Outline

1

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 non-

uniform traffic?

2/1

2/1

2/1

32

21

1211Three possiblematches, S(n):

100%). t(throughpu stable not is switch 0.0358 if so And

But

most at is served is 1 input which at rate total The

. w.p. serviced is 1 Input ) w.p.( arrivals have

both and and , time at that Assume

.)21(31121

.)21(311

)21(11)21(32

32)21(

)()(0)(0)(

21

2

22

2

32211211

-δ// - -λ

//

/-//

/-δ/

nQnQ n, L nn, L

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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

B

C

D

E

F

2

3

4

5

6

A 1

B

C

D

E

F

2

3

4

5

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Maximal Size Matching

Maximum Size Matching

A

B

C

D

E

F

1

2

3

4

5

6

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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

1

2

3

4

1

2

3

4

1

2

3

4

1

2

3

4

1: Requests

1

2

3

4

1

2

3

42: Grant

1

2

3

4

1

2

3

43: Accept/Match

uar selection

1

2

3

4

1

2

3

4

uar selection

1

2

3

4

1

2

3

4

#1

#2

Itera

tion

:

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Parallel Iterative MatchingConvergence Time

E C Nlog

E Ui N2

4i------- C # of iterations required to resolve connections=

N # of ports =

Ui # of unresolved connections after iteration i=

Number of iterations to converge:

Number of iterations to converge:

Q

k inputs with no other grant

n-k inputs with grants from others

with prob. 1 all n inputs are resolved

A. grant is accepted – all are resolvedB. grant rejected – n-k are resolved

At most (n-k)(1-k/n) are unresolved n/4

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10


PIM with a single iteration

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PIM with 4 iterations

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PIM Fairness Problems:(under inadmissible load )

1

1

21

1211

4/3

4/3

4/1

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12

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Outline



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iSLIP

1

2

3

4

1

2

3

4

1

2

3

4

1

2

3

4

1: Requests

1

2

3

4

1

2

3

42: Grant

1

2

3

4

1

2

3

43: Accept/Match

1

2

3

4

1

2

3

4

#1

#2

Round-Robin Selection

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2

3

4

1

2

3

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Round-Robin Selection

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SLIP vs. Round Robin

1. Request: each input send a request to every output i, |VOQi|>0

2. Grant: chose a request next in RR order and advance pointer beyond it.

3. Accept:chose the among the grants the one after the pointer and advance the pointer beyond.

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SLIP vs. Round Robin


2. Grant: chose a request next in RR order and advance pointer beyond it if accepted.

3. Accept:chose the among the grants the one after the pointer and advance the pointer beyond.

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iSLIP vs. Round Robin


2. Grant: chose a request next in RR order and advance pointer beyond it if accepted.

3. Accept:chose the among the grants the one after the pointer and advance the pointer beyond only if matched in 1st iteration.

in 1st iteration

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why update pointers only in the 1st round?

assume all pointers point at 1.

time 1: 1st: 1-1 is matched 2nd: 2-2 is matched

time 2 1st: 1-3 & 3-2 are

matched

time 3: 1st: 1-1 is matched 2nd: 2-2 is matched

1

3

2 2

3

1

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iSLIPProperties

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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iSLIPImplementation

Grant

Grant

Grant

Accept

Accept

Accept

1

2

N

1

2

N

State

N

N

N

Decision

log2N

log2N

log2N

ProgrammablePriority Encoder

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iSLIP Variations

L priority levels replace each pointer by L pointers

threshold SLIP

Weighted SLIP

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Outline



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Wave Front Arbiter(Tamir)

Requests Match

1

2

3

4

1

2

3

4

1

2

3

4

1

2

3

4

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Wave Front Arbiter

Requests Match

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Wave Front ArbiterImplementation

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 ArbiterWrapped WFA (WWFA)

Requests Match

N steps instead of2N-1

Outline

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