# American Institute of Mathematical Sciences

December  2021, 16(4): 513-533. doi: 10.3934/nhm.2021015

## Qualitative properties of mathematical model for data flow

 1 Computational and Applied Mathematics Group, Oak Ridge National Laboratory, 1 Bethel Valley Road, Bldg. 5700, Oak Ridge, TN 37831-6164, USA 2 Department of Mathematics, University of Tennessee, 227 Ayres Hall. 1403 Circle Drive. Knoxville TN 37996-1320, USA 3 Institut für Geometrie und Praktische Mathematik (IGPM), RWTH Aachen University, Templergraben 55, 52062 Aachen, Germany

* Corresponding author: Giuseppe Visconti

Received  November 2020 Revised  May 2021 Published  December 2021 Early access  July 2021

Fund Project: The United States Government retains and the publisher, by accepting the article for publication, acknowledges that the United States Government retains a non-exclusive, paid-up, irrevocable, world-wide license to publish or reproduce the published form of this manuscript, or allow others to do so, for the United States Government purposes. The Department of Energy will provide public access to these results of federally sponsored research in accordance with the DOE Public Access Plan (http://energy.gov/downloads/doe-public-access-plan)

In this paper, properties of a recently proposed mathematical model for data flow in large-scale asynchronous computer systems are analyzed. In particular, the existence of special weak solutions based on propagating fronts is established. Qualitative properties of these solutions are investigated, both theoretically and numerically.

Citation: Cory D. Hauck, Michael Herty, Giuseppe Visconti. Qualitative properties of mathematical model for data flow. Networks and Heterogeneous Media, 2021, 16 (4) : 513-533. doi: 10.3934/nhm.2021015
##### References:
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show all references

