A Probabilistic Quantitative Analysis of … Probabilistic Quantitative Analysis of...
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A Probabilistic Quantitative Analysis ofProbabilistic-Write/Copy-Select
Christel Baier1, Benjamin Engel2, Sascha Kluppelholz1,Steffen Marcker1, Hendrik Tews2, Marcus Volp2
1Institute for Theoretical Computer Science
2Operating-Systems Group
Technische Universitat Dresden, Germany
NASA Formal Methods Symposium (NFM’13)May 16, 2013
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Motivation
Observation: traditional locking does not scale any more
atomic operations are slow and become increasingly expensive
locking schemes will become more complex and
scalability becomes problematic on future hardware systems
Idea: Probabilistic-Write/Copy-Select (PWCS) [Mc Guire’11]
no locks, no atomic operations
make inconsistencies detectable (e.g., tags, hashes)
sufficiently high probability to find a consistent replica
Properties of PWCS
measure-based experiments [Mc Guire’11]: promising approach
promising to work with more relaxed memory models
instance of a new class of algorithms (inherent randomness)
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Motivation
Observation: traditional locking does not scale any more
atomic operations are slow and become increasingly expensive
locking schemes will become more complex and
scalability becomes problematic on future hardware systems
Idea: Probabilistic-Write/Copy-Select (PWCS) [Mc Guire’11]
no locks, no atomic operations
make inconsistencies detectable (e.g., tags, hashes)
sufficiently high probability to find a consistent replica
Properties of PWCS
measure-based experiments [Mc Guire’11]: promising approach
promising to work with more relaxed memory models
instance of a new class of algorithms (inherent randomness)
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Motivation
Observation: traditional locking does not scale any more
atomic operations are slow and become increasingly expensive
locking schemes will become more complex and
scalability becomes problematic on future hardware systems
Idea: Probabilistic-Write/Copy-Select (PWCS) [Mc Guire’11]
no locks, no atomic operations
make inconsistencies detectable (e.g., tags, hashes)
sufficiently high probability to find a consistent replica
Properties of PWCS
measure-based experiments [Mc Guire’11]: promising approach
promising to work with more relaxed memory models
instance of a new class of algorithms (inherent randomness)
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The PWCS protocol [Mc Guire’11]
Writer Replica Reader
for i=1..nE1Data1B1
E2Data2B2
.
.
.
EnDatanBn
for i=n..1
r = replica[i]; r = replica[i];
r.end tag++; ta = r.begin tag;
r.write data(); r.copy data();r.begin tag++; tb = r.end tag;
endfor if (ta == tb)
return data;
endfor
// error case
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The PWCS protocol [Mc Guire’11]
Writer Replica Reader
for i=1..nE1Data1B1
E2Data2B2
.
.
.
EnDatanBn
for i=n..1
r = replica[i]; r = replica[i];
r.end tag++; ta = r.begin tag;
r.write data(); r.copy data();r.begin tag++; tb = r.end tag;
endfor if (ta == tb)
return data;
endfor
// error case
CTMC model transition system CTMC modelmodel
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Contribution (NFM’13)
continuous-time Markov chain (CTMC) model for PWCSwith multiple writers
identify quantitative measures for the evaluation of PWCS
formalization of quantitative measures in terms ofcontinuous stochastic reward logic (CSRL)
formal quantitative analysis of PWCS using theprobabilistic model checker PRISM
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Outline
1 Motivation
2 PWCS model
3 PWCS properties
4 PWCS analysis
5 Conclusion and future work
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Continuous-time Markov chain (CTMC)
Definition (CTMC)
A CTMC is a tuple M = 〈S ,Act,R , µ〉, where
S a finite state space,
Act a finite set of action names,
R : S × Act × S → R≥0 the rate matrix of M,
µ : S → [0, 1] a distribution on S , i.e.,∑s∈S
µ(s) = 1
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Continuous-time Markov chain (CTMC)
Definition (CTMC)
A CTMC is a tuple M = 〈S ,Act,R , µ〉, where
S a finite state space,
Act a finite set of action names,
R : S × Act × S → R≥0 the rate matrix of M,
µ : S → [0, 1] a distribution on S , i.e.,∑s∈S
µ(s) = 1
Probability for sλ:α−→ s ′ ready to fire in [0, t] is
1− e−λt
Thus, the average delay of this transition is 1/λ.
