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.
In order to support the effective use of telecommunication infrastructure, the "random" behavior of traffic sources has been studied since the early days of telephony. Strange new features, like fractal like behavior and heavy tailed distributions were observed in high speed packet switched data networks in the early '90s. Since that time a fertile research aims to find proper models to describe these strange traffic features and to establish a robust method to design, dimension and operate such networks. In this paper we give an overview of methods that, on the one hand, allow us to capture important traffic properties like slow decay rate, Hurst parameter, scaling factor, etc., and, on the other hand, makes possible the quantitative analysis of the studied systems using the effective analysis approach called matrix geometric method. The presentation of this analysis approach is associated with a discussion on the properties and limits of Markovian fitting of the typical non-Markovian behavior present in telecommunication networks.
Abstract-We consider the problem of verifying quantitative reachability properties in stochastic models of concurrent activities with generally distributed durations. Models are specified as Stochastic Time Petri Nets (STPNs) and checked against Boolean combinations of interval until operators imposing bounds on the probability that the marking process will satisfy a goal condition at some time in the interval [α, β] after an execution that never violates a safety property. The proposed solution is based on the analysis of regeneration points in the model execution: a regeneration is encountered after a discrete event if the future evolution depends only on the current marking and not on its previous history, thus satisfying the Markov property. We analyze systems in which multiple generally distributed timers can be started or stopped independently, but regeneration points are always encountered with probability 1 after a bounded number of discrete events. Leveraging the properties of regeneration points in probability spaces of execution paths, we show that the problem can be reduced to a set of Volterra integral equations, and we provide algorithms to compute their parameters through the enumeration of finite sequences of stochastic state classes encoding the joint probability density function (PDF) of generally distributed timers after each discrete event. The computation of symbolic PDFs is limited to the first regeneration epoch, and the repetitive structure of the stochastic process is exploited also before the lower bound α, providing crucial benefits for large time bounds. A case study is presented through the probabilistic formulation of Fischer's mutual exclusion protocol, a well-known real-time verification benchmark.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.