Probabilistic Miss Equations: Evaluating Memory Hierarchy Performance

Probabilistic Miss Equations: Evaluating

MemoryHierarchy Performance

Universidade da Coruña

Basilio B. Fraguela

Ramón Doallo

Depto. de Electrónica e Sistemas

Emilio L. Zapata

Depto. de Arquitectura

de Computadores

Universidad de Málaga

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Introduction• Increasing gap between processor and memory speeds: bottleneck for systems performance

• Approaches to study cache behavior:– Trace-driven simulations: slow, not flexible.

– Built-in hardware counters: not flexible, no portability.

– Modeling: quick, flexible, little precision. Many models require a trace to extract some input parameters.

• We present a systematic modeling strategy that allows a fast analysis that provides good levels of accuracy.

• Supports set associative caches with LRU replacement.

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Misses Nature• Kinds of misses

– Intrinsic/compulsory: first reference to a line• There will be one per each different line accessed

– Interference: a non-first reference to a line misses• Each attempt to reuse a line will result in a miss with a given miss probability

• This probability depends on the impact on the cache of the memory regions accessed since the last access to the line.

• The portion of code executed between the last access to the line and the new access is called the reuse distance

• A line may have several reuse distances• Each reuse distance has a miss probability, estimated from the memory regions accessed during it.

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Miss Estimation

• The misses generated by a reference in a loop may be estimated by a formula that contains– The number of different lines it accesses– The number of line reuses it gives place to per possible reuse distance

– The miss probability for each of the reuse distances

• Fourth factor (external): miss probability in the first access to each line by the reference

• The formula is different for each nesting level (loop) enclosing the reference

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Example

• Assume 10 elements of A per cache line

DO J=1,5 DO I=1, 100 A(I) = I + J END DOEND DO

MissesAI ( p) 10 * p 90 * 0

MissesAJ ( p) 1* MissesAI (p) 4 * MissesAI (P(A(1:100)))

• Inner loop: 10 different lines, 90 (sure) reuses

• Outer loop: 1 first-time iteration, 4 reuses

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Initial Modeling Scope DO I0=1, N0, L0

DO I1=1, N1, L1 ... DO IZ=1, NZ, LZ

A(fA1(IA1), fA2(IA2), ..., fAdA(IAdA)) ... B(fB1(IB1), fB2(IB2), ..., fBdB(IBdB)) ... END DO ... C(fC1(IC1), fC2(IC2), ..., fCdC(ICdC)) ... END DOEND DO

AAAAAA ,,1,0,)( dxKIIf xxxxx

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Probabilistic Miss Equation

• Simplest PME form

FRi( p) LRiFR (i1)( p) (N i LRi)FR( i1)(PR (It(i,1)))

LRi 1N i 1

max{Ls / 'Aj ,1}

Line sizeStride of the reference in the loop

Number of different linesaccessed by R during theexecution of the loop innesting level i

where

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PMEs: Not so fast!

• The previous PME is only valid for references that carry no reuse with others

• Our model accurately takes into account the potential reuse between references in translation to build the PMEs

• Reuse among references that are not in translation is not modeled currently

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Reuse Among Different Nests

• Same strategy for loops in levels up to j

• The number of misses in loops jk, k>0 is a constant– the initial miss

probability is estimated in a conservative way considering the whole execution of the preceding loop

• In loop j0 this probability is an extern parameter, except for reuse iterations

• This only works well if several conditions hold

DO Ij-1=1, Nj-1, Sj-1

DO Ij0=1, Nj0, Sj0

... A(fA01(IA01), ..., fA0dA(IA0dA)) ... END DO ... DO Ij1=1, Nj1, Sj1

... A(fA11(IA11), ..., fA1dA(IA1dA)) ... END DO ... ... ... ... ... ... ... DO Ijn=1, Njn, Sjn

... A(fAn1(IAn1), ..., fAndA(IAndA)) ... END DOEND DO

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“In a K-way set with LRU replacement policy, a given line is replaced when K or more different lines mapped to its same cache set have been referenced since its last access”

Miss Probabilities: Basics (I)

• Miss probability depends on the impact on the cache of the memory regions accessed since the last access to the line to reuse

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Miss Probability: Basics (II)

