Chapter 6 Multispeed Access Designs

One-speed One-Center Design Review One-speed One-Center Design Problem: Connecting sites to a backbone node, all links with the same capacity OR

Capacitated Minimum Spanning Tree Problem (CMST) CMST problem: Given a central node N0 and a set of other nodes (N1, …, Nn), a set of weights(w1,…,wn) for each node, the capacity of a link, W, and a cost matrix Cost(i,j), find a set of trees T1, …, Tk such that each Ni belongs to exactly one Tj and each Tj contains N0.

The Esau-William Algorithm Heuristic Algorithm but guarantees the tree meets the capacity constraint Each node starts off in a tree with 1 node. Compute the tradeoff function for each node: Tradeoff(Nk)=minj Cost(Nk, Nj)-Cost(Comp(Nk),Center) If the tradeoff is negative, a merge is attractive Merge is allowed if Tradeoff for merging components A and B computes the potential savings of going to a neighbor instead of going to the center node.

Overview We need to introduce designs with multiple link types, because networks are built of a variety of different links. The cost of links is set by the market or government regulations, not any law of nature. Assume we have 3 different sorts of access links:

Multispeed Access Designs Assume the weight of the access sites is in the range of 2,400 bps to 36 Kbps, what happens if we are allowed multiple link types and still use either the Esau-Williams or Sharma’s algorithm? If we use 9.6 Kbps or 56 Kbps links, both algorithms fail because it’s impossible to fit 36 Kbps of flow on a single line and still keep the utilization under 50%. If we use 128 Kbps links, we will massively overdesign the networks for the smaller nodes. We need an algorithm that builds a tree with links of different capacity.

Continued We’ll replace the capacitated MST problem with a multispeed MST problem. Intuitively, the tree should have small-capacity links at the ends and should become “fatter” as we move toward the center of the networks, like the structure of a real tree. To define this type of tree we need a bit more notation.

Predecessor Function (definition 6.1) Pred(Root)=Root Root Pred(2)=Root 1 2 Pred(2) = Pred(Pred(5))=Pred2(5)=Root 3 4 5 Predecessor Function (definition 6.1) Pred(5)=2 A tree T rooted at a node Root can be represented uniquely by a predecessor function pred : V V on the set of vertices. The predecessor function moves 1 step closer to the root. Requirement: pred(Root) = Root pred(N) N for any other node N For any node N,  n > 0 such that pred n(N)= Root A tree is defined by the set of vertices V and edges (N, pred(N)) for all N Root

Ancestors (definition 6.2) 4 Pred(6)=4 5 6 7 8 9 Ancestors (definition 6.2) Pred2(9)=4 Given a tree T and the associated predecessor function, the ancestors of N are all the nodes N’ such that pred n (N’) = N for some n > 0 Node 5 through 9 are ancestors of node 4. A misnomer?

Given the following notations: A set of nodes N0, N1, N2, … , Nn. A set of weights (w1, w2, … , wn) for each node A set of link types L1, L2, … , Lm Capacities W1, W2, … , Wm A cost matrix C(i, j, k) that gives the cost of a link of type Lk between Ni and Nj

Multispeed CMST (definition 6.3) 4 5 6 Multispeed CMST (definition 6.3) 7 8 9 The multispeed CMST problem is to find the tree rooted at N0 and the link assignments such that (i) (ii) If N is node 4, then w(4)+w(5)+w(6)+w(7)+w(8)+w(9)< Wlink(4,pred(4)) is a minimum A set of link types L1, L2, … , Lm Clearly if m=1, this problem becomes the CMST problem

Multispeed Local Access Algorithm (MSLA) Assume the center is node 0 Assign each node the smallest link l possible to connect it to the center. For each node n, compute spare_capacity(n)=Wl -wn and set pred(n)=0 Calculate trade-offs (savings from linking site n to site i rather than linking directly to the center). And upgrade links to carry additional traffic: Tradeoffn(i)= c(n,i, l ) + Upgrade(i,wn) – c(n,0, l ) Tradeoff(n)= mini Tradeoffn(i) Add the edges as long as the tradeoffs are less than or equal to zero. Terminate when the tree is built and each edge is assigned to its link type. - Upgrade(i,wn) is the cost of adding wn units to the links that connect i and 0.