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Continuous-time Markov chain (CTMC)
Definition (CTMC)
A CTMC is a tuple M = 〈S ,Act,R , µ〉, where
S a finite state space,
Act a finite set of action names,
R : S × Act × S → R≥0 the rate matrix of M,
µ : S → [0, 1] a distribution on S , i.e.,∑s∈S
µ(s) = 1
Probability for firing sλ:α−→ s ′ in [0, t] is
P(s, α, s ′) ·(1− e−E(s)·t)
where E (s) denotes the exit rate of state s, i.e., the sum ofthe rates of all outgoing transitions of state s.
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Continuous-time Markov chain (CTMC)
Definition (CTMC)
A CTMC is a tuple M = 〈S ,Act,R , µ〉, where
S a finite state space,
Act a finite set of action names,
R : S × Act × S → R≥0 the rate matrix of M,
µ : S → [0, 1] a distribution on S , i.e.,∑s∈S
µ(s) = 1
Probability for firing sλ:α−→ s ′ in [0, t] is
λ/E (s) ·(1− e−E(s)·t)
where E (s) denotes the exit rate of state s, i.e., the sum ofthe rates of all outgoing transitions of state s.
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PWCS composed CTMC model
Product of CTMC for the writers, CTMC for the readers, andordinary (non-stochastic) transition systems for the replicas.
sλ:α−−→ s ′
〈s, x〉 λ:α−−→ 〈s ′, x〉w
λ:!a−−→ w ′, r?a−→ r ′
〈w , r , y〉 λ:a−−→ 〈w ′, r ′, y〉
x : local states of all other componentsy : local states of all readers and remaining writers and replicas
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PWCS model
consistentk currently modifiedk
damagedk
?writing startedk
?writing finishedk
?writing startedk
?writing startedk
?writing startedk
?writing finishedk
Transition system model of a replica
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PWCS model
success j
error j
idle j
readingj1
checkj1
readingjK−1
checkjK−1
.
.
.
.
.
.
readingjK check
jK
κ : reading startedjK
δ : reading finishedj1
ρ : reading startedj1
δ : reading finishedjK−1
ρ : reading startedjK−1
δ : reading finishedjK
σ : return to read idle j
ν : return to read idle j
CTMC model of a reader
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PWCS model
CTMC model of a writer
idle i writing i
1ready i
1
writing i
2ready i
2
.
.
....
writing i
K ready i
K
γ : !writing started i
1λ : !writing finished i
1
µ : !writing started i
2
λ : !writing finished i
2
µ : !writing started i
K
λ : !writing finished i
K
η : return to write idle i
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Outline
1 Motivation
2 PWCS model
3 PWCS properties
4 PWCS analysis
5 Conclusion and future work
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Interesting quantitative measures
M1: probability to successfully read the data
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Interesting quantitative measures
M1: probability to successfully read the data
M2: 99% time-quantile for successful reading
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Interesting quantitative measures
M1: probability to successfully read the data
M2: 99% time-quantile for successful reading
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Interesting quantitative measures
M1: probability to successfully read the data
M2: 99% time-quantile for successful reading
M3: fraction of time in which all replicas are damaged
M4: average time for repairing a damaged replica
M5: 99% time-quantile for repairing a damaged replicawithin time t
M6: probability to write at least c consistent replicawithin one write cycle
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Interesting quantitative measures
M1: probability to successfully read the data
M2: 99% time-quantile for successful reading
M3: fraction of time in which all replicas are damaged
M4: average time for repairing a damaged replica
M5: 99% time-quantile for repairing a damaged replicawithin time t
M6: probability to write at least c consistent replicawithin one write cycle
... on the long run ...
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Long-run behavior
Steady-state distribution
Function θ : S → [0, 1] with
θ(s)def= lim
t→∞θ(s, t) with
θ(s, t) the probability for being in state s at time t ∈ R≥0.
