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Steady State Analysis of Markov Regenerative SPN with Age Memory Policy
, 1995
"... . NonMarkovian Stochastic Petri Nets (SPN) have been developed as a tool to deal with systems characterized by non exponentially distributed timed events. Recently, some eort has been devoted to the study of SPN with generally distributed ring times, whose underlying marking process belongs to the ..."
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Cited by 20 (15 self)
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. NonMarkovian Stochastic Petri Nets (SPN) have been developed as a tool to deal with systems characterized by non exponentially distributed timed events. Recently, some eort has been devoted to the study of SPN with generally distributed ring times, whose underlying marking process belongs to the class of Markov Regenerative Processes (MRGP). We refer to this class of models as Markov Regenerative SPN (MRSPN). In this paper, we describe a computationally eective algorithm for the steady state solution of MRSPN with age memory policy and subordinated Continuous Time Markov Chain (CTMC). Key words: Stochastic Petri Nets, Generally distributed transitions, Markov regenerative processes, Preemptive resume policies. 1 Introduction In the last decade several classes of SPN's have been developed to deal with nonexponentially distributed events [7, 3]. The main reason for this is the observation that it is possible to identify a wide number of real situations in which deterministic or g...
Welldefined stochastic Petri nets, in
 Proceedings of the 4th International Workshop on Modeling, Analysis, and Simulation of Computer and Telecommunications Systems, MASCOTS ’96, IEEE Computer Society
, 1996
"... ..."
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 ..."
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Cited by 16 (12 self)
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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.
Algorithms for the Generation of StateLevel Representation of Stochastic Activity Networks with General Reward Structures
, 1995
"... Stochastic Petri nets and extensions are a popular method for evaluating a wide variety of systems. In most cases, their numerical solution requires generating a statelevel stochastic process, which captures the behavior of the SPN with respect to a set of specified performance measures. These meas ..."
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Cited by 15 (4 self)
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Stochastic Petri nets and extensions are a popular method for evaluating a wide variety of systems. In most cases, their numerical solution requires generating a statelevel stochastic process, which captures the behavior of the SPN with respect to a set of specified performance measures. These measures are commonly defined at the net level by means of a reward variable. In this paper, we discuss issues regarding the generation of statelevel reward models for systems specified as stochastic activity networks with "stepbased reward structures." Stepbased reward structures are a generalization of previously proposed reward structures that can be defined on the marking behavior of a net. While discussing issues related to the generation of the underlying statelevel reward model, we provide an algorithm to determine whether a given SAN is "well specified". A SAN is well specified if choices about which instantaneous activity completes among multiple simultaneously enabled instantaneous activities do not matter, with respect to the probability of reaching next possible stable markings, and distribution of reward obtained upon completion of a timed activity. The fact that a SAN is well specified is both a necessary and sufficient condition for its behavior to be completely probabilistically specified, and hence an important property to determine.
The Scale Factor: A New Degree of Freedom in Phase Type Approximation
"... This paper introduces a unified approach to phasetype approximation in which the discrete and the continuous phasetype models form a common model set. The models of this common set are assigned with a nonnegative real parameter, the scale factor. The case when the scale factor is strictly positiv ..."
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Cited by 13 (2 self)
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This paper introduces a unified approach to phasetype approximation in which the discrete and the continuous phasetype models form a common model set. The models of this common set are assigned with a nonnegative real parameter, the scale factor. The case when the scale factor is strictly positive results in Discrete phasetype distributions and the scale factor represents the time elapsed in one step. If the scale factor is 0, the resulting class is the class of Continuous phasetype distributions. Applying the above view, it is shown that there is no qualitative difference between the discrete and the continuous phasetype models.
Petri Nets with k Simultaneously Enabled Generally Distributed Timed Transistions
 Performance Evaluation
, 1998
"... Stochastic Petri nets have been used to analyze the performance and reliability of complex systems comprising concurrency and synchronization. Various extensions have been proposed in literature in order to broaden their field of application to an increasingly larger range of real situations. In thi ..."
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Cited by 13 (3 self)
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Stochastic Petri nets have been used to analyze the performance and reliability of complex systems comprising concurrency and synchronization. Various extensions have been proposed in literature in order to broaden their field of application to an increasingly larger range of real situations. In this paper we extend the class of Markov Regenerative Stochastic Petri Nets* (MRSPN*s), removing the restriction that at most one generally distributed timed transition can be enabled in any marking. This new class of Petri Nets, which we call Concurrent Generalized Petri Nets (CGPNs) allows simultaneous enabling of immediate, exponentially and generally distributed timed transitions, under the hypothesis that the latter are all enabled at the same instant. The stochastic process underlying a CGPN is shown to be still an MRGP. We evaluate the kernel distribution of the underlying MRGP and define the steps required to generate it automatically. The methodology described is used to assess the beh...
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 ..."
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Cited by 11 (8 self)
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. 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...
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 ..."
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Cited by 11 (5 self)
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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.
A FourthOrder Algorithm with Automatic Stepsize Control for the Transient Analysis of DSPNs
 In 7th International Conference on Petri Nets and Performance Models  PNPM97
, 1997
"... This paper presents an efficient and numerically reliable method for the transient analysis of deterministic and stochastic Petri nets. The transient behavior is described by state equations derived by the method of supplementary variables. Significant features of the proposed solution algorithm of ..."
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Cited by 11 (1 self)
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This paper presents an efficient and numerically reliable method for the transient analysis of deterministic and stochastic Petri nets. The transient behavior is described by state equations derived by the method of supplementary variables. Significant features of the proposed solution algorithm of fourth order are an automatic stepsize control and a twostage relative error control. Furthermore, a formal way of dealing with discontinuities in the transient state equations is developed. This resolves the problems posed by initially enabled deterministic transitions and also improves the accuracy of numerical results. Experiments with a queueing system with failure and repair illustrate the efficiency (with respect to both CPUtime and memory space) and the numerical quality of the new algorithm.