Input%20Queued%20Switches:%20Cell%20Switching%20vs.%20Packet%20Switching

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Input Queued SwitchesCell Switching vs. Packet
Switching

• Abtin Keshavarzian
• Joint work with
• Yashar Ganjali, Devavrat Shah
• Stanford University

2
Background
Switching Fabric

• Time is slotted
• Data units of fixed size ? cells
• Buffers at input ports (Input-Queued Switch)
• To avoid HoL blocking , virtual output queues are
  used

3
Motivation
Switch
VOQ11
Combiner
VOQ1N
Splitter
VOQN1
VOQNN

• Packets have different lengths
• Splitter module
• Combiner module (memory)
• Packet delays are more important than Cell delays
• Packet Based Scheduling algorithms

4
Outline

• Cell based algorithms review
• Stability concept
• Maximum Weight Matching algorithm
• Packet based algorithms
• Packet-Based Algorithms
• PB-MWM and its stability
• PB Algorithms Classification
• Work Conserving
• Waiting
• Waiting PB Algorithms
• Conclusion

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Notation Arrival rate
Switching Fabric

• Number of cells arrived to VOQij up to
  time n
• Number of cells departed from VOQij up
  to time n
• Number of cells queued at VOQij at time
  n
• (SLLN)
  almost surely

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Admissibility and Rate Stability

• The arrival rate matrix is
  admissible iff
• A switch under a matching algorithm is stable
  (rate stable) if, almost surely,

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MWM algorithm

• A matching
• MWM At each time slot, select the matching with
  maximum weight

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MWM Stability

• McKeown et al showed that
• MWM is stable under i.i.d. Bernoulli traffic
• Dai and Prabhakar using Fluid model technique
  showed
• MWM is stable for any admissible traffic

N. McKeown,V. Ananthram, and J. Walrand,
Achieving 100 throughput in an input-queued
switch, INFOCOM 1996, pp. 296-302.
J. G. Dai and B. Prabhakar, The throughput of
data switches with or without speedup, INFOCOM
2000, pp. 556-564.
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Outline

• Cell based algorithms review
• Stability concept
• Maximum Weight Matching algorithm
• Packet based algorithms
• Packet-Based Algorithms
• PB-MWM and its stability
• Packet Based Algorithms Classification
• Work Conserving
• Waiting
• Waiting Packet Based Algorithms
• Conclusion

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Packet-Based Switching

• Once the scheduler starts transmitting the first
  cell of a packet, it continues until the whole
  packet is received at output port

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Packet-Based Switching

• Once the scheduler starts transmitting the first
  cell of a packet, it continues until the whole
  packet is received at output port

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Packet-Based Switching

• Once the scheduler starts transmitting the first
  cell of a packet, it continues until the whole
  packet is received at output port.

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Cell-based to Packet-based

• Consider cell-based algorithm X
• At each time slot
• Busy ports middle of sending a packet
• Free ports i/o ports can be assigned freely
• PB-X
• Keep the assignments used by busy ports
• Find a sub-matching for free ports using
  algorithm X.

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Stability of PB-MWM

• PB-MWM is stable under regenerative admissible
  traffic
• Traffic is called regenerative if on average it
  requires a finite time to reach the state where
  all ports are free if it keeps using any fixed
  matching.
• Bernoulli i.i.d. is a regenerative traffic.

M.A. Marsan, A. Bianco, P. Giaccone, E. Leonardi,
and F. Nari, Packet Scheduling in Input-Queued
Cell-based switches, INFOCOM 2001, pp. 1085-1094
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Proof Outline

• Matching m(n) is k-imperfect if
• For PB-MWM
• Lemma A scheduling algorithm is rate stable if
  the average value of its weight is larger than
  maximum weight matching minus a bounded constant.
  

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Question

• CB-MWM is stable under any admissible traffic
• PB-MWM is stable under any admissible
  regenerative traffic.
• Is the regenerative condition necessary?

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Counter-example
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Counter-example
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Counter-example
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Counter-example
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Counter-example
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Counter-example
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Counter-example
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Classification of PB algorithms

• Work Conserving (non-waiting)
• No input is left unmatched when it has a packet
  for an unmatched output.
• Waiting
• Input ports may wait (do not start sending a
  packet) for infinite number of time slots.
• No work-conserving algorithm can be rate stable
  for all admissible traffic.

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PB-wMWM

• Switch runs at speedup
• Maximum packet length L
• If use usual
  PB-MWM
• Else wait till all ports are free.
• PB-wMWM is rate stable for any admissible traffic
  with known max packet length

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Modified PB-wMWM

• The packet length is not known but has bounded
  expectation
• the maximum length of packets left when
  waiting starts during lth segment
• Modified PB-wMWM is rate stable for any
  admissible traffic with bounded packet length

Segment 2
Segment 1
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Conclusion

• PB-MWM is rate stable under any admissible
  regenerative traffic.
• Work-conserving packet based algorithms can not
  be rate stable for all admissible traffics
• ? Waiting is essential
• PB-wMWM and its modified version are stable under
  any admissible traffic (with bounded mean packet
  length)
• Future work
• Find simpler algorithms that are stable for any
  admissible traffic.

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Fluid model

• number of time slots matching m
  being used up to time n