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Formalization and Strictness of Simulation Event Orderings
Transcript of Formalization and Strictness of Simulation Event Orderings
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10 May 2004 PADS04 1
Formalization and Strictness of Simulation Event Orderings
Teo Yong Meng1,2 and Bhakti Onggo2
1Singapore-Massachusetts Institute of Technology Alliance2Department of Computer ScienceNational University of Singapore
email: [email protected]: www.comp.nus.edu.sg/~teoym
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Related Work
Berry and Jefferson (1985) – critical path analysisBagrodia et al. (1991) – Space-Time algorithmBarriga et al. (1995) – incremental benchmarkJha et al. (1996) – ideal simulation protocolBalakrishnan et al. (1997) – workloadFerscha et al. (1997) – NMAP, three layers (model, protocol, platform)Liu et al. (1999) – Dartmouth Scalable Simulation Framework (DaSSF)……..
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Previous Work1. B.S.S. Onggo and Y.M. Teo, Performance Trade-off in Distributed
Simulation, Proceedings of the 6th IEEE International Workshop on Distributed Simulation and Real Time Applications, pp. 77-84, IEEE Computer Society Press, Texas, USA, October 2002.
2. Y.M. Teo, B.S.S Onggo and S.C. Tay, Effect of Event Orderings on Memory Requirement in Parallel Simulation, Proceedings of the 9th International Symposium on Modelling, Analysis and Simulation ofComputer and Telecommunication Systems, pp. 41-48, IEEE Computer Society Press, Cincinnati, USA, August 2001.
3. H. Wang, Y.M. Teo and S.C. Tay, An Analytic Method for Predicting Simulation Parallelism, Proceedings of the 33rd Annual Simulation Symposium, pp. 211-218, IEEE Computer Society Press, Washington D.C., USA, April 2000.
www.comp.nus.edu.sg/~teoym/recent-publications.htm
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Outline
IntroductionWhy? - MotivationWhat? - Performance FrameworkHow?
Formalization – Event OrderingsCharacterization – Not coveredStrictness of Event OrderingsExperimental Results
Summary
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Motivation
“The World”
A Model
Sequential /Parallel Simulator
SC1
SC2
SC3 SC4 SC6
SC5
LP1 LP3 LP4
LP2
LP6
LP5
PP1 PP2
Phys
ical
Sy
stem
Si
mul
atio
n M
odel
Si
mul
ator
PhysicalSystem
SimulationModel
Simulator /Implementation
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Parallel Simulation Performance – Layered Approach
SC1
SC2
SC3 SC4 SC6
SC5
LP1 LP3 LP4
LP2
LP6
LP5
PP1 PP2
Phys
ical
Sy
stem
Si
mul
atio
n M
odel
Si
mul
ator
ProblemParallelism
ModelParallelism
EffectiveParallelism
event orderings,
synchronization
protocols, etc.
event level parallelism
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Proposed Framework = Formalization + Characterization
Physical System
Simulation Model
Simulator
Event Orderings
Perf
Perf
Perf
Πprob
Mprob
Πord
Mord
Πsync
Msync, Mtot, Mshr, Mdst
Norm
alization
Perf
strictness (ςR), stricter
Com
parison
Poset
Perf. = performance measure
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Formalization using Poset
Simulation Model
x ⇒ y, iff (x.ts < y.ts) or (x.ts = y.ts and priority(x) < priority(y)) (Total)
x ⇒ y, iff x.ts < y.ts (Timestamp)
x ⇒ y, if (y.pred = x) or (y.ante = x) or (x.ts + W < y.ts) (Time-interval)
x ⇒ y, if (y.pred = x) or (y.ante = x) (Partial)
x ⇒ y, if (y.pred = x) or (y.ante = x) or ( ⎣x.ts/W⎦ < ⎣y.ts/W⎦ )
x ⇒ y, if (y.pred = x) or (x.ts + la < y.ts)
x ⇒ y, if (y.pred = x) or (x.ts + la < y.ts) or ( ⎣x.ts/W⎦ <⎣y.ts/W⎦ )
x ⇒ y, if y.ante = x
PhysicalSystem
x ⇒ y, iffx.ts < y.ts
Simulator
Sequential
TW Protocol
BTW Protocol
CMB Protocol
BL Protocol
Unsynchronized
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Characterization Time, Space & Strictness
Layers Time Space Strictness
Physical System prob
prob
DE
=Π
∑= <<
=m
ii
Dt
prob tQMprob
1 0)(max
Simulation
Model ordord
DE
=Π ∑= <<
=m
ii
Dt
ord tLMord
1 0)(max
Simulator sync
sync
DE
=Π
∑= <<
=n
ii
Dt
sync tBMsync
1 0)(max
syncordnorm
probtot MMMM ++=
||)}||
||||||(||{max
,
1,,0
ti
m
ititiDt
shr
B
LQM
+
+= ∑=
<≤
||)}||||||
||(||{max
,)1(,)1(
1,)1(
1 0
tjkitjki
k
jtjki
n
i Dt
dst
BL
QM
+−+−
=+−
=<≤
++
= ∑∑
tot
R
SS
=ς
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Formalization of Simulation Event Orderings
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Related Work (1)
Memory ConsistencyMemory operation orderings - sequential consistency (Lamport, 1979), Weak Ordering (Dubois, 1990), etc.
