10,271 research outputs found
Fluid Model Checking of Timed Properties
We address the problem of verifying timed properties of Markovian models of
large populations of interacting agents, modelled as finite state automata. In
particular, we focus on time-bounded properties of (random) individual agents
specified by Deterministic Timed Automata (DTA) endowed with a single clock.
Exploiting ideas from fluid approximation, we estimate the satisfaction
probability of the DTA properties by reducing it to the computation of the
transient probability of a subclass of Time-Inhomogeneous Markov Renewal
Processes with exponentially and deterministically-timed transitions, and a
small state space. For this subclass of models, we show how to derive a set of
Delay Differential Equations (DDE), whose numerical solution provides a fast
and accurate estimate of the satisfaction probability. In the paper, we also
prove the asymptotic convergence of the approach, and exemplify the method on a
simple epidemic spreading model. Finally, we also show how to construct a
system of DDEs to efficiently approximate the average number of agents that
satisfy the DTA specification
Applying Mean-field Approximation to Continuous Time Markov Chains
The mean-field analysis technique is used to perform analysis of a systems with a large number of components to determine the emergent deterministic behaviour and how this behaviour modifies when its parameters are perturbed. The computer science performance modelling and analysis community has found the mean-field method useful for modelling large-scale computer and communication networks. Applying mean-field analysis from the computer science perspective requires the following major steps: (1) describing how the agents populations evolve by means of a system of differential equations, (2) finding the emergent
deterministic behaviour of the system by solving such differential equations, and (3) analysing properties of this behaviour either by relying on simulation or by using logics. Depending on the system under analysis, performing these steps may become challenging. Often, modifications
of the general idea are needed. In this tutorial we consider illustrating examples to discuss how the mean-field method is used in different application areas. Starting from the application of the classical technique,
moving to cases where additional steps have to be used, such as systems with local communication. Finally we illustrate the application of the simulation and
uid model checking analysis techniques
Mean-Field approximation and Quasi-Equilibrium reduction of Markov Population Models
Markov Population Model is a commonly used framework to describe stochastic
systems. Their exact analysis is unfeasible in most cases because of the state
space explosion. Approximations are usually sought, often with the goal of
reducing the number of variables. Among them, the mean field limit and the
quasi-equilibrium approximations stand out. We view them as techniques that are
rooted in independent basic principles. At the basis of the mean field limit is
the law of large numbers. The principle of the quasi-equilibrium reduction is
the separation of temporal scales. It is common practice to apply both limits
to an MPM yielding a fully reduced model. Although the two limits should be
viewed as completely independent options, they are applied almost invariably in
a fixed sequence: MF limit first, QE-reduction second. We present a framework
that makes explicit the distinction of the two reductions, and allows an
arbitrary order of their application. By inverting the sequence, we show that
the double limit does not commute in general: the mean field limit of a
time-scale reduced model is not the same as the time-scale reduced limit of a
mean field model. An example is provided to demonstrate this phenomenon.
Sufficient conditions for the two operations to be freely exchangeable are also
provided
Fluid Model Checking
In this paper we investigate a potential use of fluid approximation
techniques in the context of stochastic model checking of CSL formulae. We
focus on properties describing the behaviour of a single agent in a (large)
population of agents, exploiting a limit result known also as fast simulation.
