Partial stochastic characterization of timed runs over DBM domains - PowerPoint PPT Presentation

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