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.