The defining property of a Markov process is commonly called the Markov property ; it was first stated by A. However, in the work of L. Bachelier it is already possible to find an attempt to discuss Brownian motion as a Markov process, an attempt which received justification later in the research of N.

Wiener The basis of the general theory of continuous-time Markov processes was laid by A. There are essentially distinct definitions of a Markov process. One of the more widely used is the following. Examples of continuous-time Markov processes are furnished by diffusion processes cf. Diffusion process and processes with independent increments cf.

Stochastic process with independent incrementsincluding Poisson and Wiener processes cf. Poisson process ; Wiener process. Such reasoning has led to the acceptance of the following definitions see. Then the corresponding Markov process can be taken to be right-continuous and having left limits that is, its trajectories can be chosen so.

In the theory of Markov processes most attention is given to homogeneous in time processes. Even the notation can be simplified:. A right-continuous Markov process is progressively measurable. There is a method for reducing the non-homogeneous case to the homogeneous case seeand in what follows homogeneous Markov processes will be discussed.

A Markov process is called a Fellerâ€”Markov process if the function. In the case of strong Markov processes various subclasses have been distinguished. Frequently, a physical system can be best described using a non-terminating Markov process, but only in a time interval of random length. In addition, even simple transformations of a Markov process may lead to processes with trajectories given on random intervals see Functional of a Markov process.

Guided by these considerations one introduces the notion of a terminating Markov process. A non-homogeneous terminating Markov process is defined similarly. A Markov process of Brownian-motion type is closely connected with partial differential equations of parabolic type. Kolmogorov equation :.

## Lumpability

The expectations of various functionals of diffusion processes are solutions of boundary value problems for the differential equation. Then, under certain conditions, the function. The solution of the first boundary value problem for a general second-order linear parabolic equation. More precisely, the function. At regular points the boundary values are attained by 9Quantitative analysis by means of discrete-state stochastic processes is hindered by the well-known phenomenon of state-space explosion, whereby the size of the state space may have an exponential growth with the number of objects in the model.

When the stochastic process underlies a Markovian process algebra model, this problem may be alleviated by suitable notions of behavioural equivalence that induce lumping at the underlying continuous-time Markov chain, establishing an exact relation between a potentially much smaller aggregated chain and the original one.

Marquis spa insulationHowever, in the modelling of massively distributed computer systems, even aggregated chains may be still too large for efficient numerical analysis. Recently this problem has been addressed by fluid techniques, where the Markov chain is approximated by a system of ordinary differential equations ODEs whose size does not depend on the number of the objects in the model. The technique has been primarily applied in the case of massively replicated sequential processes with small local state space sizes.

This thesis devises two different approaches that broaden the scope of applicability of efficient fluid approximations.

Glucose intravenous infusion bp 50% w/v (hameln)Fluid lumpability applies in the case where objects are composites of simple objects, and aggregates the potentially massive, naively constructed ODE system into one whose size is independent from the number of composites in the model. Similarly to quasi and near lumpability, we introduce approximate fluid lumpability that covers ODE systems which can be aggregated after a small perturbation in the parameters.

The technique of spatial aggregation, instead, applies to models whose objects perform a random walk on a two-dimensional lattice. Specifically, it is shown that the underlying ODE system, whose size is proportional to the number of the regions, converges to a system of partial differential equations of constant size as the number of regions goes to infinity.

This allows for an efficient analysis of large-scale mobile models in continuous space like ad hoc networks and multi-agent systems. Zur erweiterten Suche.

Login Registrieren. Fluid aggregations for Markovian process algebra. Tschaikowski, Max.

### Exact Fluid Lumpability for Markovian Process Algebra

Tschaikowski, Max : Fluid aggregations for Markovian process algebra. URN : urn:nbn:de:bvb Abstract Quantitative analysis by means of discrete-state stochastic processes is hindered by the well-known phenomenon of state-space explosion, whereby the size of the state space may have an exponential growth with the number of objects in the model.Formal languages with semantics based on ordinary differential equations ODEs have emerged as a useful tool to reason about large-scale distributed systems.

We present differential bisimulationa behavioral equivalence developed as the ODE counterpart of bisimulations for languages with probabilistic or stochastic semantics.

