Abstract
The discrete-continuous model is a novel contribution to mathematical modeling of pedestrian dynamics. This model is of individual type; people (particles) move in a continuous space, – in this sense the model is continuous. But the number of directions for the particles to move is limited, – in this sense the model is discrete. The model is realized in the computer evacuation module SigmaEva. This article is focused on a presenting of the model and computational aspects and the model is discussed in respect with discrete and continuous models.
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Notes
- 1.
There is unified coordinate system, and all data are given in this system.
- 2.
- 3.
- 4.
The reason why we use field S radially increasing from exit and gradient \(\varDelta S_{\alpha }\) instead of original issues is the following. Originally pure values of field S (which radially decreases from exit) are used in the probability formula in floor field models, e.g. [1, 4, 10–12, 15]. We propose to use only a value of gradient \(\varDelta S_{\alpha }\). From a mathematical view point, it yields the same result [7], but computationally this trick has a great advantage. The values of field S may be too high (it depends on the modelling space \(\varOmega \) size); in this case, \(\exp \left( k_S S_{\alpha } \right) \) can appear uncomputable. This is a significant limitation of the models. At the same time, \(0\le |\varDelta S_{\alpha }|< 1\), and a value of \(\exp \left( k_S S_{\alpha } \right) \) is computable.
- 5.
Note, function \(W(\cdot )\) “works” with nonmovable obstacles only.
- 6.
Actually this situation is impossible. Only function \(W(\cdot )\) may give (mathematic) zero to probability. If \(Norm=0\), a particle is surrounded by obstacles.
- 7.
Note (!) that positions of other particles are taken in to account for time \(t-\varDelta t\). As in cellular automata models parallel update is used here.
- 8.
Only here we operate with coordinates obtained for time t. As in floor field cellular automata models movement of all involved particles is denied with probability \(\mu \). One of candidates moves to the desired cell with probability \(1-\mu \). A person allowed to move is chosen randomly.
- 9.
Zero probabilities means very low values which are about \(10^{-4}-10^{-16}\) here.
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Kirik, E., Malyshev, A., Senashova, M. (2016). On the Evacuation Module SigmaEva Based on a Discrete-Continuous Pedestrian Dynamics Model. In: Wyrzykowski, R., Deelman, E., Dongarra, J., Karczewski, K., Kitowski, J., Wiatr, K. (eds) Parallel Processing and Applied Mathematics. Lecture Notes in Computer Science(), vol 9574. Springer, Cham. https://doi.org/10.1007/978-3-319-32152-3_50
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