Markov regenerative stochastic Petri nets

Choi, Hoon; Kulkarni, Vidyadhar G.; Trivedi, Kishor S.

doi:10.1016/0166-5316(94)90021-3

Cited by 216 publications

(115 citation statements)

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“…Non-Markovian applications may be represented using a higher order state space [55], where the level of history retained in the model needs to be determined empirically. Markov Regenerative Stochastic Petri Nets (MRSPNs) [10] may be used to represent generally distributed and deterministic transitions, whereas Fluid Stochastic Petri Nets (FSPNs) [50] may be used to keep track of the time spent in each component during testing to evaluate the impact of testing and repair of each component on application reliability. Colored Petri Nets (CPNs) [44] may be used to represent the dependencies among components that arise due to error propagation.…”

Section: Modeling Limitationsmentioning

confidence: 99%

Architecture-Based Software Reliability Analysis: Overview and Limitations

Gokhale

2007

IEEE Trans. Dependable and Secure Comput.

171

121

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Section: Modeling Limitationsmentioning

confidence: 99%

Architecture-Based Software Reliability Analysis: Overview and Limitations

Gokhale

2007

IEEE Trans. Dependable and Secure Comput.

171

121

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“…As originally observed by Choi et al [6], the underlying marking process is a Markov regenerative process under the conditions that:…”

Section: Markov Regenerative Stochastic Petri Netsmentioning

confidence: 92%

Modelling M/G/1 queueing systems with server vacations using stochastic Petri nets

Ramanath

Lakshmi

2006

ORiON

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The theory of non-Markovian stochastic Petri nets is employed in this paper to derive an alternative method for studying the steady state behaviour of the M/G/1 vacation queueing system with a limited service discipline. Three types of vacation schemes are considered, and sytems with both a finite population and those with an infinite population (but finite capacity) are considered. Simple numerical examples are also provided to illustrate the functionality of the methods and some useful performance measures for the system are obtained. Key words:Markov regenerative stochastic Petri nets, vacation queueing models, limited service discipline, single, multiple and hybrid vacation schemes. IntroductionQueueing systems in which the server is sometimes inactive (taking vacations or working elsewhere) while customers are waiting for service may find many applications in the performance modelling of computer and communication systems. For example, in many digital systems, the processor is multiplexed among a number of jobs and is hence not available all the time for a single job type. So, if we take any one job type as a reference point, the processor is alternatively busy handling that job type and absent doing work elsewhere [12].Server vacation has been used successfully to model polling systems [13], multiclass priority scheduling systems [18], maintenance models [17] and processor failure [2]. For a survey on the use of vacation models, see Doshi [7].Most of the papers on vacation models assume the source population of the queueing system to be infinite. However, Trivdei et al. [11] have studied several vacation models in which they have taken the population to be finite. They assume an exponentially * Corresponding author: School of Mathematics, Madurai Kamaraj University, Madurai 625021, India, email: kasturey@rediffmail.com † School of Mathematics, Madurai Kamaraj University, Madurai 625021, India 131 132 K Ramanath & P Lakshmi distributed service time. This paper is an extension of their work to the case where the service times are generally distributed.We obtain Markov regenerative stochastic Petri net (MRSPN) representations of the M/G/1/N queueing systems with server vacations under a limited service discipline. For an excellent introduction to the theory of M/G/1 queueing systems, see Gross and Harris [9] and Takagi [14]. The book by Takagi also contains information about queueing systems with server vacations. We consider three types of vacation schemes, namely the multiple vacation scheme, the single vacation scheme and the hybrid vacation scheme. We also show how the representations may be extended to the case where the population is infinite, but the system has a finite capacity.The chief advantages of an MRSPN representation are:1. The representation is aesthetically appealing and simple to understand, even for a layman.2. Several other features of vacation queueing systems may be included in this representation, whose inclusion in other classical models would have led to analytically intract...

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“…Due to transitions with GEN distribution, the process can accumulate memory over time, producing different classes of stochastic processes depending on the presence of regeneration points, i.e. states where all GEN timers lose their memory [2,4]. This basically depends on the conditions of persistence of GEN transitions.…”

Section: Underlying Stochastic Processmentioning

confidence: 99%

“…Analysis methods for this class of models have been finely developed relying on Markov renewal theory [4,5] and on the method of supplementary variables [9,6,10].…”

Section: Underlying Stochastic Processmentioning

confidence: 99%

“…In particular, most existing techniques develop on the so-called enabling restriction, which assumes that at most one generallydistributed transition is enabled in any reachable tangible marking, so that activity cycles of generally distributed transitions never overlap. In this case, the model underlies a Markov Regenerative Process (MRGP) which regenerates at every change in the enabling status of non-exponential timed transitions and behaves as a (subordinated) continuous time Markov chain (CTMC) between any two regeneration points [4,2,5,6]. 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.…”

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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Markov regenerative stochastic Petri nets

Cited by 216 publications

References 10 publications

Architecture-Based Software Reliability Analysis: Overview and Limitations

Architecture-Based Software Reliability Analysis: Overview and Limitations

Modelling M/G/1 queueing systems with server vacations using stochastic Petri nets

Transient analysis of non-Markovian models using stochastic state classes

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