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Long-run behavior
Steady-state distribution
Function θ : S → [0, 1] with
θ(s)def= lim
t→∞θ(s, t) with
θ(s, t) the probability for being in state s at time t ∈ R≥0.
Important:
θ is well-defined distribution on S for finite CTMCs.
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Long-run behavior
Steady-state distribution
Function θ : S → [0, 1] with
θ(s)def= lim
t→∞θ(s, t) with
θ(s, t) the probability for being in state s at time t ∈ R≥0.
Important:
θ is well-defined distribution on S for finite CTMCs.
Long-run probabilities
Let M = 〈S ,Act,R , µ〉 be a CTMC. We refer to theprobability measure obtained for the CTMCMθ = 〈S ,Act,R , θ〉.
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Conditional long-run behavior
Probability measure
Let M = 〈S ,Act,R , µ〉 be a CTMC and U ⊆ S be a set ofstates s.t. θ(U) > 0. We refer to the probability measureobtained for the CTMC MU
θ =Mν = 〈S ,Act,R , ν〉
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Conditional long-run behavior
Probability measure
Let M = 〈S ,Act,R , µ〉 be a CTMC and U ⊆ S be a set ofstates s.t. θ(U) > 0. We refer to the probability measureobtained for the CTMC MU
θ =Mν = 〈S ,Act,R , ν〉, where
ν(s) =
{0 if s ∈ S \ Uθ(s)/θ(U) if s ∈ U
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Conditional long-run behavior
Probability measure
Let M = 〈S ,Act,R , µ〉 be a CTMC and U ⊆ S be a set ofstates s.t. θ(U) > 0. We refer to the probability measureobtained for the CTMC MU
θ =Mν = 〈S ,Act,R , ν〉, where
ν(s) =
{0 if s ∈ S \ Uθ(s)/θ(U) if s ∈ U
Conditional long-run queries
Pr(Π∣∣U) : conditional long-run probability
where Π is a measurable set of infinite paths, U ⊆ S a set of
states with θ(U) > 0.
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Conditional long-run behavior
Probability measure
Let M = 〈S ,Act,R , µ〉 be a CTMC and U ⊆ S be a set ofstates s.t. θ(U) > 0. We refer to the probability measureobtained for the CTMC MU
θ =Mν = 〈S ,Act,R , ν〉, where
ν(s) =
{0 if s ∈ S \ Uθ(s)/θ(U) if s ∈ U
Conditional long-run queries
Pr(Π∣∣U) : conditional long-run probability
AccRew(♦T∣∣U) : conditional long-run accumulated reward
where Π is a measurable set of infinite paths, U ⊆ S a set of
states with θ(U) > 0. We assume Pr(♦T |U
)= 1.
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Queries for interesting long run properties
Q1: probability to successfully read a replica
Pr(¬error j U idle j
∣∣ reading started jK
)Q2: time-quantile for successful reading within time bound t
min{t : p ≤ Pr
(¬error j U≤t idle j
∣∣ reading started jK
)}
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Queries for interesting long run properties
Q3: fraction of time in which all replicas are damaged
θ(damaged1 ∧ . . . ∧ damagedK
)
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Queries for interesting long run properties
Q3: fraction of time in which all replicas are damaged
θ(damaged1 ∧ . . . ∧ damagedK
)Q4: average time for repairing a damaged replica
AccRew(♦ consistentk
∣∣ just damagedk
)
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Queries for interesting long run properties
Q3: fraction of time in which all replicas are damaged
θ(damaged1 ∧ . . . ∧ damagedK
)Q4: average time for repairing a damaged replica
AccRew(♦ consistentk
∣∣ just damagedk
)Q5: time-quantile for repairing a damaged replica within time t
min{t : p ≤ Pr
(♦≤t consistentk
∣∣ just damagedk
)}
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Queries for interesting long run properties
Q3: fraction of time in which all replicas are damaged
θ(damaged1 ∧ . . . ∧ damagedK
)Q4: average time for repairing a damaged replica
AccRew(♦ consistentk
∣∣ just damagedk
)Q5: time-quantile for repairing a damaged replica within time t
min{t : p ≤ Pr
(♦≤t consistentk
∣∣ just damagedk
)}Q6: probability to write at least c replica within one cycle
Pr(Πc
∣∣ writing started i1
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Outline
1 Motivation
2 PWCS model
3 PWCS properties
