Process algebraic non-product-forms

AbstractA generalization of the Reversed Compound Agent Theorem of Markovian process algebra is derived that yields separable, but non-product-form solutions for collections of interacting processes such as arise in multi-class queueing networks with Processor Sharing servers. It is based on an analysis of the minimal cycles in the state space of a multi-agent cooperation, which can be simply identified. The extended methodology leads to what we believe are new separable solutions and, more generally, the results represent a viable practical application of the theory of Markovian process algebras in stochastic modelling

Process algebra for performance evaluation

This paper surveys the theoretical developments in the field of stochastic process algebras, process algebras where action occurrences may be subject to a delay that is determined by a random variable. A huge class of resource-sharing systems – like large-scale computers, client–server architectures, networks – can accurately be described using such stochastic specification formalisms. The main emphasis of this paper is the treatment of operational semantics, notions of equivalence, and (sound and complete) axiomatisations of these equivalences for different types of Markovian process algebras, where delays are governed by exponential distributions. Starting from a simple actionless algebra for describing time-homogeneous continuous-time Markov chains, we consider the integration of actions and random delays both as a single entity (like in known Markovian process algebras like TIPP, PEPA and EMPA) and as separate entities (like in the timed process algebras timed CSP and TCCS). In total we consider four related calculi and investigate their relationship to existing Markovian process algebras. We also briefly indicate how one can profit from the separation of time and actions when incorporating more general, non-Markovian distributions

Extended Differential Aggregations in Process Algebra for Performance and Biology

We study aggregations for ordinary differential equations induced by fluid semantics for Markovian process algebra which can capture the dynamics of performance models and chemical reaction networks. Whilst previous work has required perfect symmetry for exact aggregation, we present approximate fluid lumpability, which makes nearby processes perfectly symmetric after a perturbation of their parameters. We prove that small perturbations yield nearby differential trajectories. Numerically, we show that many heterogeneous processes can be aggregated with negligible errors.Comment: In Proceedings QAPL 2014, arXiv:1406.156

Compositional Performance Modelling with the TIPPtool

Stochastic process algebras have been proposed as compositional specification formalisms for performance models. In this paper, we describe a tool which aims at realising all beneficial aspects of compositional performance modelling, the TIPPtool. It incorporates methods for compositional specification as well as solution, based on state-of-the-art techniques, and wrapped in a user-friendly graphical front end. Apart from highlighting the general benefits of the tool, we also discuss some lessons learned during development and application of the TIPPtool. A non-trivial model of a real life communication system serves as a case study to illustrate benefits and limitations

On Markovian Cocycle Perturbations in Classical and Quantum Probability

We introduce Markovian cocycle perturbations of the groups of transformations associated with the classical and quantum stochastic processes with stationary increments, which are characterized by a localization of the perturbation to the algebra of events of the past. It is namely the definition one needs because the Markovian perturbations of the Kolmogorov flows associated with the classical and quantum noises result in the perturbed group of transformations which can be decomposed in the sum of a part associated with deterministic stochastic processes lying in the past and a part associated with the noise isomorphic to the initial one. This decomposition allows to obtain some analog of the Wold decomposition for classical stationary processes excluding a nondeterministic part of the process in the case of the stationary quantum stochastic processes on the von Neumann factors which are the Markovian perturbations of the quantum noises. For the classical stochastic process with noncorrelated increaments it is constructed the model of Markovian perturbations describing all Markovian cocycles up to a unitary equivalence of the perturbations. Using this model we construct Markovian cocyclies transformating the Gaussian state $\rho$ to the Gaussian states equivalent to $\rho$.Comment: 27 page

Invariant, super and quasi-martingale functions of a Markov process

We identify the linear space spanned by the real-valued excessive functions of a Markov process with the set of those functions which are quasimartingales when we compose them with the process. Applications to semi-Dirichlet forms are given. We provide a unifying result which clarifies the relations between harmonic, co-harmonic, invariant, co-invariant, martingale and co-martingale functions, showing that in the conservative case they are all the same. Finally, using the co-excessive functions, we present a two-step approach to the existence of invariant probability measures

Conditional expectations associated with quantum states

An extension of the conditional expectations (those under a given subalgebra of events and not the simple ones under a single event) from the classical to the quantum case is presented. In the classical case, the conditional expectations always exist; in the quantum case, however, they exist only if a certain weak compatibility criterion is satisfied. This compatibility criterion was introduced among others in a recent paper by the author. Then, state-independent conditional expectations and quantum Markov processes are studied. A classical Markov process is a probability measure, together with a system of random variables, satisfying the Markov property and can equivalently be described by a system of Markovian kernels (often forming a semigroup). This equivalence is partly extended to quantum probabilities. It is shown that a dynamical (semi)group can be derived from a given system of quantum observables satisfying the Markov property, and the group generators are studied. The results are presented in the framework of Jordan operator algebras, and a very general type of observables (including the usual real-valued observables or self-adjoint operators) is considered.Comment: 10 pages, the original publication is available at http://www.aip.or

Quantum Non-Markovianity: Characterization, Quantification and Detection

We present a comprehensive and up to date review on the concept of quantum non-Markovianity, a central theme in the theory of open quantum systems. We introduce the concept of quantum Markovian process as a generalization of the classical definition of Markovianity via the so-called divisibility property and relate this notion to the intuitive idea that links non-Markovianity with the persistence of memory effects. A detailed comparison with other definitions presented in the literature is provided. We then discuss several existing proposals to quantify the degree of non-Markovianity of quantum dynamics and to witness non-Markovian behavior, the latter providing sufficient conditions to detect deviations from strict Markovianity. Finally, we conclude by enumerating some timely open problems in the field and provide an outlook on possible research directions.Comment: Review article. Close to published versio