Cache Related Preemption Delay for
Set-Associative Caches
Resilience Analysis
Sebastian Altmeyer, Claire Burguière, Jan Reineke
Predator Meeting, Pisa 2010
saarland
university
computer science
saarland
university
Context
computer science
Preemptive scheduling
Cache related preemption delay (CRPD):
I
I
Impact of preemption on the cache content
Overall cost of additional reloads due to preemption
T1
T2
= CRPD
= Task Activation
Altmeyer, Burguière, Reineke
CRPD for set-associative caches
Predator, Pisa 2010
2 / 16
CRPD for set-associative caches - LRU
saarland
university
computer science
CRPD computation:
I
I
preempted task: Useful Cache Blocks (UCB)
preempting task: Evicting Cache Blocks (ECB)
CRPD from UCB and ECB:
I Previous combination overestimates
⇒ Some UCBs remain useful under preemption
Altmeyer, Burguière, Reineke
CRPD for set-associative caches
Predator, Pisa 2010
3 / 16
saarland
university
Useful Cache Block - [Lee et al., 1996]
computer science
Definition (Useful Cache Block)
A memory block m at program point P is called a useful cache block, if
a) m may be cached at P
b) m may be reused at program point P 0 that may be reached from P
with no eviction of m on this path.
= hit
= miss
P
A
B
c
X
C
D
B
C
CRPDsUCB
CRPDUCB
=
CRPDsUCB
= BRT × min(|UCB(s)|, n)
s=1
Altmeyer, Burguière, Reineke
A
Cache Content:
[A, B, C, D]
CRPD for set-associative caches
n = associativity
BRT = Block Reload Time
Predator, Pisa 2010
4 / 16
saarland
university
Useful Cache Block - [Lee et al., 1996]
computer science
Definition (Useful Cache Block)
A memory block m at program point P is called a useful cache block, if
a) m may be cached at P
b) m may be reused at program point P 0 that may be reached from P
with no eviction of m on this path.
= hit
= miss
P
A
B
c
X
C
D
B
C
CRPDsUCB
CRPDUCB
=
CRPDsUCB
= BRT × min(|UCB(s)|, n)
s=1
Altmeyer, Burguière, Reineke
A
Cache Content:
[A, B, C, D]
CRPD for set-associative caches
n = associativity
BRT = Block Reload Time
Predator, Pisa 2010
4 / 16
saarland
university
Evicting Cache Blocks
[Tomiyama & Dutt, 2000]
computer science
Definition (Evicting Cache Blocks (ECB))
A memory block of the preempting task is called an evicting cache
block, if it may be accessed during the execution of the preempting
task.
Cache Content:
[A, B, C, D]
A
C
B
X
Y
Z
Cache Content:
[X , Y , Z , D]
B
A
C
= additional miss due to preemption (CRPD)
s
CRPDECB =
Altmeyer, Burguière, Reineke
0
if ECB(s) = ∅
BRT × n otherwise
CRPD for set-associative caches
Predator, Pisa 2010
5 / 16
saarland
university
Impact of the preempting task
on the preempted task
computer science
CRPD (using UCB and ECB)
CRPDUCB&ECB =
c
X
min(CRPDsUCB , CRPDsECB )
s=1
Altmeyer, Burguière, Reineke
CRPD for set-associative caches
Predator, Pisa 2010
6 / 16
saarland
university
Impact of the preempting task
on the preempted task (example)
[c, b, a, x]
0x0a
0x0b
0x0c
ECBs
= {e}
[e, c, b, a]
a
a
[a, c, b, x]
[a, e, c, b]
b
b
[b, a, c, x]
[b, a, e, c]
computer science
c
c
[c, b, a, x]
no miss
[c, b, a, e]
no miss
CRPDUCB ⇒ |UCB| = 3
CRPDECB ⇒ n = 4
CRPDUCB&ECB = min(CRPDUCB , CRPDECB ) ⇒ 3
I
Overestimation: number of additional misses= 0 < 3
Why?
