1,531 research outputs found
About the Approximation of Stochastic Petri Nets by Continuous Petri Nets: Several Regions
Reliability analysis is often based on stochastic discrete event models like Markov models or stochastic Petri nets. For complex dynamical systems with numerous components, analytical expressions of the steady state are tedious to work out because of the combinatory explosion with discrete models. Moreover, the convergence of stochastic estimators is slow. For these reasons, fluidification can be investigated to estimate the asymptotic behaviour of stochastic processes with timed continuous Petri nets. The contribution of this paper is to point out the limits of the fluidification in the context of the stochastic steady state approximation. Unfortunately, the asymptotic mean marking of stochastic and continuous Petri nets with same structure and same initial marking are mainly often different. This paper shows that this difficulty is related to the partition in regions of the reachability state space and the existence of critical region
Performance evaluation of an emergency call center: tropical polynomial systems applied to timed Petri nets
We analyze a timed Petri net model of an emergency call center which
processes calls with different levels of priority. The counter variables of the
Petri net represent the cumulated number of events as a function of time. We
show that these variables are determined by a piecewise linear dynamical
system. We also prove that computing the stationary regimes of the associated
fluid dynamics reduces to solving a polynomial system over a tropical
(min-plus) semifield of germs. This leads to explicit formul{\ae} expressing
the throughput of the fluid system as a piecewise linear function of the
resources, revealing the existence of different congestion phases. Numerical
experiments show that the analysis of the fluid dynamics yields a good
approximation of the real throughput.Comment: 21 pages, 4 figures. A shorter version can be found in the
proceedings of the conference FORMATS 201
About Dynamical Systems Appearing in the Microscopic Traffic Modeling
Motivated by microscopic traffic modeling, we analyze dynamical systems which
have a piecewise linear concave dynamics not necessarily monotonic. We
introduce a deterministic Petri net extension where edges may have negative
weights. The dynamics of these Petri nets are well-defined and may be described
by a generalized matrix with a submatrix in the standard algebra with possibly
negative entries, and another submatrix in the minplus algebra. When the
dynamics is additively homogeneous, a generalized additive eigenvalue may be
introduced, and the ergodic theory may be used to define a growth rate under
additional technical assumptions. In the traffic example of two roads with one
junction, we compute explicitly the eigenvalue and we show, by numerical
simulations, that these two quantities (the additive eigenvalue and the growth
rate) are not equal, but are close to each other. With this result, we are able
to extend the well-studied notion of fundamental traffic diagram (the average
flow as a function of the car density on a road) to the case of two roads with
one junction and give a very simple analytic approximation of this diagram
where four phases appear with clear traffic interpretations. Simulations show
that the fundamental diagram shape obtained is also valid for systems with many
junctions. To simulate these systems, we have to compute their dynamics, which
are not quite simple. For building them in a modular way, we introduce
generalized parallel, series and feedback compositions of piecewise linear
concave dynamics.Comment: PDF 38 page
CSL model checking of Deterministic and Stochastic Petri Nets
Deterministic and Stochastic Petri Nets (DSPNs) are a widely used high-level formalism for modeling discrete-event systems where events may occur either without consuming time, after a deterministic time, or after an exponentially distributed time. The underlying process dened by DSPNs, under certain restrictions, corresponds to a class of Markov Regenerative Stochastic Processes (MRGP). In this paper, we investigate the use of CSL (Continuous Stochastic Logic) to express probabilistic properties, such a time-bounded until and time-bounded next, at the DSPN level. The verication of such properties requires the solution of the steady-state and transient probabilities of the underlying MRGP. We also address a number of semantic issues regarding the application of CSL on MRGP and provide numerical model checking algorithms for this logic. A prototype model checker, based on SPNica, is also described
Ranking Functions for Vector Addition Systems
Vector addition systems are an important model in theoretical computer
science and have been used for the analysis of systems in a variety of areas.
Termination is a crucial property of vector addition systems and has received
considerable interest in the literature. In this paper we give a complete
method for the construction of ranking functions for vector addition systems
with states. The interest in ranking functions is motivated by the fact that
ranking functions provide valuable additional information in case of
termination: They provide an explanation for the progress of the vector
addition system, which can be reported to the user of a verification tool, and
can be used as certificates for termination. Moreover, we show how ranking
functions can be used for the computational complexity analysis of vector
addition systems (here complexity refers to the number of steps the vector
addition system under analysis can take in terms of the given initial vector)
Analysis of Petri Net Models through Stochastic Differential Equations
It is well known, mainly because of the work of Kurtz, that density dependent
Markov chains can be approximated by sets of ordinary differential equations
(ODEs) when their indexing parameter grows very large. This approximation
cannot capture the stochastic nature of the process and, consequently, it can
provide an erroneous view of the behavior of the Markov chain if the indexing
parameter is not sufficiently high. Important phenomena that cannot be revealed
include non-negligible variance and bi-modal population distributions. A
less-known approximation proposed by Kurtz applies stochastic differential
equations (SDEs) and provides information about the stochastic nature of the
process. In this paper we apply and extend this diffusion approximation to
study stochastic Petri nets. We identify a class of nets whose underlying
stochastic process is a density dependent Markov chain whose indexing parameter
is a multiplicative constant which identifies the population level expressed by
the initial marking and we provide means to automatically construct the
associated set of SDEs. Since the diffusion approximation of Kurtz considers
the process only up to the time when it first exits an open interval, we extend
the approximation by a machinery that mimics the behavior of the Markov chain
at the boundary and allows thus to apply the approach to a wider set of
problems. The resulting process is of the jump-diffusion type. We illustrate by
examples that the jump-diffusion approximation which extends to bounded domains
can be much more informative than that based on ODEs as it can provide accurate
quantity distributions even when they are multi-modal and even for relatively
small population levels. Moreover, we show that the method is faster than
simulating the original Markov chain
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