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## Optical Switching Networks

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### Introduction

### Optical Network Design

References

- Work by
- Ngo,Qiao,Pan, Anand, Yang
- Chu/Liu/Zhang
- Pippinger/Feldman/Friedman
- Winkler/Haxell/Rasala/Wilfong

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):Thustwo 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 limited-LWC)
- 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).

Definitions, Examples and Theoretical Results

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, Fj ) where i is a wavelength, Fj is an input fiber and Fj 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]), fbn 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.

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 cthoutput 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.

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

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.

ComplexityUsingGraph/Theoretical Representation

- (Ngo) Graph Theoretical representation.
- a) Fiber-channels get replaced by vertices
- b) Edges ~ capacity

ComplexityUsingGraph/Theoretical RepresentationExample

- 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

ComplexityUsingDAG GT RepresentationRigorous 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.

ComplexityUsingGraph/Theoretical RepresentationRigorous 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

|A|=|B|=wf and B=B1 B2 ... Bf

- 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.

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