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 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).

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, 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]), 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.

ComplexityUsing Graph/Theoretical Representation

• (Ngo) Graph Theoretical representation.

• a) Fiber-channels get replaced by vertices

• b) Edges ~ capacity

ComplexityUsing 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

ComplexityUsing 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.

ComplexityUsing 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

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

• We can now define request, request frame, route, and RNB/SNB/WSNB [w,f]-network.

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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.