Elements%20of%20Virtual%20Topology%20Design

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Chapter 8

• Elements of Virtual Topology Design

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Outlines

• Introduction
• System Architecture
• Formulation of the Optimization Problem
• Algorithm
• Experimental Results
• Summary

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Introduction

• This chapter explores design principles for
  next-generation optical wide-area networks,
  employing wavelength-division multiplexing (WDM)
  and targeted to nationwide and global coverage.
• This optical network exploits wavelength
  multiplexers and optical switches in routing
  nodes, so that an arbitrary virtual topology may
  be embedded on a given physical fiber network.
• The virtual topology, which is operated as a
  packet-switched network and which consists of a
  set of all-optical "lightpaths," is set up to
  exploit the relative strengths of both optics and
  electronics - viz., packets of information are
  carried by the virtual topology "as far as
  possible" in the optical domain using optical
  circuit switching.
• But packet forwarding from lightpath to lightpath
  is performed via electronic packet switching,
  whenever required.

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Introduction

• This chapter examines an "optical" wide-area WDM
  network which utilizes wavelength multiplexers
  and optical switches in a routing node so that an
  arbitrary virtual topology can be embedded on a
  given physical fiber network.
• The virtual topology, which is packet switched
  and which consists of a set of "all-optical
  lightpaths," is set up to exploit the relative
  strengths of both optics and electronics - viz.,
  packets of information are carried by the virtual
  topology "as far as possible" over the same
  wavelength in the optical domain (i.e., there is
  no wavelength conversion in a lightpath), but
  packet forwarding from lightpath to lightpath is
  performed via electronic packet switching,
  whenever required.
• Optical circuit switching at a node is achieved
  by using a wavelength-routing switch (WRS), which
  is capable of optically bypassing a lightpath
  from an input fiber to an output fiber, without
  any electronic processing.
• Because there is no wavelength conversion in the
  WRS, the wavelength of the lightpath stays the
  same in the output fiber as it was in the input
  fiber.

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• This architecture is a combination of the
  well-known "single-hop" and "multihop"
  approaches, and it attempts to exploit the
  characteristics of both.
• A "lightpath" in this architecture provides
  "single-hop" communication.
• However, by employing a limited number of
  wavelengths, it may not be possible to set up
  "lightpaths" between all user pairs as a result,
  "multihopping" between "lightpaths" may be
  necessary.
• In addition, when the prevailing traffic pattern
  changes, a different set of "lightpaths" forming
  a different "multihop" virtual topology may be
  more desirable.
• A networking challenge is to perform the
  necessary reconfiguration with minimal disruption
  to the network's operation.

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Introduction

• We formulate the virtual topology design problem
  as an optimization problem with one of two
  possible objective functions
• For a given traffic matrix, minimize the
  network-wide average packet delay (corresponding
  to a solution for present traffic demands), or
• Maximize the scale factor by which the traffic
  matrix can be scaled up (to provide the maximum
  capacity upgrade for future traffic demands).

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8.2 System Architecture (NSFNET)
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System Architecture

• NSFNET T1(1.544Mbps).
• Electronic packet switching
• Optical fiber
• Irregular mesh structure
• Store-and-Forward packet switching is performed
  at network node. (delay)
• Fibers transmission bandwidth is not
  exploited(T1 rate, 1 wavelength)

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ATM WDM

• T1(or E3, T3, )?ATM ? WDM
• We expect that the underlying physical network
  will continue to be a fiber plant on which our
  WDM solution will still be applicable.
• Any future technology must be incrementally
  developable. (upgraded to support WDM, packet
  switch ? wavelength-routing switches, WRS).
• How the WDM solution can be used to upgrade an
  existing ATM solution.
• How the WDM solution can support both ATM and
  non-ATM services. (not discuss in this book)

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General Problem Statement

• The problem of embedding a desired virtual
  topology on a given physical topology (fiber
  network) is formally stated below.
• We are given the following inputs to the problem
• A physical topology Gp (V, Ep) consisting of a
  weighted undirected graph,
• V is the set of network nodes, and
• Ep is the set of links connecting the nodes.
• Bidirectional.
• Links are assigned weights, which may correspond
  to physical distances between nodes.
• A network node i is assumed to be equipped with a
  Dp(i) x Dp(i) wavelength-routing switch (WRS),
  where Dp(i), called the physical degree of node
  i, equals the number of physical fiber links
  emanating out of (as well as terminating at) node
  i.
• Number of wavelength channels carried by each
  fiber M.

