Data Networks 5'3, 5'4

1
Data Networks 5.3, 5.4

• Daniel Gau
• Graduate Institute
• Information Management Dept.
• National Taiwan University
• R90057_at_im.ntu.edu.tw

2
outlines

• Coping with link failures when broadcasting
  routing information
• Flow models
• Optimal routing
• Topology design

3
Difficulties

• Topological update information must be
  communicated over links that are themselves
  subject to failure.
• One has to deal with multiple topological changes
• New topological update information may arrive as
  the old algorithm is running
• The repair of a single link. The algorithm must
  ensure that eventually the two parts agree and
  adopt the correct network topology.

4
The flooding algorithm in its pure form fails

• n is initially up, then it goes down, then up
  again.
• AC fails after the first update.

5
Assumptions

• Network links preserve the order and correctness
  of transmissions.
• Nodes maintain the integrity of messages that are
  stored in their memory.
• A link failure can be detected by both end nodes
  of the link, though not simultaneously.
• If one of the end nodes declares the link to be
  up, then within finite time, either the opposite
  end node also declares the link up or the first
  declares it down again.

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Assumptions (contd)

• A node can crash, in which case each of its
  incident links is, within finite time, declared
  down by the node at the opposite end of the link.
• The preceding assumptions are not always
  satisfied in practice, one must weigh the
  consequences of the exceptional case where they
  are not.

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Flooding The ARPANET Algorithm

• The time interval between update broadcasts by
  the node varies from 10 to 60 sec, depending on
  whether there has been a substantial change in
  any single-link average delay or not.
• Each message includes a time average of packet
  delay on each outgoing link from the node.

8
Example of difficulties with certain forms of
flooding

• Rules a node that receives a message relays it
  to all of its neighbor nodes except the one from
  which the message was received.
• This form of flooding works only if the network
  has no cycles.

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Flooding The ARPANET Algorithm (contd)

• In the ARPANET, update messages from originated
  at some node i, it checks to see if its sequence
  number is greater than the sequence number of the
  message last received from i.
• If so, the message together with its sequence
  number is stored in memory, and its contents are
  transmitted to all neighbors of j except the one
  from which the message was received
• Otherwise, the message is discarded.

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Flooding The ARPANET Algorithm (contd)

• Problems with sequence numbers
• There may be no way for the two separated
  portions to figure out the correct topology after
  they connect again by relying on the sequence
  number scheme alone
• A sequence number could be altered either inside
  a nodes memory due to malfunction, or due to an
  undetected error in transmission.

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How ARPANET resolves the problems associated with
seq

• Each update message includes an age field, which
  indicates the amount of time that the message has
  been circulating inside the network.
• In addition to the messages transmitted when a
  link status change is detected, each node is
  required to transmit update messages
  periodically.(at least one from every node every
  60 sec.)

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Flooding without periodic updates

• Method modify the previous flooding rules
• requirement The seq field must be so large
  that wraparound never occurs.
• A zero seq is allowed only when the node is
  recovering from a crash.

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Rules

• When bringing up links that have been down, the
  end nodes of the link, in addition to
  broadcasting a regular update message on all of
  their outgoing links, should exchange their
  current views of the network
• By which we mean that they should send to each
  other all the update messages stored in their
  memory that originated at other nodes together
  with the corresponding sequence numbers.

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Rules (contd)

• When node j receives an update message A that has
  originated from i, and let B be the message
  originated at node i and currently stored in node
  js memory
• AltB discarded
• AB discarded
• AgtB
• if , j stores A in its memory and sends a
  copy of A on all its incident links except the
  one on which A was received.
• if i j, node j sends to all its neighbors a new
  update message carrying the current status of all
  its outgoing links together with a seq that is
  1 plus the seq

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Why comapre A and B when the seq are the same

• Initially, both links are up. (2,3) down, (1,2)
  down, (2,3) comes up again

2
1
3
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Broadcast without Seq

• We wish to have an algorithm by which all nodes
  get to know the correct value of V after a finite
  time from the last change. V represent the
  status of the incident links of the center
• Vi each node i stores in memory a value Vi,
  which is an estimate of the correct value of V.
• each node i stores the last value received by
  each of its neighbors j, denoted as

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General rules for changing the estimate Vi of
each node i

• Suppose we have an algorithm that maintains a
  tree of directed paths from all nodes to the
  center.
• Each node except for the center has a unique
  successor s(i) in such a tree.
• Node i make Vi equal to the value last
  transmitted by the successor s(i) within a finite
  time after either the current successor transmits
  a new value or the successor itself changes.

