Results 1  10
of
36
A Characterization of the Stochastic Process Underlying a Stochastic Petri Net
 IEEE Transactions on Software Engineering
, 1994
"... Petri net ..."
DISCRETETIME MARKOVIAN STOCHASTIC PETRI NETS
, 1995
"... We revisit and extend the original definition of discretetime stochastic Petri nets, by allowing the firing times to have a “defective discrete phase distribution”. We show that this formalism still corresponds to an underlying discretetime Markov chain. The structure of the state for this process ..."
Abstract

Cited by 27 (8 self)
 Add to MetaCart
(Show Context)
We revisit and extend the original definition of discretetime stochastic Petri nets, by allowing the firing times to have a “defective discrete phase distribution”. We show that this formalism still corresponds to an underlying discretetime Markov chain. The structure of the state for this process describes both the marking of the Petri net and the phase of the firing time for of each transition, resulting in a large state space. We then modify the wellknown power method to perform a transient analysis even when the state space is infinite, subject to the condition that only a finite number of states can be reached in a finite amount of time. Since the memory requirements might still be excessive, we suggest a bounding technique based on truncation.
E.: StateDensity Functions over DBM Domains in the Analysis of NonMarkovian Models
 IEEE Trans. on SW Eng
, 2009
"... Abstract—Quantitative evaluation of models with generally distributed transitions requires the analysis of nonMarkovian processes that may be not isomorphic to their underlying untimed models and may include any number of concurrent nonexponential timers. The analysis of stochastic Time Petri Nets ..."
Abstract

Cited by 22 (17 self)
 Add to MetaCart
(Show Context)
Abstract—Quantitative evaluation of models with generally distributed transitions requires the analysis of nonMarkovian processes that may be not isomorphic to their underlying untimed models and may include any number of concurrent nonexponential timers. The analysis of stochastic Time Petri Nets (sTPNs) copes with the problem by covering the state space with stochastic classes, which extend the theory of Difference Bounds Matrix (DBM) with a state probability density function. As a core step, the analysis process requires symbolic manipulation of density functions supported over DBM domains. We characterize and engineer the critical steps of this derivation. We first show that the statedensity function accepts a continuous piecewise representation over a partition in DBMshaped subdomains. We then develop a closedform symbolic calculus of statedensity functions under the assumption that transitions in the sTPN model have expolynomial distributions over possibly bounded intervals. The calculus shows that within each subdomain, the statedensity function is a multivariate expolynomial function, and it makes explicit the way in which this form evolves and grows in complexity as the state accumulates memory through subsequent transitions. This enables an efficient implementation of the analysis process and provides the formal basis that supports the introduction of an imprecise analysis based on the approximation of statedensity functions through Bernstein Polynomials. The approximation attacks practical and theoretical limits in the applicability of stochastic state classes and devises a new approach to the analysis of nonMarkovian models, relying on approximations in the state space rather than in the structure of the model. Index Terms—Correctness verification, performance and dependability, quantitative evaluation, stochastic Time Petri nets, densetime statespace analysis, Difference Bounds Matrix, Markov Renewal Theory, approximate statespace representation, density function approximation, Bernstein polynomials. Ç 1
Using Stochastic State Classes in Quantitative Evaluation of DenseTime Reactive Systems
"... Abstract—In the verification of reactive systems with nondeterministic densely valued temporal parameters, the statespace can be covered through equivalence classes, each composed of a discrete logical location and a dense variety of clock valuations encoded as a Difference Bounds Matrix (DBM). The ..."
Abstract

Cited by 15 (12 self)
 Add to MetaCart
(Show Context)
Abstract—In the verification of reactive systems with nondeterministic densely valued temporal parameters, the statespace can be covered through equivalence classes, each composed of a discrete logical location and a dense variety of clock valuations encoded as a Difference Bounds Matrix (DBM). The reachability relation among such classes enables qualitative verification of properties pertaining events ordering and stimulus/response deadlines, but it does not provide any measure of probability for feasible behaviors. We extend DBM equivalence classes with a densityfunction which provides a measure for the probability of individual states. To this end, we extend Time Petri Nets by associating a probability densityfunction to the static firing interval of each nondeterministic transition. We then explain how this stochastic information induces a probability distribution for the states contained within a DBM class and how this probability evolves in the enumeration of the reachability relation among classes. This enables the construction of a stochastic transition system which supports correctness verification based on the theory of TPNs, provides a measure of probability for each feasible run, enables steadystate analysis based on Markov Renewal Theory. In so doing, we provide a means to identify feasible behaviors and to associate them with a measure of probability in models with multiple concurrent generally distributed nondeterministic timers.
Aggregated Stochastic State Classes in Quantitative Evaluation of nonMarkovian Stochastic Petri Nets
"... Abstract—The method of stochastic state classes provides a new approach for the analysis of nonMarkovian stochastic Petri Nets, which relies on the stochastic expansion of the graph of nondeterministic state classes based on Difference Bounds Matrix (DBM) which is usually employed in qualitative ve ..."
Abstract