We study it in the context of a Markovian process algebra. Similarly to Markovian bisimulations yielding an aggregated Markov process in the sense of the theory of lumpability, differential bisimulation yields a partition of the ODEs underlying a process algebra term, whereby the sum of the ODE solutions of the same partition block is equal to the solution of a single lumped ODE.

Differential bisimulation is defined in terms of two symmetries that can be verified only using syntactic checks. This enables the adaptation to a continuous-state semantics of proof techniques and algorithms for finite, discrete-state, labeled transition systems. For instance, we readily obtain a result of compositionality, and provide an efficient partition-refinement algorithm to compute the coarsest ODE aggregation of a model according to differential bisimulation.

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**Integrating Inference with Stochastic Process Algebra Models - Jane Hillston, Edinburgh**

Buchholz, P. Erlangen, Germany Google Scholar. Camporesi, F.We study behavioural relations for process algebra with a fluid semantics given in terms of a system of ordinary differential equations ODEs. We introduce label equivalencea relation which is shown to induce an exactly lumped fluid modela potentially smaller ODE system which can be exactly related to the original one. We show that, in general, for two processes that are related in the fluid sense nothing can be said about their relationship from stochastic viewpoint.

However, we identify a class of models for which label equivalence implies a correspondence, called semi-isomorphismbetween their transition systems that are at the basis of the Markovian interpretation. Unable to display preview.

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Ar turnENW EndNote. Buy options.Abstract Quantitative analysis by means of discrete-state stochastic processes is hindered by the well-known phenomenon of state-space explosion, whereby the size of the state space may have an exponential growth with the number of agents of the system under scrutiny.

When the stochastic process underlies a Markovian process algebra model, this problem may be alleviated by suitable notions of behavioural equivalence that induce lumping at the underlying continuous-time Markov chain, establishing an exact relation between a potentially much smaller aggregated chain and the original one.

For the analysis of massively parallel systems, however, lumping techniques may not be sufficient to yield a computationally tractable problem. Recently, much work has been directed towards forms of fluid techniques that provide a set of ordinary differential equations ODEs approximating the expected path of the stochastic process.

Unfortunately, even fluid models of realistic systems may be too large for feasible analysis. This paper studies a behavioural relation for process algebra with fluid semantics, called projected label equivalence, which is shown to yield an exactly fluid lumpable model, i. Project label equivalence relates sequential components of a process term. In general, for any two sequential components that are related in the fluid sense, nothing can be said about their relationship from the stochastic viewpoint.

We define and study a notion of well-posedness which allows us to relate fluid lumpability to the stochastic notion of semi-isomorphism, which is a weaker version of the common notion of isomorphism between the doubly labelled transition systems at the basis of the Markovian interpretation.

Location of Repository. Topics: QA75 Electronic computers. Computer science. OAI identifier: oai:eprints.

Rust screen stretchSuggested articles.In probability theorylumpability is a method for reducing the size of the state space of some continuous-time Markov chainsfirst published by Kemeny and Snell. Suppose that the complete state-space of a Markov chain is divided into disjoint subsets of states, where these subsets are denoted by t i. Both the state-space and the collection of subsets may be either finite or countably infinite.

InKatehakis and Smit discovered the Successively Lumpable processes for which the stationary probabilities can be obtained by successively computing the stationary probabilities of a propitiously constructed sequence of Markov chains. Each of the latter chains has a typically much smaller state space and this yields significant computational improvements.

## Differential Bisimulation for a Markovian Process Algebra

These results have many applications reliability and queueing models and problems. Franceschinis and Muntz introduced quasi-lumpability, a property whereby a small change in the rate matrix makes the chain lumpable. From Wikipedia, the free encyclopedia.

Laurie July [].

Mining pool softwareGehring, F. Finite Markov Chains Second ed.

Harrison and Naresh M. Probability in the Engineering and Informational Sciences. Performance Evaluation. Elsevier B. Categories : Markov processes. Namespaces Article Talk. Views Read Edit View history. Help Learn to edit Community portal Recent changes Upload file. Download as PDF Printable version.To protect your privacy, all features that rely on external API calls from your browser are turned off by default.

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