In particular, we will approximate the behaviour of a single agent with a
time-inhomogeneous CTMC which depends on the environment and on the other
agents only through the solution of the fluid differential equation. We will
prove the asymptotic correctness of our approach in terms of satisfiability of
CSL formulae and of reachability probabilities. We will also present a
procedure to model check time-inhomogeneous CTMC against CSL formulae
Learning and Designing Stochastic Processes from Logical Constraints
Stochastic processes offer a flexible mathematical formalism to model and
reason about systems. Most analysis tools, however, start from the premises
that models are fully specified, so that any parameters controlling the
system's dynamics must be known exactly. As this is seldom the case, many
methods have been devised over the last decade to infer (learn) such parameters
from observations of the state of the system. In this paper, we depart from
this approach by assuming that our observations are {\it qualitative}
properties encoded as satisfaction of linear temporal logic formulae, as
opposed to quantitative observations of the state of the system. An important
feature of this approach is that it unifies naturally the system identification
and the system design problems, where the properties, instead of observations,
represent requirements to be satisfied. We develop a principled statistical
estimation procedure based on maximising the likelihood of the system's
parameters, using recent ideas from statistical machine learning. We
demonstrate the efficacy and broad applicability of our method on a range of
simple but non-trivial examples, including rumour spreading in social networks
and hybrid models of gene regulation
Hybrid Behaviour of Markov Population Models
We investigate the behaviour of population models written in Stochastic
Concurrent Constraint Programming (sCCP), a stochastic extension of Concurrent
Constraint Programming. In particular, we focus on models from which we can
define a semantics of sCCP both in terms of Continuous Time Markov Chains
(CTMC) and in terms of Stochastic Hybrid Systems, in which some populations are
approximated continuously, while others are kept discrete. We will prove the
correctness of the hybrid semantics from the point of view of the limiting
behaviour of a sequence of models for increasing population size. More
specifically, we prove that, under suitable regularity conditions, the sequence
of CTMC constructed from sCCP programs for increasing population size converges
to the hybrid system constructed by means of the hybrid semantics. We
investigate in particular what happens for sCCP models in which some
transitions are guarded by boolean predicates or in the presence of
instantaneous transitions
Tractable diffusion and coalescent processes for weakly correlated loci
Widely used models in genetics include the Wright-Fisher diffusion and its
moment dual, Kingman's coalescent. Each has a multilocus extension but under
neither extension is the sampling distribution available in closed-form, and
their computation is extremely difficult. In this paper we derive two new
multilocus population genetic models, one a diffusion and the other a
coalescent process, which are much simpler than the standard models, but which
capture their key properties for large recombination rates. The diffusion model
is based on a central limit theorem for density dependent population processes,
and we show that the sampling distribution is a linear combination of moments
of Gaussian distributions and hence available in closed-form. The coalescent
process is based on a probabilistic coupling of the ancestral recombination
graph to a simpler genealogical process which exposes the leading dynamics of
the former. We further demonstrate that when we consider the sampling
distribution as an asymptotic expansion in inverse powers of the recombination
parameter, the sampling distributions of the new models agree with the standard
ones up to the first two orders.Comment: 34 pages, 1 figur
Complementary approaches to understanding the plant circadian clock
Circadian clocks are oscillatory genetic networks that help organisms adapt
to the 24-hour day/night cycle. The clock of the green alga Ostreococcus tauri
is the simplest plant clock discovered so far. Its many advantages as an
experimental system facilitate the testing of computational predictions.
We present a model of the Ostreococcus clock in the stochastic process
algebra Bio-PEPA and exploit its mapping to different analysis techniques, such
as ordinary differential equations, stochastic simulation algorithms and
model-checking. The small number of molecules reported for this system tests
the limits of the continuous approximation underlying differential equations.
We investigate the difference between continuous-deterministic and
discrete-stochastic approaches. Stochastic simulation and model-checking allow
us to formulate new hypotheses on the system behaviour, such as the presence of
self-sustained oscillations in single cells under constant light conditions.
We investigate how to model the timing of dawn and dusk in the context of
model-checking, which we use to compute how the probability distributions of
key biochemical species change over time. These show that the relative
variation in expression level is smallest at the time of peak expression,
making peak time an optimal experimental phase marker. Building on these
analyses, we use approaches from evolutionary systems biology to investigate
how changes in the rate of mRNA degradation impacts the phase of a key protein
likely to affect fitness. We explore how robust this circadian clock is towards
such potential mutational changes in its underlying biochemistry. Our work
shows that multiple approaches lead to a more complete understanding of the
clock
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