I
I
I
|ECB| to evict a UCB = 2
but, |ECB| = 1
One single ECB is not sufficient to evict a UCB
Altmeyer, Burguière, Reineke
CRPD for set-associative caches
Predator, Pisa 2010
7 / 16
saarland
university
Refinement
computer science
Determining max|ECB| s.t. no additional cache miss occur
m ∈ UCB
m is
4-resilient
(res(m) = 4)
Altmeyer, Burguière, Reineke
m
a1
a2
a3
m
[m, , , , , , , ]
[a3 , a2 , a1 , m, , , , ]
[m, a3 , a2 , a1 , , , , ]
CRPD for set-associative caches
Predator, Pisa 2010
8 / 16
saarland
university
Resilience analysis
computer science
Definition (Resilience)
The resilience resP (m) of memory block m at program point P is the
greatest l, such that all possible next accesses to m,
a) that would be hits without preemption,
b) would still be hits in case of a preemption with l accesses at P.
preempted task
m ∈ UCB
res(m) = 4
m
a1
a2
preempting task
[m, , , , , , , ]
ECB = {e1 , e2 , e3 , e4 }
a3
[a3 , e4 , e3 , e2 , e1 , a2 , a1 , m]
m
[m, a3 , e4 , e3 , e2 , e1 , a2 , a1 ]
if |ECB| ≤ l then the UCB is not evicted
Altmeyer, Burguière, Reineke
CRPD for set-associative caches
Predator, Pisa 2010
9 / 16
saarland
university
CRPD using resilience
computer science
CRPD (combining UCB and ECB by using resilience)
blocks contributing to CRPD
CRPD
≤
BRT
×
z
}|
{
| UCB
\
{m
|
res(m)
=
|ECB|}
|
| {z } |
{z
}
useful
Altmeyer, Burguière, Reineke
CRPD for set-associative caches
remain useful
Predator, Pisa 2010
10 / 16
saarland
university
CRPD using resilience - example
[c, b, a, x]
0x0a
0x0b
0x0c
ECBs
= {e}
[e, c, b, a]
I
I
I
a
a
[a, c, b, x]
[a, e, c, b]
b
b
[b, a, c, x]
[b, a, e, c]
computer science
c
c
[c, b, a, x]
no miss
[c, b, a, e]
no miss
|ECB| = 1
res(a) = res(b) = res(c) = 1
res
CRPDUCB&ECB
= BRT × |UCB \ {m | res(m) = |ECB| }| = 0
Instead of: CRPDUCB&ECB = min(CRPDUCB , CRPDECB ) = 3 × BRT
Altmeyer, Burguière, Reineke
CRPD for set-associative caches
Predator, Pisa 2010
11 / 16
saarland
university
Evaluation Setting
computer science
Cachesize 8KB
8 ways
32 sets
linesize 32 bytes
LRU caches
Testcases: Mälardalen benchmark suite:
Altmeyer, Burguière, Reineke
CRPD for set-associative caches
Predator, Pisa 2010
12 / 16
saarland
university
Evaluation Benchmarks
Task
minmax
insertsort
fibcall
fac
bs
bsort100
ns
matmult
fir
crc
select
qsort-exam
sqrt
qurt
Altmeyer, Burguière, Reineke
Code Size
608B
384B
256B
256B
320B
544B
576B
864B
928B
1216B
1280B
1440B
3680B
4160B
computer science
Cache Util.
7.4%
4.7%
3.1%
3.1%
3.9%
6.6%
7%
10.5%
11.3%
14.8%
15.6%
17.6%
44.9%
50.8%
CRPD for set-associative caches
|UCB|
4
5
5
6
8
10
11
12
22
35
37
42
101
118
Predator, Pisa 2010
13 / 16
saarland
university
Evaluation
computer science
preempted by fibcall (#ECBs= 8)
100
10
1
min
in
f
max sertso ac
rt
bs
bso
n
rt10 s
0
mat
f
mult ir
crc
sele
qso
sqrt
xam
crc
sele
qso
sqrt
xam
ct
rt-e
preempted by qurt (#ECBs= 121)
100
10
1
min
in
f
max sertso ac
rt
resilience
Altmeyer, Burguière, Reineke
bs
bso
n
rt10 s
0
tan
mat
f
mult ir
UCB & ECB
CRPD for set-associative caches
ct
rt-e
#UCBs
Predator, Pisa 2010
14 / 16
saarland
university
Conclusions
computer science
UCB and ECB analyses:
I
pessimistic overapproximation of the CRPD
Resilience analysis:
I
I
regain some precision
reduce pessimism
Resilience analysis:
I
I
simple data-flow analyses
similar to UCB analysis for LRU
Altmeyer, Burguière, Reineke
CRPD for set-associative caches
Predator, Pisa 2010
15 / 16
saarland
university
Further reading
computer science
Altmeyer, S. & Burguière, C. (2009).
