048866: Packet Switch Architectures

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048866 Packet Switch Architectures

• Scheduling in Input-Queued Switches
• Uniform Traffic
• Birkhoff-von Neumann

• Dr. Isaac Keslassy
• Electrical Engineering, Technion
• isaac_at_ee.technion.ac.il
• http//comnet.technion.ac.il/isaac/

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Where We Are

• We introduced IQ switches
• We saw that HoL blocking reduces throughput
• We got tools from queueing theory to analyze more
  complex queueing systems

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Where We Are

• We will now study input-queued switches with VOQs
  (Virtual Output Queues)
• No HoL blocking
• But we need good scheduling algorithms to obtain
  100 throughput

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History

• 1. Karol et al., 1987
• HoL Blocking Throughput limited to 58 for
  Bernoulli IID uniform traffic.

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History

• 2. Tamir and Frazier, 1988
• VOQs remove HoL blocking, increase throughput

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History

• 3. Anderson et al., 1993
• MSM analogy to MSM (Maximum Size Matching) in
  bipartite graph

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History

• 4. McKeown et al., 1995
• MWM MSM (Maximum Size Matching) does not
  guarantee 100 throughput. MWM (Maximum Weight
  Matching) does.
• 5. Chuang et al., 1998
• CIOQ IQ can emulate OQ with speedup 2.
• 6. Chang et al., 1999
• BvN A schedule implementing a Birkhoff-von
  Neumann decomposition gets 100 throughput.

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History

• 7. Leonardi et al., 2000 Dai and Prabhakar,
  2000
• Maximal IQ can get 100 throughput with speedup
  2 using maximal matchings. For instance, WFA
  Tamir and Chi, 1993, PIM Anderson et al.,
  1993, iSLIP McKeown et al., 1993.
• 8. Andrews and Zhang, 2001
• Network A network of MWM switches is unstable
• 9. Chang et al., 2002
• LBR A Load-Balanced Router provides 100
  throughput without scheduling.

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Achieving 100 throughput

• Switch model
• Uniform traffic
• Technique Uniform schedule (easy)
• Non-uniform traffic, but known traffic matrix
• Technique Non-uniform schedule (Birkhoff-von
  Neumann)
• Unknown traffic matrix
• Technique Lyapunov functions (MWM)
• Faster scheduling algorithms
• Technique Speedup (maximal matchings)
• Technique Memory and randomization (Tassiulas)
• Technique Twist architecture (buffered crossbar)
• Accelerate scheduling algorithm
• Technique Pipelining
• Technique Envelopes
• Technique Slicing
• No scheduling algorithm
• Technique Load-balanced router

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Head-of-Line Blocking
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Virtual Output Queues
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VOQs How Packets Move
VOQs
Scheduler
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Question do more lanes help?

• Answer it depends on the scheduling

Head of Line Blocking
VOQs with Bad Scheduling
Good Scheduling? Ayalon depends on traffic
matrix
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Basic Switch Model
S(n)
Q11(n)
A11(n)
D11(n)
1
1
A1(n)
A1N(n)
D1N(n)
AN1(n)
DN1(n)
AN(n)
N
N
ANN(n)
DNN(n)
QNN(n)
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Notations Arrivals

• Aij(n) packet arrivals at input i for output j
  at time-slot n
• Aij(n) 0 or 1
• ?ijEAij(n) arrival rate
• ??ij traffic matrix
• AAij(n) admissible iff
• For all i, ?j ?ij lt 1 no input is
  oversubscribed
• For all j, ?i ?ij lt 1 no output is
  oversubscribed

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Notations Schedule

• Qij(n) queue size of VOQ (i,j)
• QQij(n)
• Sij(n) whether the schedule connects input i to
  output j
• Sij(n) 0 or 1
• No speedup each input is connected to at most
  one output, each output to at most one input
• We will assume that each input is connected to
  exactly one output, and each output to exactly
  one input? SSij(n) permutation matrix

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Scheduling Algorithm

• What it does determine S(n)
• How
• Either using traffic matrix ?,
• Or, in most cases, using queue sizes Q(n)
  (because ? unknown)
• Objective 100 throughput
• So that lines are fully utilized
• Secondary objective minimize packet
  delays/backlogs

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What is 100 throughput?

• Work-conserving scheduler
• Definition If there is one or more packet in the
  system for an output, then the output is busy.
• An output queued switch is work-conserving.
• Each output can be modeled as an independent
  single-server queue.
• If l lt m then EQij(n) lt C for some C.
• Therefore, we say it achieves 100 throughput.
• For fixed-sized packets, work-conservation also
  minimizes average packet delay. (Q What happens
  when packet sizes vary?)
• Non work-conserving scheduler
• An input-queued switch is, in general, non
  work-conserving.
• Q What definitions make sense for 100
  throughput?

