Load Balanced Birkhoffvon Neumann Switches

1
Load Balanced Birkhoff-von Neumann Switches

• Cheng-Shang Chang, Duan-Shin Lee and Chi-Yao Yue
• presented by
• Prashanth Pappu

2
High Performance Switches

• Non-blocking crossbar
• Fixed time slot, fixed size cell
• Parallelism, memory speed line rate.
• Quadratic complexity but concentrated in a single
  chip set.
• Centralized scheduler

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Centralized Schedulers

• VOQs to avoid HOL blocking.
• Equivalent to finding a matching on a bipartite
  graph (Anderson et al)
• McKeown et al. 100 throughput with MWM.
• 10Gb/s line rate implies 40 ns for scheduling.
• Maximal size matching algorithms (PIM, iSLIP)
• More ports and faster line rates makes it harder
  to implement scheduling algorithms.

4
Overview

• New scheduling algorithm (based on Birkhoff-von
  Neumann decomposition) On service guarantees
  for input buffered crossbar switches a capacity
  decomposition approach by Birkhoff and von
  Neumann, IEEE IWQoS99.
• Birkhoff-von Neumann switches are not practical.
• Load balanced Birkhoff-von Neumann switch Load
  Balanced Birkhoff-von Neumann Switches, Part I
  One-stage Buffering, Computer Communications.
• Mis-sequencing problem and solutions Load
  Balanced Birkhoff-von Neumann Switches, Part II
  Multi-stage Buffering, Computer Communications
  and I. Keslassy and N. Mckweon Maintaining
  Packet Order in Two Stage Switches, IEEE
  Infocom, 2002.
• Providing guaranteed rate services (The actual
  paper!) Providing guaranteed rate services in
  the load balanced Birkhoff-von Neumann Switches,
  IEEE Infocom 2003.
• Talk presents only algorithms results proofs.

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Birkhoff-von Neumann Switch

• Crossbar configuration is a permutation matrix,
  P.
• 4x4 switch, input 1-output 4, input 2-output 1
  etc.
• Input rate matrix is admissible.

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Birkhoff-von Neumann Switch

• Can any admissible rate matrix be serviced?
• How do we map rate matrix to a sequence of
  permutation matrices? (Change in pov as opposed
  to finding matching on bipartite graph)
• Express as convex combination of permutation
  matrices.
• Obtain the decomposition and schedule each
  permutation matrix proportional to its weight.

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Birkhoff-von Neumann Switch

• (von Neumann 1953) Transform the doubly
  substochastic rate matrix to a doubly stochastic
  matrix.
• (Birkhoff 1946) Decompose doubly stochastic rate
  matrix to weighted sum of permutation matrices.
• (PGPS) Use simple packet scheduling algorithm
  (WFQ) to determine which permutation matrix
  should be used to configure crossbar.

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Example

• von-Neumann conversion.
• Pivots around (1,2), (2,1), (2,2) etc.
• There are other (fairer) ways to obtain this
  conversion.

R
R
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Example
0.4
0.4
R
0.2
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Not practical

• Birkhoff-von Neumann decomposition is non-trivial
  with O(N4.5) complexity, though required only
  when rates change.
• Need to know rate matrix.
• Memory O(N2) permutation matrices.
• Does not support multicast.
• Solution Load balanced Birkhoff-von Neumann
  switch.

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Load balanced BvN switch

• We know decomposition is easy for uniform
  Bernouli i.i.d traffic.
• Use a first stage that load balances traffic to
  second stage!
• First stage uses permutation matrices generated
  from a one-cycle permutation matrix. (Input i
  connects to output (ni) modulo N at time n.)

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Second stage (Switching)

• Traffic from first stage is instantly transferred
  to buffers at second stage.
• With balanced traffic, second stage can also use
  a deterministic sequence of cyclical permutation
  matrices. (Input j is connected to output ((n-j)
  modulo N) at time n.)
• Both stages are identical, complexity of
  scheduling algorithm O(1).
• Low hardware complexity.
• 100 throughput (under a mild technical condition)

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Uniform pareto bursty traffic, N16
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Load balanced BvN switch (multi-buffered)

• Problem of mis-sequencing of packets.
• Packets are distributed on arrival times no
  bound on a resequencing buffer.
• Use load-balancing and re-sequencing buffers.
• Load-balancing based on flows and not according
  to arrival times.

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FCFS with jitter control

• Flow splitter sends packets from same flow in
  round robin fashion to the N VOQs.
• Causes packets of same flow to be split almost
  evenly among inputs of second stage.
• Jitter control at second stage delays each packet
  to its maximum delay (targeted departure time is
  obtained from corresponding OQ switch)
• Flows entering second stage are time-shifted
  versions of original ones.

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FCFS with jitter control

• Delay of a packet is bounded by sum of delay
  through the corresponding OQ switch and (N-1)Lmax
  NMmax.
• Essentially delay lt 2N for unicast traffic.
• Size of load balancing buffer bounded by NLmax.
• Size of re-sequencing buffer bounded by NMmax.
• Lmax (Mmax) is the maximum number of flows at an
  input (output).

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Guaranteed Rate Services

• Load balanced BvN switch provides best effort
  service.
• How do we provide service guarantees?
• Earliest Deadline First (EDF) based scheme.

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EDF based scheme.

• Same architecture as FCFS scheme with jitter
  control.
• Targeted departure time is departure time of
  corresponding link with capacity equal to the
  guaranteed rate of the flow.
• Packets served in EDF order at output buffer.

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EDF scheme

• Every packet of a guaranteed rate flow has a
  delay bound targeted departure time (N-1)Lmax
  NMmax.
• Resequencing and load balancing buffer bounded by
  NMmax.

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Not discussed

• Full Frames First an algorithm that prevents
  packets from being mis-sequenced. (Will be
  discussed in next paper presentation Scaling
  Internet routers using optics)
• Frame based scheme for guaranteed rate services
  algorithm based on FFF for providing rate
  guarantees.