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How Fast and Fat Is Your Probabilistic Model Checker? an experimental performance comparison David N. Jansen 3,1, Joost-Pieter Katoen 1,2, Marcel Oldenkamp 2, Mariëlle Stoelinga 2, Ivan Zapreev 1,2 1 MOVES Group, RWTH Aachen University 2 FMT Group, University of Twente, Enschede 3 ICIS, Radboud University, Nijmegen

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Probabilistic Model Checker Probabilistic Model Checking Probabilistic SystemProbabilistic Requirement Probabilistic ModelProbabilistic Formula YesNo Probability ETMCC MRMC PRISM (sparse) PRISM (hybrid) YMER VESTA

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Why Are Probabilities Useful? system performance uncertainty in the environment randomized (networking) algorithms abstract from large populations

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Probabilistic Model Checking... What is inside? –temporal logics + model checking –numerical and optimisation techniques from performance and operations research Where is it used? –powerful tools –applications: distributed systems, security, biology, quantum computing... Problem: Which tool to choose?

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Probabilistic Model Checker Probabilistic Models Probabilistic SystemProbabilistic Requirement Probabilistic ModelProbabilistic Formula YesNo Probability Discrete time Markov chains Continuous time Markov chains automata transitions are probabilistic timing for CTMC: Prob(wait time ≤ t) = 1 – e – t

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Synchronous Leader Election nodes in a ring elect a leader –each node selects random number as id –passes it around the ring (synchronously) –if unique id, node with maximum unique id is leader [Itai & Rodeh 1990]

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Synchronous Leader Election 1 3 2 1 5 5 4 2

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5 4 1 2 2 3 5 14 1 2 2 3 5 1 51 2 2 3 5 1 5 4 1 3 2 1 5 5 4 2

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Models Discrete time Markov chain –transitions are fully probabilistic –timing is irrelevant Continuous time Markov chain –transitions are fully probabilistic –and timing also: Prob(wait time ≤ t) = 1 – e – t

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Probabilistic Model Checker Probabilistic Formulas Probabilistic SystemProbabilistic Requirement Probabilistic ModelProbabilistic Formula YesNo Probability Reachability Bounded Reachability Steady-State Property extensions of CTL

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Probabilistic Model Checker Probabilistic Model Checkers Probabilistic SystemProbabilistic Requirement Probabilistic ModelProbabilistic Formula YesNo Probability Choices made Three examples Overall evaluation

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Tools ETMC C numerical CTMCJava MRMCDTMC + CTMCC PRISM hybrid DTMC + CTMC MTBDD for transition matrix C(++ ) and Java PRISM sparse DTMC + CTMC sparse transition matrix VESTA statist CTMC, reachabilityJava YMERCTMC, bounded reachC(++ )

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Modelling PRISM model.tra format model PRISM YMER model ETMCCMRMCYMERVESTA model adapt syntax informal description

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Selected Benchmarks Synchronous Leader Electiondiscrete time Randomized Dining Philosophersdiscrete time Birth–Death Processdiscrete time Tandem Queuing Networkcontinuous time Cyclic Server Pollingcontinuous time

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Experiment Relevance Repeatable Verifiable Significant Encapsulated

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Experiment 1 Reachability Cyclic Polling Server: server cycles over n stations and serves each one in turn –e.g. teacher walks through class, each pupil may ask a question busy 1 P ≥1 (true U poll 1 ) If station 1 is busy, the server will poll it eventually

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Analysis ETMCCslow, out of memory PRISMonly symbolic sparse=hybrid MRMCfastest tool for small models VESTAexcessive number of samples, slow YMERnot implemented

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PRISM: MTBDD Size Multi-Terminal BDD = data structure for transition matrix size heavily depends on model large MTBDD slow Model# states# MTBDD nodes Synchronous leader election 500.0001.000.000. Cyclic polling7.000.000< 3.000.

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CPS versus SLE runtime 7.077.888 states 2.745 MTBDD nodes 458.847 states 1.131.806 MTBDD nodes

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VESTA: simulation problem actual probability close to bound P ≥p (...) estimate is almost always in [p– ,p+ ] some irregularity stops the simulation 0.95 Prob(yes actual Prob≥p) Prob(actual Prob≥p yes)

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depends heavily on MTBDD size depends heavily on MTBDD size depends heavily on MTBDD size Result Overview: Timing ETMCCMRMCPRISM hybrid PRISM sparse VESTAYMER unb’nded U –++/++ –/0 N/A

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Result Overview: Memory ETMCCMRMCPRISM hybrid PRISM sparse VESTAYMER unb’nded U –++/++ 0/+ N/A MTBDD size varies heavily almost independent from model size

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Experiment 2 Bounded Reachability Tandem Queueing Network –two queues after each other P <0.01 (true U ≤2 full ) Is the probability that the system gets full in 2 time units small? checkin counter security check

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Analysis ETMCCslow, out of memory PRISMsparse=faster, hybrid=smaller MRMCfast for small models VESTA ok if you can afford statistical errors YMER best choice if you can afford statistical errors

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Result Overview: Timing ETMCCMRMCPRISM hybrid PRISM sparse VESTAYMER unb’nded U –++/++ –/0 N/A bounded U –+0/++/+++++

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Result Overview: Memory ETMCCMRMCPRISM hybrid PRISM sparse VESTAYMER unb’nded U –++/++ 0/+ N/A bounded U –++/++++++

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Experiment 3 Steady State Property Tandem Queuing Network S >0.2 ( P >0.1 (X 2nd queue full ) ) In equilibrium, the probability to satisfy is > 0.2 P >0.1 (X 2nd queue full ) P >0.1 (X... )

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Analysis ETMCCslow, out of memory PRISM sparse=faster hybrid=slightly smaller MRMCfastest VESTAnot implemented YMERnot implemented

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Simulating Steady State? simulation of bounded reachability has clear stopping criterion simulation of unbounded reachability reachability with very large bound simulation of steady state? never stops

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Result Overview: Timing ETMCCMRMCPRISM hybrid PRISM sparse VESTAYMER unb’nded U –++/++ –/0 N/A bounded U –+0/++/+++++ steady state –++0/++ N/A

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Result Overview: Memory ETMCCMRMCPRISM hybrid PRISM sparse VESTAYMER unb’nded U –++/++ 0/+ N/A bounded U –++/++++++ steady state –++/+++ N/A

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Nested Formulas we also checked nested properties P ≥0.8 (P ≥0.9 (true U ≤100 n 70 ) U n 50 ) not detailed here

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Result Overview: Timing ETMCCMRMCPRISM hybrid PRISM sparse VESTAYMER unb’nded U –++/++ –/0 N/A bounded U –+0/++/+++++ steady state –++0/++ N/A nested –++0/++– –– – N/A based on a single property only: did not terminate

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Result Overview: Memory ETMCCMRMCPRISM hybrid PRISM sparse VESTAYMER unb’nded U –++/++ 0/+ N/A bounded U –++/++++++ steady state –++/+++ N/A nested –++/+++ N/A

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Conclusions ETMCCworst, only small models MRMCfastest for small models PRISM hybrid fast if MTBDD is small PRISM sparse fast VESTArather slow, statistical errors YMER slim & very fast, only bounded reach, few statistical errors

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