Transient analysis of Markov regenerative stochastic Petri nets: a comparison of approaches

German, Reinhard; Logothetis, D.; Trivedi, Kishor S.

doi:10.1109/pnpm.1995.524320

Cited by 38 publications

(31 citation statements)

References 12 publications

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“…In the case of a single realization of a non Markovian distribution at any time, successful methods have been proposed [2] both for the transient and steady state analysis, especially in the Stochastic Petri Net (SPN) modelling framework. Let us cite, for instance, the method of supplementary variables [3,4] or the method of the subordinated Markov chains [5].…”

Section: Introductionmentioning

confidence: 99%

A New Approach to the Evaluation of Non Markovian Stochastic Petri Nets

Haddad

Mokdad

Moreaux

2006

Petri Nets and Other Models of Concurrency - ICATPN 2006

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Abstract. In this work, we address the problem of transient and steadystate analysis of a stochastic Petri net which includes non Markovian distributions with a finite support but without any additional constraint. Rather than computing an approximate distribution of the model (as done in previous methods), we develop an exact analysis of an approximate model. The design of this method leads to a uniform handling of the computation of the transient and steady state behaviour of the model. This method is an adaptation of a former one developed by the same authors for general stochastic processes (which was shown to be more robust than alternative techniques). Using Petri nets as the modelling formalism enables us to express the behaviour of the approximate process by tensorial expressions. Such a representation yields significant savings w.r.t. time and space complexity.

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Section: Introductionmentioning

confidence: 99%

A New Approach to the Evaluation of Non Markovian Stochastic Petri Nets

Haddad

Mokdad

Moreaux

2006

Petri Nets and Other Models of Concurrency - ICATPN 2006

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“…As the firing time distributions of t 1 and t 2 are identical the integral in the denominator of (7) corresponding to the probability that t 1 fires first yields 1/2. Next we have to perform the time advancement and projection step according to (8) as…”

Section: Discussionmentioning

confidence: 99%

“…Due to the firing of t 1 , t 3 would get enabled but, as the firing of t 1 constitutes a regeneration point and our aim is to characterise only the regeneration period, the enabling step given in (10) is not performed. Accordingly, the density associated with Σ 5 is obtained by the usual time advancement and projection step according to (8) Having enumerated the classes from the initial marking p 1 p 2 up to the regeneration points, the entries of the kernels can be determined by the application of (13). Classes reached by regeneration contribute to the global kernel while the others to the local kernel.…”

Section: Discussionmentioning

confidence: 99%

“…Conditional probabilities evolve according to a set of generalised Markov renewal equations, formulated as a system of Volterra integral equations of the second kind [7] defined by a local and a global kernel which capture the transient probability of the states before the first regeneration point and the cumulative distribution of the time to reach the states at the first regeneration point, respectively. Under enabling restriction, both kernels can be expressed in closed-form in terms of the exponential of the matrix describing the subordinated CTMC [8] and evaluated numerically through uniformisation. Generalised Markov renewal equations can then be solved by numerical approaches in the time domain or through Laplace-Stieltjes transform.…”

Section: Introductionmentioning

confidence: 99%

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Transient analysis of non-Markovian models using stochastic state classes

Horváth

Paolieri

Ridi

et al. 2012

Performance Evaluation

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a b s t r a c tThe method of stochastic state classes approaches the analysis of Generalised Semi Markov Processes (GSMPs) through the symbolic derivation of probability density functions over supports described by Difference Bounds Matrix (DBM) zones. This makes steady state analysis viable, provided that at least one regeneration point is visited by every cyclic behaviour of the model.We extend the approach providing a way to derive transient probabilities. To this end, stochastic state classes are extended with a supplementary timer that enables the symbolic derivation of the distribution of time at which a class can be entered. The approach is amenable to efficient implementation when model timings are given by expolynomial distributions, and it can be applied to perform transient analysis of GSMPs within any given time bound. In the special case of models underlying a Markov Regenerative Process (MRGP), the method can also be applied to the symbolic derivation of local and global kernels, which in turn provide transient probabilities through numerical integration of generalised renewal equations. Since much of the complexity of this analysis is due to the local kernel, we propose a selective derivation of its entries depending on the specific transient measure targeted by the analysis.

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“…In the case of a single realization of a non Markovian distribution at any time, successful methods have been proposed (German et al 1995) both for the transient and steady state analysis. Let us cite, for instance, the method of supplementary variables (Cox 1955a, German andLindemann 1994) or the method of the subordinated Markov chains (Ajmone Marsan and Chiola 1987).…”

Section: Introductionmentioning

confidence: 99%

Performance evaluation of non markovian stochastic discrete event systems - a new approach

Haddad

Mokdad

Moreaux

2004

IFAC Proceedings Volumes

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In this work, we address the problem of transient and steady-state analysis of a stochastic discrete event system which includes non Markovian distributions with a finite support. Rather than computing an approximate distribution of the model (as done in the previous methods), we develop an exact analysis of an approximate model. The design of this method leads to a uniform handling for the computation of the transient and steadystate behaviour of the model. We have evaluated our method on a standard benchmark (the queuing model M/D/S/K). Our results demonstrate that: in most of the cases the solution of the approximate model converges quickly to the solution of the exact model, in the difficult cases (e.g. an heavy load on the queue) our method is more robust than the previous ones.

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Transient analysis of Markov regenerative stochastic Petri nets: a comparison of approaches

Cited by 38 publications

References 12 publications

A New Approach to the Evaluation of Non Markovian Stochastic Petri Nets

A New Approach to the Evaluation of Non Markovian Stochastic Petri Nets

Transient analysis of non-Markovian models using stochastic state classes

Performance evaluation of non markovian stochastic discrete event systems - a new approach

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