fundamental complexity of optical systems
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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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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!
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