Broadcast CommunicationMessage orderings - FIFO order, causal order, total order (Hadzilacos and Toueg, 1993), etcImplementation: Birman et al. (1987), Gambhire et al. (2000)
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Related Work (2)
HLA/Time Management causal order, timestamp, … (Fujimoto and Weatherly, 1996), AT/ATC (Fujimoto, 1999), Causal receive (Zhou et al., 2002)
Discrete-event simulationSimulation event orderings - partial event order, time-interval, timestamp, total event order, etc.Implementation / Simulator: Sequential (Banks et al., 2000), Time Warp (Jefferson, 1985), CMB (Chandy et al., 1979), Bounded Lag (Lubachesky, 1989), Bounded Time Warp (Turner, 1992), etc.
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Partially Ordered Set (Poset)
DefinitionA poset is a tuple (S, R) where S is a set and R is a partial order on the set S. R must be anti-reflexive, anti-symmetric and transitive.
Types of Orderings:Partial order (Dushnik, 1941)Total order (Dushnik, 1941)Interval order (Fishburn, 1988)Tolerance order (Bogart, 1995)Semi-order (Pirlot, 1997)Split semi-order (Fishburn, 1999)……………
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Simulation Event Orderings
Events x and y are comparable if x must be ordered before y or vice-versa. Otherwise, they are concurrent (non-comparable).
DefinitionA simulation event ordering is a tuple (E, SR) where E is a set of events and SR is a set of comparable events based on event order R. Event order R must be anti-reflexive, anti-symmetric, and transitive.
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Event Order: Physical System
A physical system is viewed as a network of service centers.
Events in a physical system occur in a physical time order.
DefinitionLet x be an event in a physical system and x.tthe physical time when event x happens. The event order in a physical system dictates that for all x and y (where x ≠ y), x is ordered before y if and only if x.t < y.t.
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Event Order: Simulation Model
Virtual time paradigmeach service center is modeled as a logical process (LP),an event in the physical system is modeled as a simulation event, and the physical time when it occurs is modeled as a simulation time (or timestamp)
Events can be executed/simulated in different orderings:
Total event order - event x is ordered before event y in iff (x.ts < y.ts) or (x.ts = y.ts and priority(x) < priority(y)).Partial event order - event x is ordered before event y in if (y.pred = x) or (y.ante = x). ………
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Event Order: ImplementationA (sequential, parallel, distributed) simulator maintains a specific event ordering during runtime.
Sequential SimulatorEvents are sorted in a timestamp order using a Future Event ListEvent with the smallest timestamp is executed, in case of a tie, event with higher priority will be executed →total event order
Time Warp SimulatorRollback ensures event with the smallest timestamp in an LP is committed first (y.pred = x).Aggressive cancellation by sending ant-messages ensure that an antecedent will be committed first (y.ante = x).
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Formalization based on Event Orderings - Summary
S1
S2
S3
S4
01a 2
2a 41d 6
6a 92d 12
6d
55a 7
3d 87a 10
9a 1111a 13
12a 1413a
44a 7
4d 1010a 13
10d
108d
Timestamp
43a
0 2 4 6 8 10
Serv
ice
Cen
ter
S1
S2
S3
S4
a)
b)
01a 2
2a 41d 6
6a 92d 12
6d
55a 7
3d 87a 10
9a 1111a 13
12a 1413a
44a 7
4d 1010a 13
10d
108d
43a
LP
LP
LP
LP
01a 2
2a 41d 6
6a 92d 12
6d
Timestep0 1 2 3 4 5 6 7
Logi
cal P
roce
ss
43a
44a
74d 10
10a 1310d
88a8
8a 108d10
8d
1413a13
12a
1111a
109a8
7a
73d5
5a73d 8
7a
Problem/ Physical System
Simulation Model – Partial Event Order
Implementation – Time Warp
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Formalization - Summary
Simulation Model
x ⇒ y, iff (x.ts < y.ts) or (x.ts = y.ts and priority(x) < priority(y)) (Total)
x ⇒ y, iff x.ts < y.ts (Timestamp)
x ⇒ y, if (y.pred = x) or (y.ante = x) or (x.ts + W < y.ts) (Time-interval)
x ⇒ y, if (y.pred = x) or (y.ante = x) (Partial)
x ⇒ y, if (y.pred = x) or (y.ante = x) or ( ⎣x.ts/W⎦ < ⎣y.ts/W⎦ )
x ⇒ y, if (y.pred = x) or (x.ts + la < y.ts)
x ⇒ y, if (y.pred = x) or (x.ts + la < y.ts) or ( ⎣x.ts/W⎦ <⎣y.ts/W⎦ )
x ⇒ y, if y.ante = x
PhysicalSystem
x ⇒ y, iffx.ts < y.ts
Simulator
Sequential
TW Protocol
BTW Protocol
CMB Protocol
BL Protocol
Unsynchronized
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Strictness Analysis
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Strictness of Event Orderings
Why?To compare event dependencies of different event orderingsTo quantify the degree of event dependenciesamong different event ordersTime independent
How?Stricter relationStrictness measure (ςR)
01a 2
2a 41d 6
6a 92d 12
6d
55a 7
3d 87a 10
9a 1111a 13
12a 1413a
44a 7
4d 1010a 13
10d
108d
43a
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Stricter RelationDefinition.
