Averaged kinetic models for flows on unstructured networks
Michael Herty Christian Ringhofer
Kinetic & Related Models 2011, 4(4): 1081-1096 doi: 10.3934/krm.2011.4.1081
We derive a kinetic equation for flows on general, unstructured networks with applications to production, social and transportation networks. This model allows for a homogenization procedure, yielding a macroscopic transport model for large networks on large time scales.
keywords: Network dynamics kinetic theory asymptotic analysis.
A kinetic games framework for insurance plans
Daniel Brinkman Christian Ringhofer
Kinetic & Related Models 2017, 10(1): 93-116 doi: 10.3934/krm.2017004

The dynamics of insurance plans have been under the microscope in recent years due to the controversy surrounding the implementation of the Affordable Care Act (Obamacare) in the United States. In this paper, we introduce a game between an insurance company and an ensemble of customers choosing between several insurance plans. We then derive a kinetic model for the strategies of the insurer and the decisions of the customers and establish the conditions for which a Nash equilibrium exists for some specific customer distributions. Finally, we give some agent-based numerical results for how the plan enrollment evolves over time which show qualitative agreement to "experimental" results in the literature from two plans in the state of Massachusetts.

keywords: Kinetic equations PDE modeling agent-based simulation Nash equilibrium insurance
Parameter extraction of complex production systems via a kinetic approach
Ali K. Unver Christian Ringhofer M. Emir Koksal
Kinetic & Related Models 2016, 9(2): 407-427 doi: 10.3934/krm.2016.9.407
Continuum models of re-entrant production systems are developed that treat the flow of products in analogy to traffic flow. Specifically, the dynamics of material flow through a re-entrant factory via a parabolic conservation law is modeled describing the product density and flux in the factory. The basic idea underlying the approach is to obtain transport coefficients for fluid dynamic models in a multi-scale setting simultaneously from Monte Carlo simulations and actual observations of the physical system, i.e. the factory. Since partial differential equation (PDE) conservation laws are successfully used for modeling the dynamical behavior of product flow in manufacturing systems, a re-entrant manufacturing system is modeled using a diffusive PDE. The specifics of the production process enter into the velocity and diffusion coefficients of the conservation law. The resulting nonlinear parabolic conservation law model allows fast and accurate simulations. With the traffic flow-like PDE model, the transient behavior of the discrete event simulation (DES) model according to the averaged influx, which is obtained out of discrete event experiments, is predicted. The work brings out an almost universally applicable tool to provide rough estimates of the behavior of complex production systems in non-equilibrium regimes.
keywords: conservation law Supply chains nonlinear parabolic PDE statistics. re-entrant factory numerical analysis
A kinetic model for an agent based market simulation
Dieter Armbruster Christian Ringhofer Andrea Thatcher
Networks & Heterogeneous Media 2015, 10(3): 527-542 doi: 10.3934/nhm.2015.10.527
A kinetic model for a specific agent based simulation to generate the sales curves of successive generations of high-end computer chips is developed. The resulting continuum market model consists of transport equations in two variables, representing the availability of money and the desire to buy a new chip. In lieu of typical collision terms in the kinetic equations that discontinuously change the attributes of an agent, discontinuous changes are initiated via boundary conditions between sets of partial differential equations. A scaling analysis of the transport equations determines the different time scales that constitute the market forces, characterizing different sales scenarios. It is argued that the resulting model can be adjusted to generic markets of multi-generational technology products where the innovation time scale is an important driver of the market.
keywords: agent based simulations Kinetic theory multi-generational technology products. market models
A hyperbolic relaxation model for product flow in complex production networks
Ali Unver Christian Ringhofer Dieter Armbruster
Conference Publications 2009, 2009(Special): 790-799 doi: 10.3934/proc.2009.2009.790
This paper presents a continuum - traffic flow like - model for the flow of products through complex production networks, based on statistical information obtained from extensive observations of the system. The resulting model consists of a system of hyperbolic conservation laws, which, in a relaxation limit, exhibit the correct diffusive properties given by the variance of the observed data.
keywords: hyperbolic systems relaxation methods Supply chains

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