Data-Fitted Second-Order Macroscopic Production Models

Forestier-Coste, Louis; Göttlich, Simone; Herty, Michaël

doi:10.1137/140989832

Cited by 18 publications

(13 citation statements)

References 33 publications

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“…In contrast, macroscopic models describe the average behavior of the production system in terms of part density or flow. They naturally arise from microscopic limits and are usually based on ordinary differential equations (ODEs) [15], hyperbolic partial differential equations (PDEs) [1,5,6,8,11] or a mixture of both [4,7,14]. We refer to [3] and the references therein for a comprehensive overview of macroscopic production models.…”

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confidence: 99%

Semi-Markovian capacities in production network models

Göttlich¹,

Knapp²

2017

Discrete &Amp; Continuous Dynamical Systems - B

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In this paper, we focus on production network models based on ordinary and partial differential equations that are coupled to semi-Markovian failure rates for the processor capacities. This modeling approach allows for intermediate capacity states in the range of total breakdown to full capacity, where operating and down times might be arbitrarily distributed. The mathematical challenge is to combine the theory of semi-Markovian processes within the framework of conservation laws. We show the existence and uniqueness of such stochastic network solutions, present a suitable simulation method and explain the link to the common queueing theory. A variety of numerical examples emphasizes the characteristics of the proposed approach.

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confidence: 99%

Semi-Markovian capacities in production network models

Göttlich¹,

Knapp²

2017

Discrete &Amp; Continuous Dynamical Systems - B

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“…It is of great interest to notice that some interesting nonlinear phenomena were observed in [3] when the pressure term was taken as P(ρ) = ρ. Especially, it was obtained in [4] that the delta standing wave was also involved in Riemann solutions for system (1.1) under this pressure term P(ρ) = ρ.…”

Section: + P(ρ)mentioning

confidence: 99%

“…In general, a given clearing function describes the averaged sample data, but cannot illustrate the data diffusion. In order to solve this problem, the following macroscopic production model [3] consisting of two conservation laws is proposed in the form ρ t + (ρu) x = 0, ρu 1 + P(ρ) t + ρu 2…”

Section: Introductionmentioning

confidence: 99%

The Perturbed Riemann Problem for a Macroscopic Production Model with Chaplygin Gas

Wang

Shen

2020

Bull. Malays. Math. Sci. Soc.

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The exact Riemann solutions for a macroscopic production model under the equation of state given by the Chaplygin gas are solved explicitly for all possible Riemann initial data. It is discovered interestingly that a composite hyperbolic wave is involved in Riemann solution under some specially designated initial conditions, which is made up of a rarefaction wave and a delta contact discontinuity attached on the wave front of the rarefaction wave. Furthermore, the constructions of global solutions to the perturbed Riemann problem for this system are also displayed in completely explicit forms when the initial data are taken to be three piecewise constant states under some suitable restrictive conditions by using the method of characteristics. During the process of constructing global solutions, the interactions of elementary waves are studied in detail. Moreover, it is proved rigorously that Riemann solutions are stable with respect to the specific small perturbations of Riemann initial data.

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“…By introducing a family of clearing functions, the data‐fitted second‐order macroscopic production model was proposed in the conservative form

({, \begin{matrix} ρ_{t} + {(ρ u)}_{x} = 0, \\ {(ρ u + ρ^{2} u)}_{t} + {(ρ u^{2} + ρ^{2} u^{2})}_{x} = 0 . \end{matrix})

The model was derived from discrete simulations based on sampled data in order to predict the production behavior. The transient behavior of aggregate product flow on possible large‐time scales can be captured well by using the model .…”

Section: Introductionmentioning

confidence: 99%

Singular solutions to the Riemann problem for a macroscopic production model

Sun

2017

Z Angew Math Mech

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Key words Delta standing wave, composite wave, vacuum state, wave interaction, Riemann problem, hyperbolic conservation law. MSC (2000) 35L65, 35L67, 35B30, 76N10This paper is devoted to the Riemann problem for the data-fitted second-order macroscopic production model when one of the Riemann initial data is at a vacuum state or at rest. The Riemann solutions are constructed completely by using the continuous method with respect to the initial data, which include some non-classical waves such as the delta standing wave and the superposition of a shock wave and a contact discontinuity etc. In addition, an example is given to explain the wave interaction problem when the vacuum state is involved.

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Data-Fitted Second-Order Macroscopic Production Models

Cited by 18 publications

References 33 publications

Semi-Markovian capacities in production network models

Semi-Markovian capacities in production network models

The Perturbed Riemann Problem for a Macroscopic Production Model with Chaplygin Gas

Singular solutions to the Riemann problem for a macroscopic production model

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