Optical
Switching Networks
Presentation by
Joaquin Carbonara
References
• Work by
–
–
–
–
Ngo,Qiao,Pan, Anand, Yang
Chu/Liu/Zhang
Pippinger/Feldman/Friedman
Winkler/Haxell/Rasala/Wilfong
Introduction
Statement of the Problem
About Optical Networks
• Wavelength-routed all-optical WDM networks are
considered to be candidates for the next generation
wide-area Backbone networks [Chlamtac,Ganz
and Karmi, 1992 and Mukherjee, 2000]
• Wavelength Routed Network: wavelength routers
connected by fiber links (each being able to
support wavelength channels by supporting
WDM)
• WXC can be uni/multicast. OXC can be used
between processors in a parallel or distributed
system.
About Optical networks
• In a dynamic wavelength-routed WDM
network, limitations of the network may
result in some light-paths requests not being
satisfied.
• Goal: design all-optical networks that
minimizes blocking.
About Optical Networks
• Wavelength Continuity Constraint (which
makes Optical nets different than circuit-switched
telephone nets): Thus two light paths that share a
common fiber link should not be assigned the
same wavelength.
• Solution: Wavelength converters.
About Optical Networks
• Switching speed is the bottleneck at the core
of the optical network infrastructure
[Singhal and Jain, 2002]
• Goal: design cost-effective WXC that are
fast and easily scalable.
Design analysis
• RNB (rearrangeable non-blocking): a set of
requests submitted at once can be satisfied by the
network.
• SNB (Strictly non-blocking): a new request can be
satisfied without changing current request paths.
• WSNB (Wide sense non-blocking): a new request
can be satisfied using an (on-line) algorithm.
• SNB --> WSNO --> RNB
Design Analysis
• Cost of components is important.
• Number of different components:
– (de)multiplexors (MUX/DEMUX)
– Wavelenght converters (full-FWC or limitedLWC)
– Semiconductor Optical Amplifiers (SOA)
– Optical add-drop multiplexors (OADM)
– Arrayed Waveguide Grating Routers (AWGR)
Design Analysis
• Theoretical results help understand and
design networks
– Complexity is important (as a function of size)
• Size: number of edges in graph theoretical
representation
• Depth: number of edges in longest path of graph
theoretical representation.
Design tools
• Mathematical modeling
– Graph Theory; Theory of Discrete
Mathematics/Combinatorics; Functions
(Real/Integer valued, one or more variables);
Linear/Multilinear Algebra.
• In mathematics you don't understand things. You just get used to them.
von Neumann, Johann (1903 - 1957)
•
Mathematicians are a species of Frenchmen: if you say something to
them they translate it into their own language and presto! it is
something entirely different.
Goethe (German writer), Maxims and Reflexions, (1829)
Design tools
• Advantages of mathematical modeling:
– Many tools available since Mathematics is an
old and well established discipline
– True statements are backed by proofs (100%
guaranteed--if used properly).
– Math language is practically universal. This
guarantees a larger audience .
– Math organizes knowledge extremely well.
Design tools
• Disadvantages of Mathematical modeling
– It is hard to fit reality into a “nice” Theory
– Theory requires organized abstract thinking-not a very popular activity
Design Tools
• Other tools include simulation and analysis
(I will not talk about these tools).
Optical Network Design
Definitions, Examples and
Theoretical Results
Heterogeneous WDM
Cross-Connect
Components:
Wavelength Converters
• Wavelength converters: take as input wavelengths
coming on different fibers and can be programmed
to modify the wavelength and output modified
wavelength.
• To reduce cost, researchers have
– Used Limited Range Wavelength converters (LWC)
instead of Full Range Wavelength converters (FWC)
– Share wavelength converters among fiber links.
• Notation: LWC(A,B) takes inputs from set A and
produces outputs from set B.
Components:
AWGR
• Arrayed Waveguide Grating Routers:
–
–
–
–
–
Passive devices: reroute channels inside fibers
Easily available and inexpensive
Take m inputs and have m outputs fibers
Process wavelengths 0 to m-1
Wavelength i at input fiber j gets routed to the
same wavelength at output fiber (i-j)mod m.
Request Model
(understanding Nets blocking properties)
• Model 1 -- (, F, F): Requests are of the form (i,
Fj, Fj  ) where i is a wavelength, Fj is an input
fiber and Fj  is an output fiber. Requests requires
only an given output fiber, but do not specify the
output wavelength.
• Model 2 -- (, F, , F): More restrictive than
Model 1 since output wavelength is also
requested.
• Note: If N satisfies M2 then it satisfies M1
WXC-RNB construction for M1
(Ngo/Pan/Qiao infocom ‘04)
• Components: Let f=2, b=3, n=4. Then it has
f demultx, fbn LWC(Bi,[n]), fb n  n-AWG,
fbn LWC([n],[bc,b(f+c)]), n multx,
nb  nb-AWG, and f multx.
WXC-RNB-1 means ...
• RNB means that any set S of valid requests will
not be blocked in the network N. While in transit
inside the network, the Wavelength Continuity
Constrain must be satisfied.
• Valid request means
– no two requests will ask for the same input wavelength
and fiber.
– the number of requests asking for the same output fiber
cannot exceed the fiber capacity.
WXC-RNB-1 and GT
• Konig’s 1916 Theorem: Let G(U,V;E) be a
bipartite graph. Then: the maximum (vertex)
degree equals the chromatic index.