##### References:
 [1] C. Appert-Rolland, P. Degond and S. Motsch, Two-way multi-lane traffic model for pedestrians in corridors, Netw. Heterog. Media, 6 (2011), 351-381.  doi: 10.3934/nhm.2011.6.351. [2] D. Armbruster, P. Degond and C. Ringhofer, A model for the dynamics of large queuing networks and supply chains, SIAM J. Appl. Math., 66 (2006), 896-920.  doi: 10.1137/040604625. [3] D. Armbruster, S. Göttlich and M. Herty, A scalar conservation law with discontinuous flux for supply chains with finite buffers, SIAM J. Appl. Math., 71 (2011), 1070-1087.  doi: 10.1137/100809374. [4] A. Aw, A. Klar, T. Materne and M. Rascle, Derivation of continuum traffic flow models from microscopic follow-the-leader models, SIAM J. Appl. Math., 63 (2002), 259-278.  doi: 10.1137/S0036139900380955. [5] R. C. Barnard, K. Huang and C. Hauck, A mathematical model of asynchronous data flow in parallel computers, IMA Journal of Applied Mathematics, 85 (2020), 865-891.  doi: 10.1093/imamat/hxaa031. [6] N. Bellomo and C. Dogbé, On the modelling crowd dynamics from scaling to hyperbolic macroscopic models, Mathematical Models and Methods in Applied Sciences, 18 (2008), 1317-1345.  doi: 10.1142/S0218202508003054. [7] H. Chan, M. J. Cherukara, B. Narayanan, T. D. Loeffler, C. Benmore, S. K. Gray and S. K. R. S. Sankaranarayanan, Machine learning coarse grained models for water, Nature Communications, 10 (2019), 379. doi: 10.1038/s41467-018-08222-6. [8] A. Chertock, A. Kurganov, A. Polizzi and I. Timofeyev, Pedestrian flow models with slowdown interactions, Mathematical Models and Methods in Applied Sciences, 24 (2014), 249-275.  doi: 10.1142/S0218202513400083. [9] C. M. Dafermos, Polygonal approximations of solutions of the initial value problem for a conservation law, J. Math. Anal. Appl., 38 (1972), 33-41.  doi: 10.1016/0022-247X(72)90114-X. [10] C. M. Dafermos, Hyperbolic Conservation Laws in Continuum Physics, vol. 325 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 2nd edition doi: 10.1007/3-540-29089-3. [11] M. Di Francesco and M. D. Rosini, Rigorous derivation of nonlinear scalar conservation laws from follow-the-leader type models via many particle limit, Archive for Rational Mechanics and Analysis, 217 (2015), 831-871.  doi: 10.1007/s00205-015-0843-4. [12] J. Dongarra, J. Hittinger, J. Bell, L. Chacón, R. Falgout, M. Heroux, P. Hovland, E. Ng, C. Webster and S. Wild, Applied Mathematics Research for Exascale Computing, Technical report, U.S. Department of Energy, Office of Science, Advanced Scientific Computing Research Program, 2014. doi: 10.2172/1149042. [13] L. C. Evans, Partial Differential Equations, American Mathematical Society, 2010. doi: 10.1090/gsm/019. [14] A. Galante and D. Levy, Modeling selective local interactions with memory, Physica D: Nonlinear Phenomena, 260 (2013), 176-190.  doi: 10.1016/j.physd.2012.10.010. [15] D. Helbing, A fluid dynamic model for the movement of pedestrians, Complex Systems, 6 (1992), 391-415. [16] H. Holden and N. H. Risebro, Models for dense multilane vehicular traffic, SIAM J. Math. Anal., 51 (2019), 3694-3713.  doi: 10.1137/19M124318X. [17] C. Murray et al., Basic Research Needs for Microelectronics: Report of the Office of Science Workshop on Basic Research Needs for Microelectronics, Technical report, USDOE Office of Science (SC)(United States), 2018. [18] J. S. Vetter et al., Extreme Heterogeneity 2018-Productive Computational Science in the Era of Extreme Heterogeneity: Report for DOE ASCR Workshop on Extreme Heterogeneity, Technical report, USDOE Office of Science (SC), Washington, DC (United States), 2018. doi: 10.2172/1473756. [19] S. Williams, A. Waterman and D. Patterson, Roofline: An insightful visual performance model for multicore architectures, Communications of the ACM, 52 (2009), 65-76.  doi: 10.1145/1498765.1498785.
Shape of the throttling functions used in the definition of the flux $\Phi$ in (13)
Left: Contour plot of the density $\rho(t,x,z)$ for a fixed time $t.$ Processed data with constant density $r>0$ is depicted in blue up to a stage of completion $z = \zeta(t,x).$ Zero data (grey) is prescribed for completion stages $z> \zeta(t,x)$, c.f. equation (28). Right: Similar plot of a density with constant values $r_1$ and $r_2$ and regions separated by functions $\zeta_1(t,x)$ and $\zeta_2(t,x)$
Domain of hyperbolicity of system (42). In both cases, the non-hyperbolic region (gray) is bounded by the $\xi_1$-axis and by $\xi_2 = \pm C \xi_1$ with slope $C = \frac{4 \epsilon \alpha \rho_* \eta}{\left(\epsilon \alpha -\rho_*\right)^2} \to 0$ as $\epsilon \to 0^+$
Density profiles for Example 1
Density profiles based on the a simulation of the constant front solution (69) with the scheme in (58). The simulations in panels (a)-(c) use $N^x = N^z = 800$ computational cells, while in panel (d), numerical solutions are computed at different refinement levels. In the top row, the analytical front is marked by circles
Density profiles based on the simulation of the $\textsf{V}$-shaped initial front (70) with the scheme in (58) using $N^x = N^z = 800$. The analytic solution is computed using (72). In the top row, the analytical front is marked by circles
Density profiles based on the simulation of the $\textsf{V}$-shaped initial front in (70), using the scheme in (58) with $N^x = N^z = 800$. The analytic solution is computed using (72). In the top row, the analytical front is marked by circles
Density profiles based on the simulation of the $\textsf{V}$-shaped initial front (70) with the scheme in (58) using $N^x = N^z = 800$. The analytical solution is computed using (74). In the top row, the analytical front is marked by circles
Density profiles for two different policies choices of $\alpha$ with the scheme in (58) using $N^x = N^z = 800$
Comparison of the quantities of interest $w_i, i = 1,2,3$ given in (78)
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