Parallel discrete-event simulation schemes with heterogeneous processing elements

To understand the effects of nonidentical processing elements (PEs) on parallel discrete-event simulation (PDES) schemes, two stochastic growth models, the restricted solid-on-solid (RSOS) model and the Family model, are investigated by simulations. The RSOS model is the model for the PDES scheme governed by the Kardar-Parisi-Zhang equation (KPZ scheme). The Family model is the model for the scheme governed by the Edwards-Wilkinson equation (EW scheme). Two kinds of distributions for nonidentical PEs are considered. In the first kind computing capacities of PEs are not much different, whereas in the second kind the capacities are extremely widespread. The KPZ scheme on the complex networks shows the synchronizability and scalability regardless of the kinds of PEs. The EW scheme never shows the synchronizability for the random configuration of PEs of the first kind. However, by regularizing the arrangement of PEs of the first kind, the EW scheme is made to show the synchronizability. In contrast, EW scheme never shows the synchronizability for any configuration of PEs of the second kind.

Figure 1

Plots of $W(N,t)$ for the RSOS model on random networks. (a) $\eta$ on each node is randomly assigned as $\eta =0.5$ or $\eta =1$ with equal probability. (b) $\eta$ on each node is a random number drawn from the interval $[0,1]$. Insets: Plots of $W_{\mathrm{sat}}\left(N\right)$ against $1/N$.

Figure 2

Plots of $W(N,t)$ for the RSOS model on SF networks. (a) $\eta$ on each node is randomly assigned as $\eta =0.5$ or $\eta =1$ with equal probability. (b) $\eta$ on each node is a random number drawn from the interval $[0,1]$. Insets: Plots of $W_{\mathrm{sat}}\left(N\right)$ against $1/N$.

Figure 3

(a) Plots of $W(N,t)$ against $t$ for the RSOS model with randomly arranged bimodal PEs in $1d$. The solid line denotes $W(N,t)\sim t^{0.7}$ and the dashed line corresponds to $W(N,t)\sim t^{1/3}$. (b) Plots of $W_{\mathrm{sat}}\left(N\right)$ obtained from the data in panel (a) against $N$. The solid line denotes $W_{\mathrm{sat}}\left(N\right)\sim N$ and the dashed line is $W_{\mathrm{sat}}\left(N\right)\sim N^{1/2}$. (c) Scaling collapse of $W(N,t)$ for the RSOS model with the uniform distribution of $\eta$. $\alpha =1$ and $z=2$ are used for the scaling collapses.

Figure 4

Plot of $W(N,t)$ against $t$ for Family model with randomly arranged bimodal $\eta$ (a) on random networks, (b) on SF networks with $\gamma =2.5$, and (c) on $1d$ substrate. The solid lines represent the relation $W(N,t)\sim t$. (d) Morphology of the Family model for $N=32$. Black squares represent the PEs with $\eta =0.5$. Unmarked PEs have $\eta =1.0$.

Figure 5

(a) A configuration of regularized arrangement (RA) of PEs with the bimodal $\eta$ on $1d$ substrate. (b) A pair of neighboring PEs merges into a new single super PE (rectangle) with the renormalized ${\eta }^{*}$.

Figure 6

Scaling collapse of $W(N,t)$ for the Family model with the RA in Fig. 5 using $\alpha =1/2$ and $z=2$.

Figure 7

Schematic illustration of an RA on networks. (a) The big open circles denote nodes of a base network with $N_0$ nodes. Each node is replaced by the connected pair of a PE with $\eta =0.5$ (small filled circle) and one with $\eta =1.0$ (small open circle). (b) The dashed ellipse represents the node of the base network. The original link between $I$ and $J$ in the base network is replaced in the following way. Let $i$ be a randomly selected PE between the two PEs in $I$ and $j$ be the PE with ${\eta }_i\ne {\eta }_j$ in $J$. Then the link between $i$ and $j$ replaces the link between $I$ and $J$.

Figure 8

(a) Plot of $W(N,t)$ for the Family model on random networks against $t$ when the RA in Fig. 7 is applied. (b) Plot of $W(N,t)$ against $t$ on SF networks under the RA in Fig. 7. Insets: Plot of $W_{\mathrm{sat}}\left(N\right)$ against $N$.