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Information broadcast over a tree rooted at the
center that holds the value V
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SPTA (Shortest Path Topology Algorithm)

• The tree construction algorithm and the
  topological change broadcast procedure are
  integrated into a single algorithm
• The data structure each node i stores
• The Main Topology Table
• Stores the believed status of every link
• Official table used by the routing algorithm at
  the node
• The Port Topology Table
• There is one such table for each neighbor node j.

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Data structures of SPTA at node i
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Communication Rules

• When a link status entry of a nodes main
  topology tables changes, the new entry is
  transmitted on each of its operating incident
  links.
• When a failed link comes up, each of its end
  nodes transmits its entire main topology table to
  the opposite end node over that link. Upon
  reception these entries, the opposite end node
  enters the new link status in its main and port
  topology tables.

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Topology Table Update Rules

• When an incident link of a node fails, the failed
  status is entered into the nodes main and port
  topology tables.
• When receiving a link status message from a
  neighbor, it enters the message into the port
  topology table associated with the neighbor
• When entries changes according to the above
  situation, node i update its main topology table
  by using the following algorithm.

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Main Topology Update Algorithm

• At the start of Kth iteration, there is a set of
  nodes . Each nodes m in has a label,
  which is the ID number of the neighbor of i that
  lies on the path from i to m that has a minimum
  number of links.
• Example, , each node in it is directly
  connected to i and the label is their own id.
• At Kth iteration, will be augmented by new
  nodes to form or the algorithm terminates.

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Main Topology Update Algorithm (contd)

• Status of links ( at least end node in but
  no node in ) is entered into Main Topology
  Table.
• (m,n) m ? , n ? , m or
  n(or both) ?
• For notational completeness

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Main Topology Update Algorithm (contd)

• Step 1
• Link (m,n) ? , let m ? , let ms label
  j
• Copy status of (m,n) from port topology to
  .
• If status is up and n ? , give to node n the
  label j.

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Main Topology Update Algorithm (contd)

• Step 2
• Let be the nodes labeled in step 1.
• If is empty, the algorithm terminates
• Otherwise, set and do
  iteration

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Example
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Example (contd)
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Properties of the main topology update algorithm

• the set of nodes m with the property that
  in the topology finally obtained, there is a
  path of k or fewer up links connecting m with i.
• (kgt1) all paths of up links connecting
  one of the end nodes of l with node i has no
  fewer than k links, and there is exactly one such
  path with k links.
• When an incident link (i,j) fails, then the
  information from port topology is in effect
  disregarded.

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Operation of the SPTA for a single-link status
change
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Operation of the SPTA for a multi-link status
change
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Operation of the SPTA for a multi-link status
change
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Chapter 5.4

• Flow models, optimal routing, and topological
  design

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Routing Optimization

• cost function for optimization
• flow of link (i,j)
• A frequently used formula
• Another cost function

is the propagation delay
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Routing Optimization (contd)

• A computational study Vas79 has shown that it
  typically make little difference whether the cost
  function of the previous slide is used for
  routing optimization.

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Formulate a problem of optimal routing

• the input traffic arrival rate for each
  pair w (i,j)
• W set of all OD pairs
• set of all directed paths connecting the
  origin and destination nodes of OD pair w
• flow (data units/sec) of path p

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Formulate a problem of optimal routing (contd)

• The collection of all path flows
• must satisfy the
  constraints
• for all w ? W,
• for all p ? , w ? W

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Schema representation
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Formulate a problem of optimal routing (contd)

• The total flow of link (i,j) is the sum of
  all path flows traversing the link

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Formulate a problem of optimal routing (contd)

• Subject to for all w ? W
• for all p ? , w
  ? W

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Problem

• Algorithms for its solution are given in Section
  5.6 and 5.7.
• The underlying hypothesis here is that one
  achieves reasonably good routing by optimizing
  the average levels of link traffic without paying
  attention to other aspects of the traffic
  statistics.

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Routing based on queue length information
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Overview of Topological Design Problems

• Concept
• Given a set of traffic demands, we want to put
  together a network that will service these
  demands at minimum cost while meeting some
  performance requirements.

