Introduction partially stochastic Time Petri Nets Characterization of symbolic runs Partial stochastic characterization of timed runs over DBM domains Laura Carnevali Lorenzo Ridi Enrico Vicario Dipartimento di Sistemi e Informatica Università di Firenze PMCCS-9 September 18, 2009 - Eger, Hungary 1 / 21
Introduction partially stochastic Time Petri Nets Characterization of symbolic runs Outline Introduction 1 The addressed problem: an intuition Contribution Related work partially stochastic Time Petri Nets 2 Characterization of symbolic runs 3 Domain of timings along a symbolic run Timing boundaries enlargement Partial stochastic characterization of timings 2 / 21
Introduction The addressed problem: an intuition partially stochastic Time Petri Nets Contribution Characterization of symbolic runs Related work The addressed problem: an intuition Continuous-Time Discrete-Events Model non-deterministic timings; controllable timings are bounded within continuous intervals; non-controllable timings are chosen by the system within a predictable range, following a given probability distribution. (input/output transitions, actions/endogenous events) [3 , 20] t 1 Controllable events [10 , 25] t 2 [0 , 10] [5 , 25] t 3 t 4 Non-controllable environment 3 / 21
Introduction The addressed problem: an intuition partially stochastic Time Petri Nets Contribution Characterization of symbolic runs Related work The addressed problem: an intuition The system can execute along different firing sequences (symbolic runs); the actual sequence is determined by values assumed by timers. [3 , 20] t 1 t 1 Controllable events [10 , 25] t 2 t 2 [0 , 10] [5 , 25] t 4 t 3 t 3 t 4 Non-controllable environment 0 25 t 1 t 1 t 3 t 4 t 1 t 2 t 3 t 2 4 / 21
Introduction The addressed problem: an intuition partially stochastic Time Petri Nets Contribution Characterization of symbolic runs Related work The addressed problem: an intuition Problem : force the system to run along a selected sequence. controllable timers can be assigned arbitrary values; success still depends upon values of non-controllable timers. The problem has a qualitative and a quantitative aspect: t c 1 identification of the range of valuations for controllable timers that can let the system run along the selected sequence ( qualitative problem); t c 2 t c f ( t c 1 , t c 2 ) 1 evaluation of the success probability for every choice of controllable timers ( quantitative problem). t c 2 5 / 21
Introduction The addressed problem: an intuition partially stochastic Time Petri Nets Contribution Characterization of symbolic runs Related work An introductory example [3 , 20] t 1 Controllable events ? t 1 [10 , 25] ? t 2 t 2 [0 , 10] [5 , 25] t 4 t 3 t 3 t 4 0 25 Non-controllable environment Problem : select values for t 1 4 concurrent transitions; and t 4 so as to make t 1 , t 2 : controllable transitions; possible/maximize the t 3 , t 4 : non-controllable probability to execute the transitions; sequence ρ = t 3 , t 1 , t 2 , t 4 . 6 / 21
Introduction The addressed problem: an intuition partially stochastic Time Petri Nets Contribution Characterization of symbolic runs Related work Contribution partially stochastic Time Petri Nets combines non-deterministic selection of controllable timers and stochastic sampling of non-controllable timers. evaluation of the execution probability of any firing sequence: support : set of controllable choices that can let the system execute along the sequence; function : distribution of the success probability as a function of values given to controllable timers. 7 / 21
Introduction The addressed problem: an intuition partially stochastic Time Petri Nets Contribution Characterization of symbolic runs Related work Related work Real-Time test case sensitization L. Carnevali, L. Sassoli, E. Vicario: ETFA ’07 qualitative approach: all timers are non-deterministic. application in testing of real-time software (Linux RTAI). stochastic Time Petri Nets G. Bucci, R. Piovosi, L. Sassoli, E. Vicario: QEST ’05 L. Carnevali, L. Sassoli, E. Vicario: Trans. on Software Engineering, September 2009. quantitative evaluation: all timers are stochastic. Test case execution optimization on Timed Automata M. Jurdi´ nsky, D. Peled, H. Qu: FATES ’05 N. Wolowick, P . D’Argenio, H. Qu: ICST ’09 non-controllable timers are uniformly distributed. 8 / 21