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General Problem Statement

• An N x N traffic matrix,
• where N is the number of network nodes, and the
  (i, j)-th element is the average rate of traffic
  flow from node i to node j.
• Note that the traffic flows may be asymmetric,
  i.e., flow from node i to node j may be different
  from the flow from node j to node i..
• The number of wavelength-tunable lasers
  (transmitters) and wavelength tunable filters
  (receivers) at each node.

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Virtual topology

• A virtual topology Gv (V,Ev) as another graph
• the out-degree of a node is the number of
  transmitters at that node and
• the in-degree of a node is the number of
  receivers at that node.
• Each link between a pair of nodes in the virtual
  topology corresponds to a direct all-optical
  "lightpath" between the corresponding nodes in
  the physical topology.
• (Noting that each such link of the virtual
  topology may be routed over one of several
  possible paths on the physical topology, an
  important design issue is "optimal routing" of
  all lightpaths so that the constraint on having a
  limited number of wavelengths per fiber is
  satisfied.

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Virtual topology design

• A wavelength assignment for lightpaths, such that
  if two lightpaths share a common physical link,
  they must necessarily employ different
  wavelengths.
• The size and configuration of the WRSs at the
  intermediate nodes.
• Communication between any two nodes now takes
  place by following a path (a sequence of
  lightpaths) from the source node to the
  destination node on the virtual topology.
• Each intermediate node in the path must perform
• (1) an opto-electronic conversion,
• (2) electronic routing (possibly ATM switching,
  if needed), and (3) electrooptic forwarding onto
  the next lightpath.

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Modified Physical topology
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Virtual topology
Physical link
Virtual link
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Node Switch architecture
Optical component
Electronic component
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Node Switch architecture

• The nodal switching architecture consists of an
  optical component and an electronic component.
• The optical component is a wavelength-routing
  switch (WRS),
• optically bypass some lightpaths, and
• can locally terminate some other lightpaths by
  directing them to node's electronic component.
• The electronic component is an electronic packet
  router (which may be an ATM switch),
• serves as a store-and-forward electronic overlay
  on top of the optical virtual topology.

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Hypercube embedding studied in MRBM94
5 wavelengths
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Hypercube embedding studied in MRBM94
Virtual path from CA1 to NE
7 wavelengths
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8.3 Formulation of the Optimization Problem

• We formulate the problem as an optimization
  problem, using principles from multicommodity
  flow for physical routing of lightpaths and
  traffic flow on the virtual topology, and using
  the following notation
• s and d used as subscript or superscript denote
  source and destination of a packet, respectively,
• i and j denote originating and terminating nodes,
  respectively, in a lightpath, and
• m and n denote endpoints of a physical link that
  might occur in a lightpath.

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Given

• Number of nodes in the network N.
• Maximum number of wavelengths per fiber M (a
  system-wide parameter).
• Physical topology Pmn, where
• Pmn Pnm 1 if and only if there exists a
  direct physical fiber link between nodes m and n,
  where m, n 1, 2, 3, . . ., N
• Pmn Pnm 0 otherwise (i.e., fiber links are
  assumed to be bidirectional).
• Distance matrix, viz., fiber distance dmn from
  node m to node n.
• For simplicity in expressing packet delays, dmn
  is expressed as a propagation delay (in time
  units).
• Note that dmn dnm since fiber links are
  bidirectional, and dmn 0 if Pmn 0.
• Number of transmitters at node i Ti (Ti ?_
  1).
• Number of receivers at node i Ri (Ri ? 1).

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Given

• Traffic matrix ?sd which denotes the average rate
  of traffic flow from node s to node d, with ?sd
  0 for s, d 1, 2, . . ., N.
• (packet per second)
• Capacity of each channel C (normally expressed
  in bits per second, but converted to units of
  packets per second by knowing the mean packet
  length).

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Variables

• Virtual topology The variable Vij 1 if there
  exists a lightpath from node i to node j in the
  virtual topology Vij 0 otherwise.
• Note that this formulation is general since
  lightpaths are not necessarily assumed to be
  bidirectional. (Vij 1 ?Vji 1)
• Traffic routing The variable ?ijsd denotes the
  traffic flowing from node s to node d and
  employing Vij as an intermediate virtual link.
• Note that traffic from node s to node d may be
  bifurcated(??) with different components taking
  different sets of lightpaths.
• Physical topology route The variable pmnij 1
  if the fiber link Pmn is present in the lightpath
  for virtual link Vij pmnij 0 otherwise.
• Wavelength color The variable ckij 1 if a
  lightpath from originating node i to terminating
  node j is assigned the color k, where k 1, 2, .
  . ., M ckij 0 otherwise.