• Chromatic index= minimum number of
colors needed to edge color G so that
adjacent edges use different colors.
About Konig’s Theorem
Back to WXC-RNB-1...
• Represent the network as a bipartite graph
G(U,V;E) for the sole purpose of
determining a non-blocking route for each
request:
– The set U corresponds to the set of input bands
(there are fb of them)
– The set V corresponds to the set of output fibers
(there are f of them)
Graph of WXC-RNB-1
• Represent the network as a bipartite graph
G(U,V;E):
– Request (p, Fq, Fj)  edge (ui,vj)
where i = qb+ p/n
• By a simple variation of Konig’s theorem, the
graph G is colorable with n x b colors (label each
color with a tuple (c,d)), 1≤c ≤n and 1≤d ≤b, in
such a way that edges sharing a vertex in U have
different first color component.
Routing in WXC-RNB-1
• The basic idea is this:
1. [request (p, Fq, Fj)]  [edge (ui,vj)]  [color
(c,d)]
2. Then Route p so that it ends up in the cth output
line of its stage-1 AWGR.
3. Working from the other end, we want the request
to end in Fj. There are b fibers demuxing to it. We
can see that if the stage-2 LWC routes the
wavelength to its dth line of its demuxer, the
desired output is obtained.
Routing in WXC-RNB-1
• The basic idea is this (cont.):
4. The properties of the coloring inherited from
Konig’s theorem guaranteed non-blockiness.
Example (Ngo/Pan/Qiao):
WXC-RNB in Model-2
Other interesting results related
to non-blocking networks
• Strictly non-blocking networks are highly
desirable. It is difficult to build such
networks that are cost efficient.
• An interesting result (Ngo):
WXC-SNB-1 if and only if WXC-SNB-2
Haxel/Rasala/Wilfong/Winkler’s work on
WDM Cross-connects
On the news...
Optical Network Complexity
Graph Theoretical representations, Bounds,
minimizing the number of components.
Examples and theoretical results.
Complexity:
Minimizing the Number of LWC
• Results related to using the least possible
number of LWC on a uni/multicast network:
– Define LWC(d) when LWC can convert i to j
iff |i-j|≤d.
– Consider Homogenous Model-2 of requests
with w wavelengths and f fibers (HM2(w, f)).
– Want to study statistic m1(w,f,d) = least number
of LWC(d) needed if HM2(w, f) is SNB.
Complexity of WDM networks
(unicast) m1(w,f,d) even w (Ngo/Pan/Yang)
Complexity of WDM networks
(unicast) m1(w,f,d) odd w (Ngo/Pan/Yang)
Complexity:
Size and Depth using GT representation
• (Ngo) Using the DAG model (Directed
Acyclic Graphs) we can establish a formal
definition of size and depth of a network.
• Size: number of edges in the graph
• Depth: number of edges in the longest path.
Complexity
Using Graph/Theoretical Representation
• (Ngo) Graph
Theoretical
representation.
• a) Fiber-channels get
replaced by vertices
• b) Edges ~ capacity
Complexity
Using Graph/Theoretical Representation
Example
• Size of the network is
number of edges.
• Depth is longest path.
• It uses 2 2x2 AWG, 4
FWC 2 multiplexors
and 2 demultiplexors
• DAG=Directed
Acyclic Graph
Graph/Theoretical Representation
(Winkler/Haxell/Rasala/Wilfong)
Dynamic bipartite graphs
Complexity
Using DAG GT Representation
Rigorous Setting Model-2
• DAG model networks as follows:
– (n1,n2)-network is a DAG N=(V,E;A,B)
V=vertices, E=edges, A=inputs, B=outputs,
n1= |A|, n2=|B|.
• We can now define request, request frame, route,
RNB/SNB/WSNB network.
• Key idea: requests’ path must be disjoint to be
(simultaneously) realizable.
Complexity
Using Graph/Theoretical Representation
Rigorous Setting Model-1
• DAG model networks as follows:
– [w,f]-network is a DAG N=(V,E;A,B)
V=vertices, E=edges, A=inputs,
B=outputs
U
|A|=|B|=wf and B=B1 U B2 U ... UBf

• We can now define request, request frame,



route, and RNB/SNB/WSNB [w,f]-network.
Complexity:
DAG size
• Let an n-network be a Homogeneous Network
with n inputs and outputs. If the output is further
divided into f bands of size w (needed for M-2) we
call it a [w,f]-network.
• The smallest number of edges (size) for it to be
SNB, RNB, and WSNB is sc2(w,f), rc2(w,f) and
wc2(w,f) (Model 2), or sc1(n), rc1(n) and wc1(n)
(Model 2).
• rc1(n)≤wc1(n) ≤ sc1(n),
Complexity:
Results from DAG model
• M1 is less restrictive than M2 since M2 requests specify an
output wavelength. The following result shows that in the
SNB case there is no difference in cost between models:
Complexity:
RNB [w,f]-networks
• The size function has known estimates in
this case:
Complexity
• Advantages of having bounds:
– Number of edges can be related to network cost
– Theoretical results are often the only way to
gain experience with abstract systems.
Examples may be too poor or difficult to
concoct.
Complexity
• Other results include different ways of
create “atomic” networks, and operations to
create larger networks from smalles
– Left and right union
– The -product
Future Work
• Expansion of current models using different
models with the goal of eliminating blockiness
while reducing cost.
• Search for better bounds on the current statistics.
• Search for new meaningful statistics (is size and
depth the only ones that matter?) on GT
representations.