenting the Backbone

• Suppose 1. We have completed an initial backbone

design2. 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

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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

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AMENTOR Algorithm

• Find articulation points a1 …ak and 2-connected components C1 … Cm

– If there are none, the network is 2-connected already

• Build an auxiliary graph G– Nodes of G correspond to ai and Cj

• Thus there are k + m nodes in G• If ar is in Cs than there is an edge in G between ar and Cs

– G is a tree – if there were a cycle in G, all components in the cycle would collapse into one 2-connected component

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AMENTOR Algorithm (2)

• Give each node ai a weight of 0 and each node Cj 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 Cj to Ch is the number of 2-connected

components traversed in going from a node in Cj to a node in Ch

• Adding an edge will collapse all these components into a single 2-connected component

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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 = costN(n1,n2)/distG(C1,C2)• Add link between pair with lowest merit

• Return to beginning of algorithm

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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 …

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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

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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

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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 ….

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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

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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

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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

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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

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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

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Equipment Constraints

• Equipment limitations may have significant effect on network design– Degree constraints– Processing constraints

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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

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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

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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 Cap = 1000167

167

333333 333 333

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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

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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 ”

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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

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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

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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

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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– 2k possible combinations …

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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

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Reliability Constraints (2)

• Let tree have nodes N1, N2, …, Nn and links L1, L2, …Ln-1

• Let failure probabilities be pi and pj* respectively

• Probability network is working is then

(1-pi)(1-pj*)

= (1-p)n(1-p*)n-1 if probabilities are uniform• Obviously tends to 0 when n is large

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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 = pnpe1pe2

– Parallel reduction• Replace two parallel edges with single edge, probability of

working = 1- (1-pe1)(1-pe2)

= pe1 + pe2 - pe1 pe2

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Reducing a Graph

A B C

FED

0.8 0.8

0.8 0.8

0.9 0.9 0.9

Assume nodes have reliability = 1.0

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Reducing a Graph

B C

FED

0.72

0.8

0.8 0.8

0.9 0.9

Apply series reduction to A

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Reducing a Graph

B

FED

0.720.72

0.8 0.8

0.9

Next, apply series reduction to C

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Reducing a Graph

B

E

0.576 0.5760.9

Next, apply series reduction to D and F

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Reducing a Graph

B

E

0.982

Next, apply parallel reduction

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Assignment

• Cahn 10.13

• Read Cahn Chapter 11