4 PWCS analysis
5 Conclusion and future work
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Selected parameters and scenarios
Common parameters
time ratewrite duration 2 λ = 0.5read duration 1 δ = 1other 0.01 µ = ρ = σ = ν = 100
Selected scenarios
frequent reads moderate readsmoderate writes moderate writes
time rate time rateidle time (writer) 20 γ = 0.05 200 γ = 0.005idle time (reader) 2 κ = 0.5 20 κ = 0.05
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Selected parameters and scenarios
Common parameters
time ratewrite duration 2 λ = 0.5read duration 1 δ = 1other 0.01 µ = ρ = σ = ν = 100
Selected scenarios
frequent reads moderate readsmoderate writes moderate writes
time rate time rateidle time (writer) 20 γ = 0.05 200 γ = 0.005idle time (reader) 2 κ = 0.5 20 κ = 0.05
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Results
Q1: probability to successfully read the datamoderate reads, moderate writes
1 2 3 4 5no. of replica K
0.75
0.80
0.85
0.90
0.95
1.00pro
babili
ty p
1 writer
2 writers
3 writers
4 writers
5 writers
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Results
Q1: probability to successfully read the datafrequent reads, moderate writes
1 2 3 4 5no. of replica K
0.75
0.80
0.85
0.90
0.95
1.00pro
babili
ty p
1 writer
2 writers
3 writers
4 writers
5 writers
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Results
Q2: time-quantile for successful readingmoderate reads, moderate writes, 5 replicas
0 2 4 6 8 10 12time bound t
0.0
0.2
0.4
0.6
0.8
1.0pro
babili
ty p
1 writer
2 writers
3 writers
4 writers
5 writers
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Results
Q2: time-quantile for successful readingfrequent reads, moderate writes, 5 replicas
0 2 4 6 8 10 12time bound t
0.0
0.2
0.4
0.6
0.8
1.0pro
babili
ty p
1 writer
2 writers
3 writers
4 writers
5 writers
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Results
Q3: time fraction in which all replicas are damagedmoderate reads, moderate writes
1 2 3 4 5no. of replica K
0.000
0.005
0.010
0.015
0.020
0.025
0.030
0.035
0.040pro
babili
ty p
2 writers
3 writers
4 writers
5 writers
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Results
Q3: time fraction in which all replicas are damagedfrequent reads, moderate writes
1 2 3 4 5no. of replica K
0.00
0.05
0.10
0.15
0.20
0.25
0.30
0.35pro
babili
ty p
2 writers
3 writers
4 writers
5 writers
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Results
Q5: time-quantile for repairing a damaged replica within time tttmoderate reads, moderate writes, 5 replicas
0 50 100 150 200 250 300 350 400time bound t
0.0
0.2
0.4
0.6
0.8
1.0pro
babili
ty p
2 writers
3 writers
4 writers
5 writers
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Results
Q5: time-quantile for repairing a damaged replica within time tttfrequent reads, moderate writes, 5 replicas
0 5 10 15 20 25 30 35 40time bound t
0.0
0.2
0.4
0.6
0.8
1.0pro
babili
ty p
2 writers
3 writers
4 writers
5 writers
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Outline
1 Motivation
2 PWCS model
3 PWCS properties
4 PWCS analysis
5 Conclusion and future work
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Conclusion and future work
Conclusion
CTMC model for PWCS with multiple writers
identification of quantitative measures for the evaluation of PWCS
formalization of quantitative measures in terms of CSRL queries
formal quantitative analysis of PWCS using PRISM
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Conclusion and future work
Conclusion
CTMC model for PWCS with multiple writers
identification of quantitative measures for the evaluation of PWCS
formalization of quantitative measures in terms of CSRL queries
formal quantitative analysis of PWCS using PRISM
Future work
comparative quantitative analysis with alternative protocols
stronger object consistency in PWCS (e.g., multiple objects)
other synchronization primitives (e.g., barriers)
formal methods for quantile and (conditional) long run properties
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