In Proceedings of the 21st Euromicro Conference on Real-Time Systems (ECRTS ’09) pp.
109–118, IEEE Computer Society.
Lee, C.-G., Hahn, J., Min, S. L., Ha, R., Hong, S., Park, C. Y., Lee, M. & Kim, C. S. (1996).
In RTSS’96 p. 264, IEEE Computer Society.
Negi, H. S., Mitra, T. & Roychoudhury, A. (2003).
In CODES+ISSS’03 ACM.
Reineke, J. (2008).
Caches in WCET Analysis.
PhD thesis, Universität des Saarlandes, Saarbrücken.
Staschulat, J. & Ernst, R. (2007).
ACM Trans. on Embedded Computing Sys. 6, 25.
Tan, Y. & Mooney, V. (2004).
In SCOPES’04 pp. 182–199,.
Tomiyama, H. & Dutt, N. D. (2000).
In CODES’00 ACM.
Altmeyer, Burguière, Reineke
CRPD for set-associative caches
Predator, Pisa 2010
16 / 16
saarland
university
m
m
(a)
Altmeyer, Burguière, Reineke
m
m
2-resilient
m
m is not useful 2-resilient
computer science
2-resilient
0-resilient 0-resilient
l-resilience analysis
m
(b)
CRPD for set-associative caches
Predator, Pisa 2010
16 / 16
saarland
university
CPRD using ECB: Pitfall
computer science
0x08
0x09
0x0a
0x0b
[b, a, 9, 8]
ECBs
= {e}
[e, b, a, 9]
8
8∗
[8, b, a, 9]
[8, e, b, a]
9
9∗
[9, 8, b, a]
[9, 8, e, b]
a
a∗
[a, 9, 8, b]
[a, 9, 8, e]
b
b∗
[b, a, 9, 8]
0 misses
[b, a, 9, 8]
4 misses
|UCB(s)| = 4
|ECB(s)| = 1
n=4
number of additional misses= 4
Altmeyer, Burguière, Reineke
CRPD for set-associative caches
Predator, Pisa 2010
16 / 16
saarland
university
Upper-bound on the CRPD - direct-mapped caches
computer science
using UCB [Lee et al., 1996]:
CRPDUCB = BRT · |{si | ∃m ∈ UCB : m mod c = si }|
using ECB [Tomiyama & Dutt, 2000]:
CRPDECB = BRT · |{si | ∃m ∈ ECB : m mod c = si }|
using UCB and ECB [Negi et al., 2003, Tan & Mooney, 2004]:
CRPDUCB&ECB
= BRT · |{si |
∃m ∈ UCB : m mod c = si
∧∃m0 ∈ ECB : m0 mod c = si }|
Altmeyer, Burguière, Reineke
CRPD for set-associative caches
Predator, Pisa 2010
16 / 16
saarland
university
CRPD for FIFO: Pitfalls
ECBs
= {x}
[b, a]
[x, b]
a
a∗
[b, a]
[a, x]
e∗
e∗
[e, b]
[e, a]
computer science
b
b∗
[e, b]
[b, e]
c∗
c∗
[c, e]
[c, b]
e
e∗
[c, e]
2 misses
[e, c]
5 misses
|UCB(s)| = 2
|ECB(s)| = 1
n=2
But: number of additional misses= 3
Altmeyer, Burguière, Reineke
CRPD for set-associative caches
Predator, Pisa 2010
16 / 16
saarland
university
CRPD for PLRU: Pitfalls
1
a
1
b
ECBs
= {x, y}
c
d
a
y
b
c
c
a
a
b
d
c
b
a
d
c
b
a
y
1
b
d
c
b
a
y
1
b
d
c
d
b
a
a
c
c
0 misses
d
1
0
1
y
b
d*
1
c
0
1
0
a*
0
1
c
d
0
d
1
1
b
0
1
x
d
0
1
y
b
1
a
0
1
c*
1
x
d
1
b
0
1
1
y
0
b*
0
x
d
0
c
0
0
d*
0
1
b
0
1
1
0
a
0
d
1
computer science
b
1
d
a
5 misses
c
|UCB(s)| = 4
|ECB(s)| = 2
n=4
But: number of additional misses= 5
Altmeyer, Burguière, Reineke
CRPD for set-associative caches
Predator, Pisa 2010
16 / 16