Cited by 11 (5 self)
 Add to MetaCart
(Show Context)
Abstract—The method of stochastic state classes provides a new approach for the analysis of nonMarkovian stochastic Petri Nets, which relies on the stochastic expansion of the graph of nondeterministic state classes based on Difference Bounds Matrix (DBM) which is usually employed in qualitative verification. In so doing, the method is able to manage multiple concurrent nonexponential (GEN) transitions and largely extends the class of models that are amenable to quantitative evaluation. However, its application requires that every cycle in the graph of nondeterministic state classes visits at least a regeneration point where all GEN transitions are newly enabled. In particular, this rules out models whose nondeterministic class graph includes cycles within a Continuous Time Markov Chain (CTMC) subordinated to the activity period of one or more GEN transitions. In this paper, we propose an extension that overcomes this limitation by aggregating together classes that are reached through firings that do not change the enabling status of GEN transitions. This enlarges the class of models that can be analysed through the method of stochastic state classes and makes it become a proper extension of the class of models that satisfies the so called enabling restriction. Index Terms—nonMarkovian stochastic Petri nets, stochastic time Petri nets, steady state analysis, stochastic state classes.
Markov Regenerative Stochastic Petri Nets with Age Type General Transitions
 Application and Theory of Petri Nets (16th International Conference), Lecture Notes in Computer Science
, 1995
"... . Markov Regenerative Stochastic Petri Nets (MRSPN) have been recently introduced in the literature with the aim of combining exponential and nonexponential ring times into a single model. However, the realizations of the general MRSPN model, so far discussed, require that at most a single nonexpo ..."
Abstract

Cited by 10 (8 self)
 Add to MetaCart
. Markov Regenerative Stochastic Petri Nets (MRSPN) have been recently introduced in the literature with the aim of combining exponential and nonexponential ring times into a single model. However, the realizations of the general MRSPN model, so far discussed, require that at most a single nonexponential transition is enabled in each marking and that its associated memory policy is of enabling type. The present paper extends the previous models by allowing the memory policy to be of age type and by allowing multiple general transitions to be simultaneously enabled, provided that their enabling intervals do not overlap. A nal completely developed example, that couldn't have been considered in previous formulations, derives the closed form expressions for the transient state probabilities for a queueing system with preemptive resume (prs) service policy. Key words: Markov regenerative processes, Stochastic Petri Nets, Queueing systems with preemptive resume service, Transient analys...
On phased delay stochastic Petri nets: Definition and an application
 In Proceedings 9th International Workshop on Petri Nets and Performance Models  PNPM01. IEEE Computer Society
, 2001
"... We present a novel stochastic Petri net formalism where both discrete and continuous phasetype firing delays can appear simultaneously in the same model. By capturing nonMarkovian behavior in discrete or continuous time, as appropriate, the formalism affords higher modeling fidelity. Alone, discre ..."
Abstract

Cited by 10 (2 self)
 Add to MetaCart
(Show Context)
We present a novel stochastic Petri net formalism where both discrete and continuous phasetype firing delays can appear simultaneously in the same model. By capturing nonMarkovian behavior in discrete or continuous time, as appropriate, the formalism affords higher modeling fidelity. Alone, discrete or continuous phasetype Petri nets have simple underlying Markov chains, but mixing the two complicates matters. We show that, in a mixed model where discretetime transitions are synchronized, the underlying process is semiregenerative and we can employ Markov renewal theory to formulate stationary or timedependent solutions. Also noteworthy are the computational tradeoffs between the socalled embedded and subordinate Markov chains, which we employ to improve the overall solution efficiency. We present a preliminary stationary solution method that shows promise in terms of time and space efficiency and demonstrate it on an aeronautical data link system application. 1.
Dependability Modeling & Evaluation of MultiplePhased Systems using DEEM
 IEEE TRANSACTIONS ON RELIABILITY
, 2004
"... ... This paper describes the modeling methodology and the solution procedure implemented in DEEM, a dependability modeling and evaluation tool specifically tailored for Multiple Phased Systems. It also describes its use for the solution of representative MPS problems. DEEM relies upon Deterministic ..."
Abstract

Cited by 8 (2 self)
 Add to MetaCart
(Show Context)
... This paper describes the modeling methodology and the solution procedure implemented in DEEM, a dependability modeling and evaluation tool specifically tailored for Multiple Phased Systems. It also describes its use for the solution of representative MPS problems. DEEM relies upon Deterministic and Stochastic Petri Nets as the modeling formalism and on Markov Regenerative Processes for the model solution. When compared to existing generalpurpose tools based on similar formalisms, DEEM o#ers advantages on both the modeling side (submodels neatly model the phasedependent behaviors of MPS), and on the evaluation side (a specialized algorithm allows a considerable reduction of the solution cost and time). Thus, DEEM is able to deal with all the scenarios of MPS that have been analytically treated in the literature, at a cost which is comparable with that of the cheapest ones, completely solving the issues posed by the phasedbehavior of MPS
NonExponential Stochastic Petri Nets: an Overview of Methods and Techniques
 In To be published in: Computer Systems Science & Engineering
, 1997
"... The analysis of stochastic systems with nonexponential timing requires the development of suitable modeling tools. Recently, some eort has been devoted to generalize the concept of Stochastic Petri nets, by allowing the ring times to be generally distributed. The evolution of the PN in time beco ..."
Abstract

Cited by 8 (4 self)
 Add to MetaCart
The analysis of stochastic systems with nonexponential timing requires the development of suitable modeling tools. Recently, some eort has been devoted to generalize the concept of Stochastic Petri nets, by allowing the ring times to be generally distributed. The evolution of the PN in time becomes a stochastic process, for which in general, no analytical solution is available. The paper surveys suitable restrictions of the PN model with generally distributed transition times, that have appeared in the literature, and compares these models from the point of view of the modeling power and the numerical complexity. Key words: Stochastic Petri Nets, Nonexponential Distributions, Phasetype Distributions, Markov and Semimarkov Reward Models, Markov Regenerative Processes, Queueing Systems with Preemption. 1 Introduction The usual denition of Stochastic Petri Net (SPN) implies that all the timed activities associated to the transitions are represented by exponential random ...