Transcript CS244a: An Introduction to Computer Networks

Outline
Why Maximal and not Maximum
 Definition and properties of Maximal Match
 Parallel Iterative Matching (PIM)
 iSLIP
 Wavefront Arbiter (WFA)

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Why doesn’t maximizing instantaneous
throughput give 100% throughput for nonuniform traffic?
 11   12  1 / 2  
Three possible
matches, S(n):
 21  1 / 2  
 32  1 / 2  
Assume that at time n, L11(n)  0 , L12(n)  0, and Q21 (n) and Q32 (n) both
have arrivals(w.p. (1/ 2-δ ) 2 )  Input 1 is servicedw.p. 2 / 3.

 
The total rate at whi chinput1 is served is at most 2 / 3  (1/ 2- ) 2  1 1  (1/ 2   ) 2

 1  1/ 3  (1/ 2   ) 2 .
But λ1  1-2  1 - 1/ 3  (1/ 2 -δ ) 2 .
And so if   0.0358switch is not stable (throughput  100%).
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Maximal Matching
A maximal matching is one in which each
edge is added one at a time, and is not
later removed from the matching.
 i.e. no augmenting paths allowed (they
remove edges added earlier).
 No input and output are left
unnecessarily idle.

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Example of Maximal Size Matching
A
1
A
1
A
1
B
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B
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B
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C
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C
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D
4
C
D
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E
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D
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F
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E
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E
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Maximal
Size Matching
Maximum
Size Matching
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Maximal Matchings
In general, maximal matching is simpler to
implement, and has a faster running time.
 A maximal size matching is at least half
the size of a maximum size matching.
 A maximal weight matching is defined in
the obvious way.
 A maximal weight matching is at least half
the weight of a maximum weight matching.

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Outline
Definition and properties of Maximal Match
 Parallel Iterative Matching (PIM)
 iSLIP
 Wavefront Arbiter (WFA)

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Parallel Iterative
Matching
uar selection
uar selection
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f1: Requests
f2: Grant
f3: Accept/Match
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#2
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Parallel Iterative Matching
Convergence Time
Number of iterations to converge:
N2
E U i   ------4i
E C   log N
C = # of iterations required to resolve connections
N = # of ports
U i = # of unresolved connections after iteration i
Number of iterations to converge:
with prob. 1 all n inputs are resolved
A.
B.
grant is accepted – all are resolved
grant rejected – n-k are resolved
k inputs with no
other grant
Q
n-k inputs with
grants from others
At most k(1-k/n) are unresolved  n/4
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Parallel Iterative Matching
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Parallel Iterative Matching
PIM with a
single iteration
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Parallel Iterative Matching
PIM with 4
iterations
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PIM Fairness Problems:
(under inadmissible load )
 11   12  1
 11  1 / 4
 12  3 / 4
 21  1
 21  3 / 4
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Outline
Definition and properties of Maximal Match
 Parallel Iterative Matching (PIM)
 iSLIP
 Wavefront Arbiter (WFA)

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iSLIP
Round-Robin Selection
Round-Robin Selection
#1
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F1: Requests
F2: Grant
F3: Accept/Match
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#2
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SLIP vs. Round Robin
1.
2.
3.
Request: each input send a request to
every output i, |VOQi|>0
Grant: chose a request next in RR order
and advance pointer beyond it.
Accept:chose the among the grants the
one after the pointer and advance the
pointer beyond.
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SLIP vs. Round Robin
1.
2.
3.
Request: each input send a request to
every output i, |VOQi|>0
Grant: chose a request next in RR order
and advance pointer beyond it if accepted.
Accept:chose the among the grants the
one after the pointer and advance the
pointer beyond.
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iSLIP vs. Round Robin
1.
2.
3.
Request: each input send a request to
every output i, |VOQi|>0
Grant: chose a request next in RR order
and advance pointer beyond it if accepted.
in 1st iteration
Accept:chose the among the grants the
one after the pointer and advance the
pointer beyond only if matched in 1st
iteration.
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why update pointers only in the
1st round?


assume all pointers
point at 1.
time 1:



time 2


1st: 1-1 is matched
2nd: 2-2 is matched
1
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1st: 1-3 & 3-2 are matched
time 3:


1st: 1-1 is matched
2nd: 2-2 is matched
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iSLIP
Properties
Random under low load
 TDM under high load
 Lowest priority to MRU
 1 iteration: fair to outputs
 Converges in at most N iterations. On
average < log2N
 Implementation: N priority encoders
 Up to 100% throughput for uniform i.i.d.
traffic

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iSLIP
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iSLIP
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Programmable
Priority Encoder
N
N
iSLIP
Implementation
1
1
Grant
Accept
2
2
Grant
Accept
log2N
log2N
State
Decision
N
N
Grant
N
Accept
log2N
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iSLIP Variations


L priority levels

replace each pointer by L pointers

threshold SLIP
Weighted SLIP
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Outline
Definition and properties of Maximal Match
 Parallel Iterative Matching (PIM)
 iSLIP
 Wavefront Arbiter (WFA)

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Wave Front Arbiter
(Tamir)
Requests
Match
1
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Wave Front Arbiter
Requests
Match
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Wave Front Arbiter
Implementation
1,1
1,2
1,3
1,4
2,1
2,2
2,3
2,4
3,1
3,2
3,3
3,4
4,1
4,2
4,3
4,4
Simple combinational
logic blocks
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Wave Front Arbiter
Wrapped WFA (WWFA)
N steps instead of
2N-1
Requests
Match
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