048866: Packet Switch Architectures

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048866 Packet Switch Architectures
MSM (Maximum Size Matching) and MWM (Maximum
Weight Matching)

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

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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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Unknown Traffic Matrix

• We want to maximize throughput
• Traffic matrix unknown ? cannot use BvN
• Idea maximize instantaneous throughput
• In other words transfer as many packets as
  possible at each time-slot
• Maximum Size Matching (MSM) algorithm

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Maximum Size Matching (MSM)

• MSM maximizes instantaneous throughput
• MSM algorithm among all maximum size matches,
  pick a random one

Q11(n)gt0
Maximum Size Match
QN1(n)gt0
Bipartite Match
Request Graph
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Question

• Is the intuition right?
• Answer No, there is a counter-example for which,
  in a given VOQ (i,j), ?ij lt ?ij

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Counter-example
Consider the following non-uniform traffic
pattern, with Bernoulli IID arrivals
Three possible matches, S(n)
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Simulation of simple 3x3 example
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Idea Use Lyapunov
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Some definitions
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Some definitions
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Problem
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Maximum Weight Matching (MWM)
S(n)
Q11(n)
A11(n)
A1(n)
D1(n)
1
1
A1N(n)
LQF MWM Algorithm
AN1(n)
AN(n)
DN(n)
ANN(n)
N
N
QNN(n)
Q11(n)
Maximum Weight Match
QN1(n)
Bipartite Match
Request Graph
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Outline of Proof
Note proof based on paper by McKeown et al.
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Proof

• Lets prove
• First, well work with the approximate Lyapunov
  function
• Well assume that there exists some ? such that
• For all i, ?j ?ij 1-?
• For all j, ?i ?ij 1-?
• In other words, ? (1-?) ?m with ?m doubly
  stochastic

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Proof
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Proof
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Proof

• We worked with the approximate Lyapunov function
• Now, lets work with the real Lyapunov function,
  and show that

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End of Proof
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Review of Proof
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Review of Proof
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LQF (Longest Queue First)

• LQF is the name given to the maximum weight
  matching, where weight wij(n) Lij(n).
• But the name is so bad that people keep the name
  MWM!
• LQF doesnt necessarily serve the longest queue.
• LQF can leave a short queue unserved
  indefinitely.
• However, MWM-LQF is very important theoretically
  most (if not all) scheduling algorithms that
  provide 100 throughput for unknown traffic
  matrices are variants of MWM!

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LQF (Longest Queue First)

• Question what if or
• What if weight wij(n) Wij(n) (waiting time)?
• Preference is given to cells that have waited a
  long-time.
• Is it stable?
• We call the algorithm OCF (Oldest Cell First).
• Remember that it doesnt guarantee to serve the
  oldest cell!

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OCF (Oldest Cell First)
Cij(n)
Cij(nl)
Wij(n)
n
nl
tij(n)
Cij(n)
Cij(nl)
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Rough outline of proof
Expectation given W(n)
Note full proof in paper by McKeown et al.
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Implementing MSM

• How can we find maximum size matches?
• We do so by recasting the problem as a network
  flow problem

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Network Flows
a
c
Source s
Sink t
b
d

• Let G V,E be a directed graph with capacity
  cap(v,w) on edge v,w.
• A flow is an (integer) function, f, that is
  chosen for each edge so that
• We wish to maximize the flow allocation.

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A maximum network flow exampleBy inspection
a
c
Source s
Sink t
b
d
Step 1
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A maximum network flow example
Step 2
a
c
10, 10
Source s
Sink t
10, 10
1
10, 10
1
10, 1
b
d
10, 1
1, 1
Flow is of size 101 11
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Ford-Fulkerson Method of Augmenting Paths

• Set f(v,w) -f(w,v) on all edges.
• Define a Residual Graph, R, in which res(v,w)
  cap(v,w) f(v,w)
• Find paths from s to t for which there is
  positive residue.
• Increase the flow along the paths to augment them
  by the minimum residue along the path.
• Keep augmenting paths until there are no more to
  augment.

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Example of Residual Graph
a
c
10, 10
10, 10
1
10, 10
s
t
10
1
10
b
d
1
Flow is of size 10
Residual Graph, R
res(v,w) cap(v,w) f(v,w)
a
c
10
10
10
1
t
s
10
1
10
b
d
1
Augmenting path
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Example of Residual Graph
Step 2
a
c
10, 10
s
t
10, 10
1
10, 10
1
10, 1
b
d
10, 1
1, 1
Flow is of size 101 11
Residual Graph
a
c
10
s
t
10
10
1
1
1
1
b
d
9
1
9
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Complexity of network flow problems

• In general, it is possible to find a solution by
  considering at most V.E paths, by picking
  shortest augmenting path first.
• There are many variations, such as picking most
  augmenting path first.

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Finding a maximum size match

• How do we find the maximum size match?

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Network flows and bipartite matching
A
1
B
2
Sink t
Source s
3
C
4
D
5
E
6
F

• Finding a maximum size bipartite matching is
  equivalent to solving a network flow problem with
  capacities and flows of size 1.

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Example Maximum Size MatchingFord-Fulkerson
method
Residual Graph for first three paths
A
1
B
2
t
s
3
C
4
D
5
E
6
F
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Example Maximum Size MatchingFord-Fulkerson
method
Residual Graph for next two paths
A
1
B
2
t
s
3
C
4
D
5
E
6
F
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Example Maximum Size MatchingFord-Fulkerson
method
Residual Graph for augmenting path
A
1
B
2
t
s
3
C
4
D
5
E
6
F
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Example Maximum Size MatchingFord-Fulkerson
method
Residual Graph for last augmenting path
A
1
B
2
t
s
3
C
4
D
5
E
6
F
Note that the path augments the match no input
and output is removed from the match during the
augmenting step.
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Example Maximum Size MatchingFord-Fulkerson
method
Maximum flow graph
A
1
B
2
t
s
3
C
4
D
5
E
6
F
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Example Maximum Size MatchingFord-Fulkerson
method
Maximum Size Matching
A
1
B
2
3
C
4
D
5
E
6
F
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LPF (Largest Port First)
Note full proof in paper by Mekkitikul and
McKeown
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LPF