The asymptotic scaling properties of conservative algorithms for parallel discrete-event simulations (e.g.: for spatially distributed parallel simulations of dynamic Monte Carlo for spin systems) of one-dimensional systems with system size $L$ is studied. The particular case studied here is the case of one or two elements assigned to each processor element. The previously studied case of one element per processor is reviewed, and the two elements per processor case is presented. The key concept is a simulated time horizon which is an evolving non equilibrium surface, specific for the particular algorithm. It is shown that the flat-substrate initial condition is responsible for the existence of an initial non-scaling regime. Various methods to deal with this non-scaling regime are documented, both the final successful method and unsuccessful attempts. The width of this time horizon relates to desynchronization in the system of processors. Universal properties of the conservative time horizon are derived by constructing a distribution of the interface width at saturation.

Non-equilibrium surface growth for competitive growth models in (1+1) dimensions, particularly mixing random deposition (RD) with correlated growth process which occur with probability $p$ are studied. The composite mixtures are found to be in the universality class of the correlated growth process, and a nonuniversal exponent $\delta$ is identified in the scaling in $p$. The only effects of the RD admixture are dilations of the time and height scales which result in a slowdown of the dynamics of building up the correlations. The bulk morphology is taken into account and is reflected in the surface roughening, as well as the scaling behavior. It is found that the continuum equations and scaling laws for RD added, in particular, to Kardar-Parisi-Zhang (KPZ) processes are partly determined from the underlying bulk structures. Nonequilibrium surface growth analysis are also applied to a study of the static and dynamic load balancing for a conservative update algorithm for Parallel Discrete Event Simulations (PDES). This load balancing is governed by the KPZ equation. For uneven load distributions in conservative PDES simulations, the simulated (virtual) time horizon (VTH) per Processing Element (PE) and the imulated time horizon per volume element $N_{v}$ are used to study the PEs progress in terms of utilization. The width of these time horizons relates to the desynchronization of the system of processors, and is related to the memory requirements of the PEs. The utilization increases when the dynamic, rather than static, load balancing is performed.