MSLA Example Table 6.1 Use the links shown in Table 6.1 at a 50% utilization: Define D96 as link type L0 Define D56 as link type L1 Define F128 as link type L2 L0 L1 L2 Figure 6.1 L0 L0 Assume N0 is center. We have 4 access nodes and their weights in Figure 6.1: L1 L1 Initial State (Utilization=0.5) spare_capacity(1)=0.5*56000-20000=8000 spare_capacity(2)=0.5*9600-2400=2400 spare_capacity(3)=0.5*56000-9600=18400 spare_capacity(4)=0.5*9600-4800=0 Initial state

Initial state State 2 State 3 L0 L1 L2 L1 L1 State 2: N2 is furthest away from N0. It is closer to N4. For N2 to go through N4, require (4,0) to upgrade from 9.6 Kbps to 56 Kbps. Upgrade(4,2400)= c(4,0,1) – c(4,0,0) ; Tradeoff2(4)=c(2,4,0)+(c(4,0,1)–c(4,0,0)) – c(2,0,0) Positive, not pick. Let N4 goes through N3,no upgrade is needed. It’s the best tradeoff. w3=w3+w4=9600+4800=14400 spare_capacity(3)=18400-4800=13600 L0 State 2 L0 L1 L1 L0 State 3: Next, the most attractive tradeoff is route N2 through N3. Again no upgrade is needed. w3=w3 + w2 =14400 + 2400 =16800 spare_capacity(3)=13600-2400=11200 L0 State 3 L1 L1

State 3 Final design spare_capacity(3)=13600-2400=11200 L0 L1 L2 L1 spare_capacity(3)=13600-2400=11200 spare_capacity(1)=8000 (initial) L1 Finally, connect N3 to N1 and increase (1,0) to 128 Kbps link. Spare_capacity(1)=0.5*128000-16800-20000=27200 L0 L0 Final design L1 My question: why not use state 3 ? L2 Reason: Final design might make good use of the economy of scale offered by the higher speed links.

A realistic example of MSLA Algorithm: We have 20 nodes in Squareworld and the weights of the nodes are generated according to the above TABLE TRAFDIST. Weights of nodes are shown in the TABLE SITES. Note that the weight of N0 is normalized so that it sums to the traffic from all the other sites. To simplify the mathematics, we assume that every line can be used to 100% of capacity.

Esau Williams: 20 nodes with 9.6Kbps links Cost = $26,963 Only 9 sites share links to N0, more like a star.

Esau Williams: 20 nodes with 56Kbps links Cost = $30,160 A nice tree structure, but the cost is higher because out on the periphery of the network there is too much capacity.

MSLA: 20 nodes with multispeed links Cost = $22,760 the best There is a central D56 tree and a peripheral D96 tree

Chapter 7 MultiCenter Local-Access Design

What happens if there are multiple centers What happens if there are multiple centers? Must build a forest instead of a tree. So what is a forest? Definition 7.1 A forest F = ( V,E ) is a simple graph without cycles. Note: a forest need not be connected.

Notations: A set of backbone sites (B0, …, Bm) = B A set of access nodes (N1, … , Nn) = N A set of weights (w1, … , wn) for each access node A upper limit of weight, W. A cost matrix Cost(i,j) giving the costs between each backbone/access pair of sites.

MultiCenter Local Access Problem (MCLA) Definition 4.2 The multicenter local – access problem is to find a set of trees T1, … , Tk such that (1) Exactly 1 backbone site belongs to each tree (2) (3) is a minimum

An example Circle  3 backbone nodes X, Y and Z Square  17 access nodes A, B, C and D, etc

Comments: This problem is a bit more complex than the single-center one Suppose we have n access nodes that we want to partition into 3 sets. The number of possible partitions is : Each partition of the access sites results in 3 capacitated MST problems each of which can be attacked by the Esau-Williams algorithm Even for the modest number 17, the complexity is daunting!