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Constraints
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Constraints
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Constraints
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Constraints
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Objective
Propagation delay on link mn from lightpath ij
queuing delay and trans. delay on lightpath ij
(M/M/1)
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Formulation

• Conflict-free routing
• Two lightpaths that share a fiber should not be
  assigned the same wavelength.
• If we assume shortest path routing of the
  lightpaths over the physical topology, then the
  pmnij values become deterministic. If , in
  addition, neglect queuing delay, the optimization
  problem is Equ.(8.15) reduces to

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Algorithms

• The optimization problem in Section 8.3 is
  NP-hard, since several subproblems of this
  problem are NP-hard.
• The problem of optimal virtual topology design
  can be partitioned into the following four
  subproblems, which are not necessarily
  independent
• determine a good virtual topology, viz., which
  nodal transmitter should be directly connected to
  which nodal receiver,
• route the lightpaths over the physical topology,
• assign wavelengths optimally to the various
  lightpaths (this problem has been shown to be
  NP-hard in ChGK93, chapter 10), and
• route packet traffic on the virtual topology (as
  in any packet-switched network).

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Previous Works

• The problem of designing optimal virtual
  topologies has been studied before BaFG90,
  LaAc91.
• Our formulation is more general in the sense that
  we accommodate many of the physical connectivity
  constraints which were not considered earlier.
• In general, the optimal virtual topology problem
  has been conjectured to be NP-hard, which means
  that the problem cannot be solved optimally for
  large problem sizes, unless one resorts to some
  form of exhaustive search.
• One instance of this problem has been formulated
  as a mixed integer linear program which gets
  difficult to solve with increasing problem size
  LaAc91.
• Accordingly, heuristic approaches have been
  employed to solve these problems BaFG90, LaAc91.

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Previous Works

• Related work on these problems can be found in
  ChGK93, MRBM94, RaSi95, ZhAc94.
• Embedding of a packet-switched virtual topology
  on a physical fiber plant in a switched network
  was first introduced in ChGK93, and this
  network architecture was referred to as a
  lightnet.
• Some algorithms to embed a hypercube virtual
  topology were provided in ChGK93, MRBM94.
• The work in RaSi95 proposes a virtual topology
  design for packet-switched networks. The average
  hop distance is minimized, which automatically
  increases the network traffic supported.
• The work in RaSi95 uses the physical topology
  as a subset of the virtual topology, employing
  algorithms for maximizing the throughput subject
  to bounded delay characteristics.

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Solution Method

• We employ an iterative approach consisting of
  simulated annealing to search for a good
  virtual topology. (sub-problem 1) in conjunction
  with the flow deviation algorithm for optimal
  routing of packet traffic on the virtual topology
  (sub-problem 4).
• Start with a random configuration (virtual
  topology) and try to find a good virtual topology
  through simulated annealing by using
  node-exchange (similar to branch-exchange
  LaAc91) techniques.
• Then, scale up the traffic matrix to ascertain
  the maximum throughput that can be accommodated
  by the virtual topology, using flow deviation for
  packet routing over the virtual topology.
• For a given traffic matrix, the flow-deviation
  algorithm minimizes the network-wide packet delay
  by properly distributing the flows on the virtual
  links (to reduce the effect of large queueing
  delays).

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Local search algorithms

• In many optimization problems, the path to the
  goal is irrelevant the goal state itself is the
  solution
  
• State space set of "complete" configurations
• Find configuration satisfying constraints,
• e.g.,
• (1) find optimal configuration (e.g., TSP),
  or,
• (2) find configuration satisfying constraints
  (n-queens)
• In such cases, we can use local search algorithms
• keep a single "current" state, try to improve it

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Iterative improvement

• Optimization problem.
• Objective function.
• In such cases, can use iterative improvement
  algorithms keep a single current state, and
  try to improve it.

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Example n-queens

• Put n queens on an n n board with no two queens
  on the same row, column, or diagonal.
• Complete configuration (states)

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Hill-climbing search

• Problem depending on initial state, can get
  stuck in local maxima
  

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Hill-climbing search

• "Like climbing Everest in thick fog with amnesia"
  

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Local Minima Problem

• Question How do you avoid this local minima?