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Overview (contd)

• Given conditions
• Geographical location of a collection of
  devices(terminals) that need to communicate with
  each other
• A traffic matrix giving the input traffic flow
  from each terminal to every other terminal

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Overview (contd)

• What we want to design
• Topology of a communication subnet to service
  the traffic demands of the terminals
• Location of the nodes
• Choice of links
• Capacity of each link
• Local access network i.e., the collection of
  communication lines that will connect the
  terminals to entry nodes of the subnet

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illustration
Local access network
Subnetwork
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Design Objectives

• Keep the average delay per packet or message
  below a given level
• Satisfy some reliability constraint to guarantee
  the integrity of network service in the face of a
  number of link and node failure
• Minimize a combination of capital investment and
  operating costs while meeting the above
  objectives 1 and 2.

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Subnet design problem

• Assumptions
• Local access network has been designed
• Matrix of input traffic flow for every pair of
  subnet nodes is known
• We want to select the capacity and flow of each
  link so as to meet some delay and reliability
  constraints while minimizing costs.
• This turns to be a difficult combinational
  problem as illustrated by the following problem

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Subnet design problem (contd)

• The problem is to choose the capacity of each
  link (i,j), so as to minimize the linear cost
• Subject to (average
  delay constraint)
• is a known positive price per unit
  capacity
• ? is the total arrival rate into the network

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Subnet design problem (contd)

• When the flows are known, the problem is to
  minimize the linear cost
• over the capacities
• We introduce a Lagrange multiplier ?, and form
  the Lagrangian function

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Subnet design problem (contd)

• Set the partial derivatives to zero
• Solving for gives

(1)
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Subnet design problem (contd)

• Substituting in the constraint equation, we can
  obtain
• And from the last equation

(2)
(3)
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Subnet design problem (contd)

• Last equation is substituted in Eq. (1) yields
  the optimal solution
• After rearranging

(4)
(5)
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Subnet design problem (contd)

• We can get the optimal cost

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Subnet design problem (contd)

• There is a tendency for many of the flows
  and corresponding capacities to be zero
• Is unrealistic because , in practice, capacity
  costs are not linear
• The capacity of a line is typically equal in both
  directions, whereas this is not necessarily true
  for the optimal capacities of the above equation

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Heuristic methods for capacity assignment

• Assumptions
• The nodes of the network and the input traffic
  flow for each pair of nodes are known.
• A routing model has been adopted
• There is a delay constraint must be met.
• There is a reliability constraint must be met.
• There is a cost criterion

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Steps of the iteration

• Assign Flows by means of some routing algorithm
• Check Delay
• If go to step 3 D can be, for example,
• Else go to step 5
• Check Reliability
• If the constraints are not met, go to step 5
• Else go to step 4

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Steps of the iteration(contd)

• Check Cost Improvement
• If the cost of the trial topology is less than
  the cost of the current best topology, replace it
• Generate a New Trial Topology
• Use some heuristic to change one or more
  capacities of the current best topology,
  obtaining a trial topology. Go to step 1

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Step5 Create New Topology

• Branch exchangeone link is deleted and another
  link is added
• Saturated cut method(how to choose the links to
  be added and deleted)
• Identify a partition of the nodes into two sets
  N1 and N2 such that the links joining N1 and N2
  are highly utilized.
• It seems plausible that adding a link between a
  node in N1 and a node in N2 will help reduce the
  high level of utilization across the cut.

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Saturated cut method

• Prepare a list of all undirected links (i,j),
  sorted in order of decreasing utilization as
  measured by
• Find a link k such that
• If all links above k on the list are eliminated,
  the network remains connected
• If k is eliminated together with all links above
  the list, the network separates in two
  disconnected component N1 and N2

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Saturated cut method (contd)

• Remove the most underutilized link in the
  network, and replace it with a new link that
  connects a node in the component N1 and a node in
  the component N2.

