Fundamental Complexity of Optical Systems

Hadas Kogan, Isaac Hadas Kogan, Isaac KeslassyKeslassy

Technion (Israel)Technion (Israel)

Router – schematic representationRouter – schematic representation

Problem - electronic routers do not scale to optical speeds:

Access to electronic memory is slow and power consuming.

Data conversions are power consuming as well.

Electronicto optic

Electronicto optic

…

Lookup Switching

Optic to electronic

Optic to electronic

…

Buffering

Router

Power consumption per chassisPower consumption per chassis

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1990 1993 1996 1999 2002 2003 2004

Po

wer

(kW

)

[Nick McKeown, Stanford]

There has to be some future alternative!

How about an optical router?How about an optical router? No electronic memory bottleneck No O/E/O conversions

BUT:An optical router is thought to be too complex.

Is it?

Optical router complexityOptical router complexity

Objective: quantify the fundamental complexity of an optical router

Two types of fundamental complexity: Construction complexity: number of

basic optical components needed (e.g., 2x2 optical switches)

Control complexity: frequency of optical switch reconfigurations

Main contributionsMain contributions Define fundamental complexity in

general optical constructions: Control complexity Construction complexity

Find lower and upper bounds on these costs.

Construct optical router with minimum complexity.

OutlineOutline Background Control complexity (# switch

reconfigurations) Definition Bounds

Construction complexity (# switches) Definition Optimally constructed constructions

Two possible ways to “store” lightTwo possible ways to “store” light

To slow/stop light.

BUT: requires gas environments with tight temperature and pressure constraints, and currently seems impractical.

Use optical switches and fiber delay lines.

.

Buffer

Buffer

An optical memory cell:

(a) writing the packet

(b) circulating the packet

(c) reading the packet

How do we store light?How do we store light?

11 1(a) (b) (c)

We’ve presented a buffer capable of storing one optical packet.

A naive optical queue with buffer A naive optical queue with buffer BB

The number of 22 switches needed for the naive construction is B.

Could be less than B when several packets can share the same line (with different line lengths).

1 1 1 1 1

What we want: an ideal routerWhat we want: an ideal router An output-queued push-in-first-out

(OQ-PIFO) switch.

OQ - Arriving packets are placed immediately in the queue of size B at their destination output.

PIFO – packets departure ordering is according to their priority.

Input 1

Input N

… …

Output 1

Output N

What we want: an ideal routerWhat we want: an ideal router Why it is ideal:

OQ: Work conserving implies best throughput and minimal delay.

PIFO: Enables FIFO, strict priorities, WFQ… But – up to N packets destined to the

same output: Speed-up for switch Speed-up for queue PIFO is hard to implement.

How do we do it in optics?How do we do it in optics?

If packets are destined to different outputs: Switching: optical switch NxN with O(NlnN)

2x2 optical switches ([Shannon ’49], [Benes ’67]).

Buffering: optical PIFO queue B 2x2 optical switches ([Sarwate & Anantharam ’04]).

Input 1

Input N

…

…

Output 1

Output N

O( BlnB)

PIFO

PIFOOQ

1B

1B1

12

2

33

Output 2

Output 3

Control complexityControl complexity

Generalization to systemsGeneralization to systems An optical system - a network element

that has input links, output links and inner states, and is built with optical 2x2 switches and FDLs.

Inner states - the different settings of the system elements.External states – distinguishable possible system outputs.

DefinitionDefinition Control complexity – a measure of the

minimal expected number of switch reconfigurations.

Example: 4 inputs, 4 outputs,3 external states:

What is the control complexity of an optical system with these states?

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1

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0.5

0.25

0.25

1234

1234

21433

412

Link to codingLink to coding

Source symbols:A1 – w.p. 0.5A2 – w.p. 0.25A3 – w.p. 0.25

A 2x2 switch A binary digitState entropy Source entropy

??? Minimizing expected code length

Coding results should apply also to switching!

CodingSwitching0.

5

0.25

0.25

1234

1234

21433

412

DefinitionsDefinitions A super switch:

Passive and active controls – for each state, a control is called passive if its value is irrelevant for setting that state. Otherwise, it is called active.

C

C2

C1

Example:Example: ActivePassive

Active

With coding:w.p 0.5 A1 ↔0w.p 0.25 A2↔10w.p 0.25 A3↔11

0.5

0.25

0.25

1234

1234

21433

412

C1=0

C1=1, C2=0

C1=1, C2=1

Definition – control complexityDefinition – control complexity Definition: the control complexity of an

optical system is its minimal expected number of active controls,

T – states space, - number of active controls per state

* min ( )i

i

T i tt T

C P t l

itl

it

Link to codingLink to coding

Source symbols:A1 – w.p. 0.5

A2 – w.p. 0.25

A3 – w.p. 0.25

A 2x2 switch A binary digit.States entropy Source entropy

Minimized expected code length

CodingSwitching

???Control complexity

0.5

0.25

0.25

1234

1234

21433

412

Lower boundLower boundTheorem: The control complexity is lower bounded by the entropy of the states:

Proof: Similar to the proof of expected codelength lower bound

*C H

C2

C1

* 1 1 1 3*1 *2 *22 4 4 2

C H

In the previous example:

0.5

0.25

0.25

1234

1234

21433

412

Theorem: The control complexity is upper bounded as follows:

Stages of proof: Generate Huffman coding (expected code

length ≤ H+1) . There exists a construction (using

multiplexers and distributers) of a memoryless system such that the active controls for each state are the Huffman coding of that state

A system with memory can be composed from a memoryless system using a time-space transformation.

An upper bound on the control An upper bound on the control complexitycomplexity

* 1C H

Construction complexityConstruction complexity

DefinitionDefinition Construction complexity: the minimal possible

number of 2x2 switches in the construction. Examples:

An NxN switch:

N! states, O(NlnN) switches [Shannon, ‘49], [Benes, ‘65].

A Time Slot Interchange (TSI) with time frame N:

N! states - O(lnN) switches [Jordan et. al., ‘94].812345678

N

1 2 345 6 78

12345678

1

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67

8

Construction complexityConstruction complexity Intuition: With C 2x2 switches during T

time slots, the possible number of resulting states K is upper bounded by 2CT.

Therefore: to get K states in state duration T, a lower bound on the construction complexity is given by:

* 2log KC

T

Optimally-constructed Optimally-constructed constructionsconstructions

A construction algorithm is optimally constructed if its number of 2x2 switches is equal in growth to the construction complexity.

Examples: An NxN switch: A TSI:

* 2log ( !)( ln )

1

NC N N C

* 2log ( !)(ln )

NC N C

N [Jordan et. al., ‘94].

[Benes, ‘65].

Conclusion – construction Conclusion – construction complexity of optical routerscomplexity of optical routers

Input 1

Input N

… …

Output 1

Output N

NxN switch: Θ(Nln(N)) PIFO buffer of sizeB: Θ(ln(B))

B

The construction complexity of an OQ-PIFO switch is Θ(Nln(N))+Θ(Nln(B)) = Θ(Nln(NB))

Thank you!Thank you!