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Title: Elements%20of%20Virtual%20Topology%20Design


1
Chapter 8
  • Elements of Virtual Topology Design

2
Outlines
  • Introduction
  • System Architecture
  • Formulation of the Optimization Problem
  • Algorithm
  • Experimental Results
  • Summary

3
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.

4
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.

5
  • 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.

6
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).

7
8.2 System Architecture (NSFNET)
8
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)

9
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)

10
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.

11
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.

12
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.

13
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.

14
Modified Physical topology
15
Virtual topology
Physical link
Virtual link
16
Node Switch architecture
Optical component
Electronic component
17
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.

18
Hypercube embedding studied in MRBM94
5 wavelengths
19
Hypercube embedding studied in MRBM94
Virtual path from CA1 to NE
7 wavelengths
20
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.

21
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).

22
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).

23
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.

24
Constraints
25
Constraints
26
Constraints
27
Constraints
28
Objective
Propagation delay on link mn from lightpath ij
queuing delay and trans. delay on lightpath ij
(M/M/1)
29
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

30
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).

31
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.

32
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.

33
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).

34
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

35
Iterative improvement
  • Optimization problem.
  • Objective function.
  • In such cases, can use iterative improvement
    algorithms keep a single current state, and
    try to improve it.

36
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)

37
Hill-climbing search
  • Problem depending on initial state, can get
    stuck in local maxima

38
Hill-climbing search
  • "Like climbing Everest in thick fog with amnesia"

39
Local Minima Problem
  • Question How do you avoid this local minima?

40
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
41
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

42
Modify Hill-Climbing
  • Sideway move
  • Stochastic hill climbing
  • First-choice hill climbing.
  • Random-restart hill climbing

43
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.

44
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.

45
Simulated annealing in practice
  • set T
  • optimize for given T
  • lower T
  • (see Geman Geman, 1984)
  • repeat

46
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.

47
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.
-
48
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.

49
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.

50
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.

51
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.

52
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.

53
Flow Deviation method
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Experimental Results
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Results
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The End
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