Introduction partially stochastic Time Petri Nets Characterization of symbolic runs partially stochastic Time Petri Nets: Syntax psTPN = � P ; T c ; T nc ; A + ; A − ; m 0 ; EFT ; LFT ; τ 0 ; C ; F� t 3 [0 , 10] t 4 t 1 [0 , 10] [3 , 10] t 2 t 5 [4 , 8] [3 , 6] T partitioned: T c controllable, T nc non-controllable; F : T nc → F associates each non-controllable transition with a static probability distribution F t () supported in [ EFT ( t ) , LFT ( t )] : � x F t ( x ) = f t ( y ) dy 0 9 / 21
Introduction partially stochastic Time Petri Nets Characterization of symbolic runs partially stochastic Time Petri Nets: Semantics psTPN = � P ; T c ; T nc ; A + ; A − ; m 0 ; EFT ; LFT ; τ 0 ; C ; F � t 3 [0 , 10] t 4 t 1 [0 , 10] [3 , 10] t 2 t 5 [4 , 8] [3 , 6] Tokens move as in Petri Nets (logical locations); each transition t has an Earliest and a Latest Firing Time ( EFT ( t ) and LFT ( t ) ), and an initial time to fire τ 0 ( t ) . t cannot fire before it has been enabled with continuity for EFT ( t ) ; neither it can let time advance without firing after it has been enabled with continuity for LFT(t); firings occur in zero-time. 10 / 21
Introduction partially stochastic Time Petri Nets Characterization of symbolic runs partially stochastic Time Petri Nets: Analysis state s = marking + valuation of transitions times-to-fire state class S = marking + s continuous set of times-to-fire timers within the same state class range in a S Difference Bound Matrix (DBM) zone. τ i − τ j ≤ b ij Remark : every state (class) may jointly enable controllable and non-controllable transitions, thus combining stochastic and non-deterministic behavior. 11 / 21
Introduction partially stochastic Time Petri Nets Characterization of symbolic runs State class graph enumeration AE reachability relation between state classes: Definition: AE reachability relation Given two state classes S and S ′ we say that S ′ is a successor of S through t 0 iff S ′ contains all and only the states that are reachable from some state collected in S through some feasible firing of t 0 . Enumeration → Timed Transition System ( state class graph ); S 0 S 2 DBM form is closed wrt successor evaluation; S 1 S 3 S 4 symbolic runs are paths in the state class graph. 12 / 21
Introduction Domain of timings along a symbolic run partially stochastic Time Petri Nets Timing boundaries enlargement Characterization of symbolic runs Partial stochastic characterization of timings Domain of timings along a symbolic run S 0 Consider a symbolic run ρ starting from class S 0 , terminating in S N ; S 1 t n i is the instance of transition t i enabled along ρ in class S n ; associated to an absolute virtual firing time τ n i ; ••• absolute firing times feasible for ρ satisfy three kinds of constraints: S N − 1 1 model constraints; 2 disabling constraints; 3 sequence constraints. S N 13 / 21
Introduction Domain of timings along a symbolic run partially stochastic Time Petri Nets Timing boundaries enlargement Characterization of symbolic runs Partial stochastic characterization of timings Domain of timings along a symbolic run 1 Model constraints time elapsed between enabling and firing of each transition t n i fired along ρ must range within its static firing interval: i − τ ν ( n ) ι ( n ) ≤ LFT ( t i ) where t ν ( n ) EFT ( t i ) ≤ τ n ι ( n ) enables t n i i − τ ν ( n ) τ n ι ( n ) t ν ( n ) t n i fires ι ( n ) fires ( t n i is enabled) 14 / 21
Introduction Domain of timings along a symbolic run partially stochastic Time Petri Nets Timing boundaries enlargement Characterization of symbolic runs Partial stochastic characterization of timings Domain of timings along a symbolic run 2 Disabling constraints if transition t n x is enabled but not fired along ρ , its absolute firing time must be greater than the one of its disabling transition t δ ( x , n ) γ ( x , n ) : x ≥ τ δ ( x , n ) γ ( x , n ) where t δ ( x , n ) τ n γ ( x , n ) disables t n x τ δ ( x,n ) γ ( x,n ) − τ n x t δ ( x,n ) t n γ ( x,n ) fires x is enabled ( t n x is disabled) 15 / 21
Introduction Domain of timings along a symbolic run partially stochastic Time Petri Nets Timing boundaries enlargement Characterization of symbolic runs Partial stochastic characterization of timings Domain of timings along a symbolic run 3 Sequence constraints transitions must fire in the expected sequence: τ ν ( n + 1 ) ι ( n + 1 ) ≥ τ ν ( n ) ι ( n ) ∀ n ∈ [ 0 , N − 1 ] τ ν ( n ) ι ( n ) − τ ν ( n − 1) ι ( n − 1) τ ν ( n − 1) τ ν ( n ) ι ( n − 1) fires ι ( n ) fires (class S n is entered) (class S n − 1 is entered) 16 / 21
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