Design and Performance Analysis of Buffers:a Constructive Approach

Rudolf H. Mak

Dept. of Mathematics and Computer Science

Technische Universiteit Eindhoven

P.O. Box 513, 5600 MB Eindhoven, The Netherlands

E-mail: R.H.Mak@tue.nl

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Baroque Buffer

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Design issues

• Functional correctness• Preservation of order

• Behavioral correctness• Absence of deadlock

• Performance• Cycle time, throughput• Latency• Occupancy

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Constructive Approach

• Basic building blocks• 1-place buffer• 2-way split • 2-way merge

• Construction methods• Serial composition• Wagging composition• Multi-wagging

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Functional Specification

A functional specification of a system spe-cifies each output stream as a transforma-tion of (a combination of) its input streams.

• Basic stream transformers• Take operator• Drop operator

• Stream transformer calculus• Computation rules

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Baroque Buffer (functional spec)

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Sequence Functions

• map the events of a system onto discrete time-slots

• are used to:• establish absence of deadlock• analyze system performance

• are constructed using:• a fixed set of basic sequence functions• a set of sequence function transformers

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Baroque Buffer (sequence function)

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Capacity and I/O-distance

• The capacity of a buffer is the sum of the number of variables of each of its compo-nents

• The i/o-distance of a buffer is the minimal number of variables visited by any value on its passage through the buffer

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Cycle Time

Given a sequence function we define

• The individual cycle time by

• The average cycle time by

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Latency

Given a sequence function we define

• The individual latency by

• The average latency by

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Occupancy

Given a sequence function we define

• The instantaneous occupancy by

• The average occupancy by

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Queuing Formula (Little)

Let be a sequence function of a buffer with finite average latency, occupancy, and cycle time. Then

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-bounds

Let X be a buffer with capacity and i/o-distance . Then for every sequence function for X with average cycle time , the average occupancy is bounded by

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Optimality Problem

Given a pair of values (, ), design the optimal buffer with capacity and i/o-distance , where optimal means that the occupancy can attain both bounds.

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Optimal Buffer Families

• Linear buffers

• Tree-like buffers

• Rectangular buffers• Square buffers

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Rectangular Buffer (5x3)

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Ebergen’s Square FIFO

J. Ebergen, Squaring the FIFO in GasP, Async 2001

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Ebergen’s Ring Experiment

• Capacity: 74• I/o-distance: 34• Theoretical bounds:

17 57

• Experimental bounds:23 63

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Conclusions

• Simple calculus to verify functional correctness

• Accurate performance analysis based on sequence functions

• Rectangular buffers are (, )-optimal

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1-place Buffer

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2-way Split

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2-way Merge

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Serial Composition

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Wagging Composition

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Multi-wagging

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Take Operator

For integers and l, 0 k l, and an infinite stream of values A, we define the postfix-operator by

“from A take every k-th out of l values”

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Drop Operator

For integers and l, 0 k l, and an infinite stream of values A, we define the postfix-operator by

“from A drop every k-th out of l values”

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• Drop-take rule

• Take-take rule

• Complement rule

Computation Rules

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Linear Buffers

• Capacity

= n

• I/o-distance

= n

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Tree-like Buffers

• Capacity

= 32n-1 2

• I/o-distance

= 2n 1

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Rectangular Buffers

• Capacity

= mn

• I/o-distance

= m + n 1

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SF-construction

0 a#0

1 b#0

2 a#1

3 b#1

4 a#2

5 b#2

6 a#3

7 b#3

a#0

b#0

a#1

b#1

a#0

b#0

a#1

b#1

2 2,4

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Multi-split and Multi-merge