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Some common definitions of 100 throughput

1. Work-conserving
2. For all n,i,j, Qij(n) lt C,i.e.,
3. For all n,i,j, EQij(n) lt Ci.e.,
4. Departure rate arrival rate,i.e.,

weaker
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Achieving 100 throughput

• Switch model
• Uniform traffic
• Technique Uniform schedule (easy)
• Non-uniform traffic, but known traffic matrix
• Technique Non-uniform schedule (Birkhoff-von
  Neumann)
• Unknown traffic matrix
• Technique Lyapunov functions (MWM)
• Faster scheduling algorithms
• Technique Speedup (maximal matchings)
• Technique Memory and randomization (Tassiulas)
• Technique Twist architecture (buffered crossbar)
• Accelerate scheduling algorithm
• Technique Pipelining
• Technique Envelopes
• Technique Slicing
• No scheduling algorithm
• Technique Load-balanced router

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Uniform Traffic

• Definition ?ij? for all i,j
• i.e., all input-output pairs have same traffic
  rate
• Condition for admissible traffic ? lt 1/N
• Example Bernoulli traffic
• ??/N
• Arrivals at input i are Bernoulli(?) and i.i.d.

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Algorithms that give 100 throughput for uniform
traffic

• Nearly all algorithms in literature can give 100
  throughput when traffic is uniform
• For example
• Uniform cyclic.
• Random permutation.
• Wait-until-full simulations.
• Maximum size matching (MSM) simulations.
• Maximal size matching (e.g. WFA, PIM, iSLIP)
  simulations.

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Uniform Cyclic Scheduling
Each (i,j) pair is served every N time slots
Geom/D/1
l?/N lt 1/N
1/N
Stable for ? lt 1
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Wait-until-full

• We dont have to do much at all to achieve 100
  throughput when arrivals are Bernoulli IID
  uniform.
• For example, simulation suggests that the
  following algorithm leads to 100 throughput.
• Wait-until-full
• If any VOQ is empty, do nothing (i.e. serve no
  queues).
• If no VOQ is empty, pick a random permutation.

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Maximum Size Matching (MSM)

• Intuition maximize instantaneous throughput
• Simulations suggest 100 throughput for uniform
  traffic.

Q11(n)gt0
Maximum Size Match
QN1(n)gt0
Bipartite Match
Request Graph
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Some simple algorithms that achieve 100
throughput
Wait until full
Uniform Cyclic
Maximal Matching Algorithm (iSLIP)
MSM
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Uniform Random Scheduling

• At each time-slot, pick a schedule uar among
• The N cyclic permutations
• Or the N! permutations
• Then P(Si,j1) 1/N
• Q why?

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Uniform Random Scheduling

• We get a Geom/Geom/1 system
• We studied the birth-death chain
• We get
• Stable when ? lt 1

?1/N
l?/N
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Achieving 100 throughput

• Switch model
• Uniform traffic
• Technique Uniform schedule (easy)
• Non-uniform traffic, but known traffic matrix
• Technique Non-uniform schedule (Birkhoff-von
  Neumann)
• Unknown traffic matrix
• Technique Lyapunov functions (MWM)
• Faster scheduling algorithms
• Technique Speedup (maximal matchings)
• Technique Memory and randomization (Tassiulas)
• Technique Twist architecture (buffered crossbar)
• Accelerate scheduling algorithm
• Technique Pipelining
• Technique Envelopes
• Technique Slicing
• No scheduling algorithm
• Technique Load-balanced router

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Non-Uniform Traffic

• Assume the traffic matrix is
• ? is admissible
• and non-uniform

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Uniform Schedule?

• What if uniform schedule?
• Each VOQ serviced at rate ? 1/N 1/4
• But arrivals to VOQ(1,2) have rate ?12 0.57
• Birth-death chain with birth rate gt death rate
  ?switch unstable!

? Need to adapt schedule to traffic matrix
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Example 1 (Trivial) scheduling to achieve 100
throughput

• Assume we know the traffic matrix, it is
  admissible, and it follows a permutation
• Then we can simply choose

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Example 2

• Assume we know the traffic matrix, and it doesnt
  follow a permutation. For example
• Then we can choose the sequence of service
  permutations
• And either cycle though it or pick randomly
• In general, if we know an admissible L, can we
  pick a sequence S(n) so that l lt m?

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Doubly Stochastic Matrices

• ? is admissible, or doubly (strictly)
  sub-stochastic
• Theorem 1 (von Neumann) There exists ??ij
  such that ? lt ? and ? is doubly stochastic ?i
  ?ij ?j ?ij 1
• Example

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Doubly Stochastic Matrices

• Fact 1 the set of doubly stochastic matrices is
  convex, compact, in Rn
• Fact 2 any convex, compact set in Rn has extreme
  points, and is equal to the convex hull of its
  extreme points (Krein-Milman Theorem)

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Doubly Stochastic Matrices

• Theorem 2 (Birkhoff) Permutation matrices are
  the extreme points of the set of doubly
  stochastic matrices
• In other words Given ?, there exists K numbers
  ?k gt0 and K permutation matrices Pk such that
• Further, K N2-2N2.

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Birkhoff-von Neumann (BvN) Scheduling

• BvN decomposition ? ? ? ? ?k, Pk
• BvN weighted random scheduling pick Pk with
  proba. ?k
• Theorem BvN scheduling achieves 100 throughput

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BvN and 100 Throughput

• Proof
• Lindleys equation
• Birth-death chain
• Birth rate P(Aij(n)1)EAij(n)?ij
• Death rate
• Birth rate lt death rate ? 100 throughput
  (ergodic)