Mean Field for Performance Models with Generally-Distributed Timed Transitions

Hayden, Richard A.; Horváth, Illés; Telek, Miklós

doi:10.1007/978-3-319-10696-0_8

Cited by 6 publications

(9 citation statements)

References 20 publications

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“…We use Poisson representation (see e.g. [19]) for the dynamics of the process. Since the vertices are not identical (as in Kurtz [20] or [19]), we need to be cautious to handle them separately.…”

Section: Convergence Of the Stochastic Processmentioning

confidence: 99%

“…[19]) for the dynamics of the process. Since the vertices are not identical (as in Kurtz [20] or [19]), we need to be cautious to handle them separately. Accordingly, let N N ss ′ ,i (t) denote a family of independent Poisson processes with rate 1.…”

Section: Convergence Of the Stochastic Processmentioning

confidence: 99%

“…Many different types of processes have been examined on large networks. One of the most important applications is infection spreading models such as the SI (susceptible-infected), SIS (susceptible-infected-susceptible) and SIR (susceptible-infected-recovered) models, but there have been applications for wireless sensor networks [6], push-pull gossip protocols [3], peer-to-peer software updates [19], biological systems [7] or molecular biology [28]. A common trait is that as the network size increases, a complete analysis becomes computationally infeasible; instead, mean-field approximations become relevant.…”

Section: Introductionmentioning

confidence: 99%

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Local-density dependent Markov processes on graphons with epidemiological applications

Keliger¹,

Horváth²,

Takács³

2020

Preprint

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We investigate local-density dependent Markov processes on a class of large graphs sampled from a graphon, where the transition rates of the vertices are influenced by the states of their neighbors. We show that as the average degree converges to infinity, the evolution of the process in the transient regime converges to the solution of a set of non-local integro-partial differential equations. We also provide rigorous derivation for the epidemic threshold in the case of the Susceptible-Infected-Susceptible (SIS) process on such graphons.

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Section: Convergence Of the Stochastic Processmentioning

confidence: 99%

Section: Convergence Of the Stochastic Processmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Local-density dependent Markov processes on graphons with epidemiological applications

Keliger¹,

Horváth²,

Takács³

2020

Preprint

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“…Moreover, the mean-field approach has recently been expanded to a class of models with both Markovian and deterministically-timed transitions, as introduced for generalized semi-Markov processes in [50]; and generally-distributed timed transitions for population generalized semi-Markov processes [52]. In addition, the extension towards hybrid Markov population models has recently been made in [92] and [91].…”

Section: Beyond Kurtz's Theoremmentioning

confidence: 99%

“…Even though there is an increasing body of work that deals with relaxing the conditions of the convergence theorem, e.g., [91], [50], [52], it would be very interesting to see whether it can be applied to other kinds of model, for example, such including non-determinism.…”

Section: Future Directionsmentioning

confidence: 99%

Model-checking mean-field models : algorithms & applications

Kolesnichenko¹

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Large systems of interacting objects are highly prevalent in today's world. Such system usually consist of a large number of relatively simple identical objects, and can be observed in many different field as, e.g., physics (interactions of molecules in gas), chemistry (chemical reactions), epidemiology (spread of the infection), etc. In this thesis we primarily address large systems of interacting objects in computer science, namely, computer networks. Analysis of such large systems is made difficult by the state space explosion problem, i.e., the number of states of the model grows exponentially with the number of interacting objects. In this thesis we tackle the state-space explosion problem by applying meanfield approximation, which was originally developed for models in physics, like the interaction of molecules in a gas. The mean-field method works by not considering the state of each individual object separately, but only their average, i.e., what fraction of the objects are in each possible state at any time. It allows to compute the exact limiting behaviour of an infinite population of identical objects, and this limiting behaviour is a good approximation, even when the number of objects is not infinite but sufficiently large. In this thesis we provide the theoretical background necessary for applying the mean-field method and illustrate the approach by a peer-to-peer Botnet case study. This thesis aims at formulating and analysing advanced properties of large systems of interacting objects using fast, efficient, and accurate algorithms. We propose to apply model-checking techniques to mean-field models. This allows (i) defining advanced properties of mean-field models, such as survivability, steady-state availability, conditional instantaneous availability using logic; and (ii) automatically checking these properties using model-checking algorithms. Existing model-checking logics and algorithms can not directly be applied to mean-field models since the model consist of two layers: the local level, describing the behaviour of a randomly chosen individual object in a large system, and the global level, which addresses the overall system of all viii interacting objects. Therefore, we motivate and define two logics, called Mean Field Continuous Stochastic Logic (MF-CSL), and Mean-Field Logic (MFL), for describing properties of systems composed of many identical interacting objects, on both the local and the global level. We present model-checking algorithms for checking both MF-CSL and MFL properties, and illustrated these algorithms using an extensive example on virus propagation in a computer network. We discuss the differences in the expressiveness of these two logics as well as their possible combination. Additionally, we combine the mean-field method with parameter fitting techniques in order to model real-world large systems, and obtain a better understanding of the behaviour of such systems. We explain how to build a mean-field model of the system, and how to estimate the corresponding parameter valu...

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Model Checking Two Layers of Mean-Field Models

Kolesnichenko¹,

Remke

Boer

et al. 2016

Springer Series in Reliability Engineering

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Recently, many systems that consist of a large number of interacting objects have been analysed using the mean-field method, which allows a quick and accurate analysis of such systems, while avoiding the state-space explosion problem. To date, the mean-field method has primarily been used for classical performance evaluation purposes. In this chapter, we discuss model-checking mean-field models. We define and motivate two logics, called Mean-Field Continuous Stochastic Logic (MF-CSL) and Mean-Field Logic (MFL), to describe properties of systems composed of many identical interacting objects. We present model-checking algorithms and discuss the differences in the expressiveness of these two logics and their combinations.

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Mean Field for Performance Models with Generally-Distributed Timed Transitions

Abstract: Abstract. In this paper we extend the mean-field limit of a class of stochastic models with exponential and deterministic delays to include exponential and generally-distributed delays. Our main focus is the rigorous proof of the mean-field limit.

Cited by 6 publications

References 20 publications

Local-density dependent Markov processes on graphons with epidemiological applications

Local-density dependent Markov processes on graphons with epidemiological applications

Model-checking mean-field models : algorithms & applications

Model Checking Two Layers of Mean-Field Models

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