• Miss probability = probability K or more lines have been mapped to the sets of the lines to reuse during the reuse distance

• Notice that the ratio of cache sets that have X lines is also the probability a given cache set has X lines

• We need a way to represent the distribution of the number of lines assigned to each set during the reuse distance

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Area Vectors

• Associated to data structure V we have area vector

• The area vector for each data structure is calculated separately as a functions of its access pattern

KVVVV SSSS 10

lines. received have that sets

portion theis 0 and V, from lines moreor

received have that sets of portion theis where0

iK

Ki, S

KS

iV

V

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Area Vector Example

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Miss Probability Computation

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Interference Area Vectors Calculation

• The references are analyzed in each nesting level i to count the number of points accessed in each dimension d (Nrid) and the distance between each two of them (Lrid)

• The region accessed may be described as the tuple

• In general this region describes an area with the shape of either a sequential access or an access to groups of consecutive elements separated by a constant stride.

• These two accesses (and others) have been modeled for the calculation of their corresponding interference area vectors

)),(,),,(),,(( 2211 AA dRiRidRiRiRiRiRi LNLNLN R

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Region Examples

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Area vectors union• As independent probabilities:

KiSS

)S(S

SSSS

SSSS

jiKj

ij

K

K

V

K

ijUi

K

j

K-j

iVU

VVVV

UUUU

0,)SS(

)SS(

:as defined is SS addition, their from

resulting vector area the, and

vectorsarea Given two

)(

10

10

VU

0 00VU

VU

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Consideration of the Relative Positions

• For each pair of data structures A and B, their overlapping coefficient Sol(A,B) is calculated: portion of cache sets that may contain lines belonging to both structures.

• Before adding the interference area vector generated by one of them in order to calculate the miss probability in the accesses to the other one, it is scaled using this factor

• If both references are sequential and are in translation (their indices only differ in added constants), a simple algorithm with total precision is applied

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Memory Performance Analysis Tool: MEPAT

• The model was integrated in Polaris– FORTRAN codes with references with affine indexes can be analyzed

– Predicts the behavior of caches with an artitrary size, line size and associativity

– Complemented with the Delphi CPU model– Optimization module: optimal tile size selection

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MEPAT Structure

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Validation with SPECfp95 and Perfect

Benchmarks

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Prediction vs Measurementin SPECfp95

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Prediction vs Measurement

in Perf. Bench.

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Typical Miss Ratio Errors

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Modeling Times(O200, R10000 180MHz)

Aditional 0.2 to 2.5 seconds for syntactical analysis, etc.

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Prediction vs HW Counters for Blocked Matrix Product (I)

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Prediction vs HW Counters for Blocked Matrix Product (II)

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Optimal Tile Size Search in the SPECfp95 and Perf. Bench. Codes

• Uses memory model + Delphi CPU model• Environment: Origin 200 with R10000

– Processor parameters known for Delphi– Good compiler: MIPSpro 7.3.1.1m

• Objective: generate code faster than that of the production compiler replacing the tile sizes it has chosen by those proposed by the model

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Problems in the Experiment

• In these codes only the last row and/or column is reused, rather than the whole tile

• Base addresses of the data structures are not available

• The data sets of several of the codes fit in the second level cache (1 MB)

• The execution time of some of the loops modified is too small to be meaningful, so the whole application was measured

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MEPAT vs MIPSpro

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Related Work

• Other models:– Ghosh and col.: CME (Cache Miss Equations), linear Diophantine equations.

– Vera and Xue: statistical sample of CMEs– Chatterjee and col: Presburger formulae– Harper and col.: cache footprints

• Other prediction tools:– Delphi– SPLAT

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Conclusions

• General strategy for the modeling of the memory hierarchy behavior

• Good precision: average miss ratio prediction error about 0.1%

• Very fast: milliseconds for SPECfp95 codes

• Complemented with CPU model to predict real execution times

• Competitive with a good production compiler

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Current / Future Work

• Trying/optimizing in different platforms– successful Pentium IV experiments

• Extension to model codes with conditionals

• Further extension to model indirections– SIGMETRICS’98, Europar’98

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Application to tile size selection: Pentium 4 @ 2GHz

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Application to tile size selection:

Itanium 2 @ 1.5GHz