Let (E, SR1) and (E, SR2) be two event orderings on the same set of events E. Event order R1 is stricter than R2 if for any E, SR2 ⊆ SR1. An event order R1 is incomparableto event order R2 if we can find two sets of events E1 and E2, such SR2 ⊆ SR1 is true for E1 but SR2 ⊆ SR1 is not true for E2.
Lemma 3.1. The properties of a stricter relation are:if R1 is stricter than R2 and R2 is stricter than R1, then R1 = R2 (anti-symmetric).if R1 is stricter than R2 and R2 is stricter than R3, then R1 is stricter than R3 (transitive).
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Strictness
The strictness of an event order R ( ςR) is defined as
where and is the size of the set of comparable (or non-concurrent) events ordered by R and the total event order respectively.
Strictness of an event ordering ranges from 0 when SR = ∅, and 1 when R is the total event order.
tot
R
SS
RS totS
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Strictness Example
Set of events E = { 103a , 11
4a , 115a , 12
2d , 136a , 13
3d , 147a ,
154d , 15
5d , 168a }, hence ||E|| = 10.
|| Stot || = ||E|| × (||E||-1) / 2 = 45
SR = {( 114a , 13
6a ), ( 136a , 15
4d ), ( 114a , 15
4d ), ( 103a , 12
2d ), ( 122d , 13
3d ),
( 133d , 16
8a ), ( 103a , 13
3d ), ( 103a , 16
8a ), ( 122d , 16
8a ), ( 122d , 13
6a ),
( 103a , 13
6a ), ( 103a , 15
4d ), ( 122d , 15
4d ), ( 133d , 14
7a ), ( 103a , 14
7a ),
( 122d , 14
7a ), ( 103a , 15
5d ), ( 122d , 15
5d ), ( 133d , 15
5d ), ( 115a , 14
7a ),
( 147a , 15
5d ), ( 155d , 16
8a ), ( 115a , 15
5d ), ( 115a , 16
8a ), ( 147a , 16
8a )},
hence || SR || = 25
ςR = || SR || / || Stot || = 0.56
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Comparison of Strictness of Event Orderings
Tot : Total CMB : Chandy-Misra-Bryant Partial : Partial TS : Timestamp BL : Bounded Lag Unsync : Unsynchronized TI : Time-interval BTW : Bounded Time Warp
TOT
BL
TS
TI
BTW
CMB
Unsync
Causality Line Maintains Causality
Ignores Causality
Sequential Simulator
Parallel Simulators
Totally Unordered
Partial
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Instrumentation
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Benchmarks
Open system - MIN(n×n, ρ) [Teo&Tay95]Closed system - PHOLD(n×n, m) [Fujimoto90]
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Strictness Analysis (1)
MIN (n, 0.8)
0.0
0.2
0.4
0.6
0.8
1.0
n=8 n=16 n=24 n=32
Problem Size
Str
ictn
ess
TotalTSTI(1)TI(2)CMBPartial
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Strictness Analysis (2)
PHOLD (n, 4)
0.0
0.2
0.4
0.6
0.8
1.0
n=8 n=16 n=24 n=32
Problem Size
Str
ictn
ess
TotalTSTI(1)TI(2)CMBPartial
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Strictness Analysis (3)
0.00.10.20.30.40.50.60.70.80.91.0
MIN(8x8, 0.8) PHOLD(8x8, 4)
Benchmarks
Stric
tnes
s Physical SystemCMB event orderCMB protocol (4PPs)CMB protocol (8PPs)
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Summary
A unified simulation performance framework based on event orderings.
Formalization of event orderings based on poset
Performance characterizationtime (event parallelism)space (memory usage)event dependencies- stricter relation, strictness (ςR)
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Ongoing Work
Parallel simulation -> performance & capability
Performance – scalability (fixed workload (Amdahl’s law ‘67), scaled workload (Gustafson’s Law ‘87), memory bounded (Sun & Ni ’93), ….)
Capability -> framework for understanding performance of grid-enabled simulation
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Forthcoming …
“A Framework for Formalization and Characterization of Simulation Performance”,B.S.S. Onggo, PhD Thesis, Department of Computer Science, National University of Singapore, 2004.
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Thank You
www.comp.nus.edu.sg/~teoym/recent-publications.htm