Nearest-Neighbor Esau-Williams (NNEW) For each b in B, let Sb={ nN | Cost(n,b) < Cost(n,b’) b’B} If n is equidistant between several backbone nodes, add n to one Sb at random. Use Esau-Williams to construct a capacitated MST on each set bSb. Example: A definitely belongs to X (since X is not only the closest backbone node to A; it is almost the closest node to A) B, C and D not clear

Creditability Test Let us look at two failed examples. Test: reattach the leaves to a different tree and see if it reduces the cost. The creditability of NNEW is not good Let us look at two failed examples.

Example 1: a 10-site bad design

Example 2: another 10-site bad design

What do we deduce from the 2 examples? Design Principle 7.1 In local-access design with multiple centers, the location of the other access nodes cannot be ignored when deciding which access nodes should home to which center.

MultiCenter Esau-Williams algorithm (MCEW or Kershenbaum-Chou) A variant of the original Esau-Williams algorithm Recall that, in Esau-Williams algorithm, we calculate the tradeoff as the saving by linking Ni to Nj instead of linking it directly to the center. Tradeoff(Ni)=minjCost(Ni, Nj) - Cost(Comp(Ni),Center) MCEW algorithm replaces the tradeoff function as: Tradeoff(Ni)=minjCost(Ni, Nj) - Cost(Comp(Ni),Center(Ni)) Initially, we set Center(Ni) to be the closest center. As node Ni is merged with node Nj, update Center(Ni)=Center(Nj).

NNEW vs. MCEW Only a slight cost advantage of using MCEW as opposed to NNEW. MCEW is far more creditable than NNEW.

Practical Suggestions From GÖdel’s theorem: Given any set of algorithms, it is always possible to formulate a problem for which they provide no good solution. Design Principle 7.2 The designer needs to be inventive and agile when dealing with unusual constraints.

Some access trees contain too many nodes. EW tests only if the combined weight of the two components doesn’t exceed the upper bound weight_limit. Solution: add additional size_limit constraints that prohibit the merge of two components with too many nodes.

Some access trees contain too many hops. Solution: add depth checking constraint, i.e., depth-limit the tree built by EW Each site maintains a value depth[ni] Initially set to 1, update when we evaluate the tradeoff between n1 and n2, Depth[n2] = max (depth[n2], depth[n1] +1) and compare against threshold.

Some site in the access tree has too many links Solution: add degree constraint, or a “valence” constraint. Initialize the valence of each site to 1, when we accept the merge from n1 to n2 then we increase the valence of n2 by 1. Do not accept merges that violates the constraint.

A central site has too many links Solution 1: Modify the tradeoff function as Tradeoff(Ni)=minj Cost(Ni,Nj)-a*Cost(Comp(Ni),Center)) where a > 1. By adjusting a we limit the links that connect to a center. Solution 2: If we have multiple centers and EW has overloaded a given site, use NNEW or MCEW with initial assignment of centers overridden to low utilization center.

Some site fails too often If a given site is known to have availability problems, it shouldn’t be an interior point. Solution: Mark it as not being able to be set as a predecessor of other sites.

To overcome such constraints without the ability to rewrite programs is hopeless! Design Principle 7.3 Designers need to be able to modify algorithms to deal with unusual constraints. This may necessitate adding a programmer to the design team but it will be well worth the effort and expense. The programmer needs to know the code and to understand the idea that is being carried out in the algorithm. This requires a deeper understanding than can be obtained from reading the comments in the existing code.

HW #7 Due Date: July 04 Design a connected topology that connects nodes 1, …, 4 to center 0. The link types are listed below. Cost tables are listed on the next slide. NODE WEIGHT 1 20,000 2 2,400 3 9,600 4 4,800 Link Type Capacity 4.8 K 1 28 K 2 64 K

  1 2 3 4 - 5 7 6   1 2 3 4 - 10 15 12   1 2 3 4 - 20 25 22   Link 2   Link 1   Link 0 Cost Tables

THE END Thank you!