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Consequences of the Occasional Ascents
desired effect
Help escaping the local optima.
adverse effect
(easy to avoid by keeping track of best-ever
state)
Might pass global optima after reaching it
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Hill-climbing search 8-queens problem

• h number of pairs of queens that are attacking
  each other, either directly or indirectly
• h 17 for the above state

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Modify Hill-Climbing

• Sideway move
• Stochastic hill climbing
• First-choice hill climbing.
• Random-restart hill climbing

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Simulated annealing basic idea

• From current state, pick a random successor
  state
• If it has better value than current state, then
  accept the transition, that is, use successor
  state as current state
• Otherwise, do not give up, but instead flip a
  coin and accept the transition with a given
  probability (that is lower as the successor is
  worse).
• So we accept to sometimes un-optimize the value
  function a little with a non-zero probability.

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Simulated annealing

• Kirkpatrick et al. 1983
• Simulated annealing is a general method for
  making likely the escape from local minima by
  allowing jumps to higher energy states.
• The analogy here is with the process of annealing
  used by a craftsman in forging a sword from an
  alloy.
• He heats the metal, then slowly cools it as he
  hammers the blade into shape.
• If he cools the blade too quickly the metal will
  form patches of different composition
• If the metal is cooled slowly while it is shaped,
  the constituent metals will form a uniform alloy.

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Simulated annealing in practice

• set T
• optimize for given T
• lower T
• (see Geman Geman, 1984)
• repeat

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Simulated annealing in practice

• set T
• optimize for given T
• lower T (see Geman Geman, 1984)
• repeat
• Geman Geman (1984) if T is lowered
  sufficiently slowly (with respect to the number
  of iterations used to optimize at a given T),
  simulated annealing is guaranteed to find the
  global minimum.
• Caveat this algorithm has no end (Geman
  Gemans T decrease schedule is in the 1/log of
  the number of iterations, so, T will never reach
  zero), so it may take an infinite amount of time
  for it to find the global minimum.

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Simulated annealing algorithm

• Idea Escape local extrema by allowing bad
  moves, but gradually decrease their size and
  frequency.

Note goal here is to maximize E.
-
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Simulated Annealing

• Simulated annealing (along with genetic
  algorithms) has been found to provide good
  solutions for complex optimization problems
  (AaKo89).
• In the simulated annealing process, the algorithm
  starts with an initial random configuration for
  the virtual topology.
• Node-exchange operations are used to arrive at
  neighboring configurations.

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Node-exchange operation

• In a node-exchange operation,
• Neighboring configurations which give better
  results (lower average packet delay) than the
  current solution are accepted automatically.
• Solutions which are worse than the current one
  are accepted with a certain probability which is
  determined by a system control parameter.

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SA

• The probability with which these failed
  configurations are chosen, however, decreases as
  the algorithm progresses in time so as to
  simulate the "cooling" process associated with
  annealing.
• The probability of acceptance is based on a
  negative exponential factor and is inversely
  proportional to the difference between the
  current solution and the best solution obtained
  so far.
• The initial stages of the annealing process
  examine random configurations in the search space
  so as to obtain different initial starting
  configurations without getting stuck at a local
  minimum as in a greedy approach.
• However, as time progresses, the probability of
  accepting bad solutions goes down, and the
  algorithm settles down into a minimum after
  several iterations.
• The state become "frozen" when there is no
  improvement in the objective function of the
  solution even after a large number of iterations.
  For further information on simulated annealing,
  see AaKo89.

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Flow Deviation Algorithm

• L.Fratta, M.Gerla, and L.Kleinrock. The Flow
  Deviation Method An Approach to
  Store-and-forward Network Design, Networks, 3,
  pp. 97-133, 1973.
• http//www.elet.polimi.it/upload/martigno/qos/node
  11.html
• Flow Deviation Method which allows to determine
  the optimal routing of all the flows entering the
  network at different source/destination pairs.
• By properly adjusting link flows, the flow
  deviation algorithm FrGK73 provides an optimal
  algorithm for minimizing the network-wide average
  packet delay.

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FDA

• Traffic from a given source to a destination may
  be bifurcated.
• If the flows are not balanced, then excessively
  loading of particular channel may be lead to
  large delays on that channel, thus have a
  negative influence on the network-wide average
  packet delay.
• The algorithm is based on the notation of
  shortest-path flows which first calculates the
  linear rate of increase in the delay with an
  infinitesimal(????) increase in the flow on any
  particular channel.
• This length or cost rates are used to pose a
  shortest-path flow problem.
• The resulting paths represent the cheapest
  paths on which some of the flow may be deviated.
• An iteration algorithm determines how much of the
  original flow needs to be deviated. The algorithm
  continues until a certain performance tolerance
  level is reached.

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Flow Deviation method
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Experimental Results
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Results
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The End