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# Qualitative properties of mathematical model for data flow

• * Corresponding author: Giuseppe Visconti
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.

Mathematics Subject Classification: 35L65, 35L40.

 Citation: • • Figure 1.  Shape of the throttling functions used in the definition of the flux $\Phi$ in (13)

Figure 2.  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 $\zeta_2(t,x)$

Figure 3.  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 $\epsilon \to 0^+$

Figure 4.  Density profiles for Example 1

Figure 5.  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

Figure 6.  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

Figure 7.  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

Figure 8.  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

Figure 9.  Density profiles for two different policies choices of $\alpha$ with the scheme in (58) using $N^x = N^z = 800$

Figure 10.  Comparison of the quantities of interest $w_i, i = 1,2,3$ given in (78)

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