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Example
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Step3 Check Reliability

• K-connected
• We say two nodes i and j in an undirected graph
  are k-connected if there is a path connecting i
  and j in every subgraph obtained by deleting
  (k-1) nodes other than i and j together with
  their adjacent arcs from the graph.
• Arc-connectivity
• The minimum number of arcs that must be deleted
  before the graph becomes disconnected

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The definition of k-connectivity
Graph(b) is 2-connected
Node 1 and 2 are 6-connected 2 and 3 are
2-connected 1 and 3 are 1-connected Graph(a) is
1-connected
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Check Reliability (contd)

• It can be shown that i and j are k-connected if
  and only if either i and j are connected with an
  arc or there are at least k node-disjoint paths
  connecting i and j.
• Its possible to calculate the number of
  node-disjoint paths connecting two nodes i and
  j.This can be done by setting up and solving a
  max-flow problem

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Check Reliability (contd)
To find the number of node-disjoint paths from A
to F
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Check Reliability (contd)

• One way to check k-connectivity of a graph is to
  check k-connectivity of every pair of nodes.
• Another particular method by Kleitman
• Choose an arbitrary node and check
  k-connectivity between that node and every other
  node.
• Delete and its adjacent arcs

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Check Reliability (contd)

• Another particular method by Kleitman
• Choose another node and check
  (k-1)-connectivity between that node and every
  other node.
• Continue in this manner until node is
  checked to be 1-connected to every remaining
  node, or (k-i)-connectivity of some node , i
  0,1,....k-1, to every remaining node can not be
  verified.

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illustration
1
3
2
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Spanning tree topology design

• For some networks where reliability is not a
  serious issue, a spanning tree topology may be
  appropriate.
• A potential difficulty arises when there is a
  constraint on the amount of traffic that can be
  carried by any one link.
• A modified Kruskal or Prim-Dijkstra algorithms
  can be used.

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Essau-Williams algorithm

• A special case of the constrained MST(minimum
  weight spanning tree)
• All traffic must go through the central node, so
  it can be assumed that there is input traffic
  only from the noncentral nodes to the central
  node and the reverse.
• Its a branch exchange heuristic

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Essau-Williams algorithm steps

• Starts with the spanning tree where the central
  node is directly connected with each of the N
  other nodes. (assumed to be a feasible solution)
• At each successive iteration, a link (i,0)
  connecting some node i with the central node 0 is
  deleted from the current spanning tree, and a
  link (i,j) is added

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Essau-Williams algorithm steps(contd)

• These links are chosen so that
• No cycles formed
• The capacity constraints of all the links of the
  new spanning tree are satisfied
• The saving in link weight obtained by
  exchanging (i,0) with (i,j) is positive and is
  maximized over all nodes i and j for which 1 and
  2 are satisfied.

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Essau-Williams algorithm steps(contd)

• The algorithm terminates when there are no nodes
  i and j for which requirements 1 to 3 are
  satisfied when (i,0) is exchanges with (i,j).

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Illustration of the Essau-Williams
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Local Area Network Design problem

• Assumptions a communication subnet is available
• Objective to design a network that connects a
  collection of terminals with known demands to the
  subnet
• Its difficult to recommend a global design
  strategy without knowledge of the given practical
  situation.

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Concentrator problem

• There are n sources located at known geographical
  points. Each source may be a gateway of a local
  area network that connects several terminals.
• The problem is to connect the n sources to m
  concentrators that are themselves connected to a
  communication subnet.

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Concentrator problem example
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Concentrator problem (contd)

• the cost of connecting source i to
• concentrator j

If source i is connected to concentrator j
otherwise
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Concentrator problem (contd)

• The total cost of a source assignment specified
  by a set of variables is
• Assume that each source is connected to only one
  concentrator

(1)
for all sources i (2)
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Concentrator problem (contd)

• There is a maximum number of sources that
  can be handled by concentrator j.
• The problem is to select the variables so as to
  to minimize the cost (1) subject to the
  constraints (2) and (3)

for all concentrators j (3)
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Concentrator problem (contd)

• If we change by
• the problem becomes a linear transportation
  problem that can be solved by very efficient
  algorithms such as the simplex method. It will be
  shown that the solution obtained from the simplex
  method will be integer.

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Further discussion

• When the location of the concentrator is not
  fixed, and is subject to optimization.
• There are m potential concentrator sites and
  there is a cost for locating a concentrator
  at site j.

If a concentrator is located at site j
where
otherwise
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Presentation end
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Tendency for low connectivity

• We want to choose the capacities C1,..., Cn so as
  to minimize the sum C1... Cn while dividing the
  input rate ? among the n links so as to meet an
  average delay constraint
• The optimal solution is the minimal connectivity
  topology to meet the delay constraint. (3.1
  and 3.3)