Fundamental Complexity of Optical Systems

1
Fundamental Complexity of Optical Systems

• Hadas Kogan, Isaac Keslassy
• Technion (Israel)

2
Router schematic representation
Router
Optic to electronic
Electronic to optic


Optic to electronic
Electronic to optic

• 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.

3
Power consumption per chassis
There has to be some future alternative!
Nick McKeown, Stanford
4
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?

5
Optical 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

6
Main 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.

7
Outline

• Background
• Control complexity ( switch reconfigurations)
• Definition
• Bounds
• Construction complexity ( switches)
• Definition
• Optimally constructed constructions

8
Two 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
9
How do we store light?
An optical memory cell (a) writing the
packet (b) circulating the packet (c) reading
the packet
(a)
(b)
(c)
1
1
1
Weve presented a buffer capable of storing one
optical packet.
10
A naive optical queue with buffer B
1
1
1
1
1

• The number of 2?2 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).

11
What 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.

12
What 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.

13
How do we do it in optics?
PIFO
1
B
OQ
Input 1
Output 1
1
2
Output 2
1

Output 3
3

3
Input N
Output N
2
PIFO
1
B

• 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).

14
Control complexity

15
Generalization 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.

16
Definition

• 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?

17
Link 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
• State entropy Source entropy
• ??? Minimizing expected code length
• Coding results should apply also to switching!

Coding
Switching
18
Definitions

• 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
19
Example
C10
C11, C20
C11, C21
With coding w.p 0.5 A1 ?0 w.p 0.25 A2?10 w.p
0.25 A3?11
20
Definition 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

21
Link 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

Coding
Switching
???
Control complexity
22
Lower bound

• Theorem The control complexity is lower
• bounded by the entropy of the states
• Proof Similar to the
• proof of expected code
• length lower bound

• In the previous example

23
An upper bound on the control complexity

• Theorem The control complexity is upper bounded
  as follows
• Stages of proof
• Generate Huffman coding (expected code length
  H1) .
• 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.

24
Construction complexity

25
Definition

• 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.

26
Construction 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

27
Optimally-constructed constructions

• 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

Benes, 65.
Jordan et. al., 94.
28
Conclusion construction complexity of optical
routers
B
NxN switch T(Nln(N))
PIFO buffer of sizeB T(ln(B))

• The construction complexity of an OQ-PIFO switch
  is T(Nln(N))T(Nln(B)) T(Nln(NB))

29
Thank you!