We study a system consisting of $n$ particles, moving forward in jumps on the real line. Each particle can make both independent jumps, whose sizes have some distribution, or ``synchronization'' jumps, which allow it to join a randomly chosen other particle if the latter happens to be ahead of it. The mean-field asymptotic regime, where $n\to\infty$, is considered. As $n\to\infty$, we prove the convergence of the system dynamics to that of a deterministic mean-field limit (MFL). We obtain results on the average speed of advance of a ``benchmark'' MFL (BMFL) and the liminf of the steady-state speed of advance, in terms of MFLs that are traveling waves. For the special case of exponentially distributed independent jump sizes, we prove that a traveling wave MFL with speed $v$ exists if and only if $v\ge v_*$, with $v_*$ having simple explicit form; this allows us to show that the average speed of the BMFL is equal to $v_*$ and the liminf of the steady-state speeds is lower bounded by $v_*$. Finally, we put forward a conjecture that both the average speed of the BMFL and the exact limit of the steady-state speeds, under general distribution of an independent jump size, are equal to number $v_{**}$, which is easily found from a ``minimum speed principle.'' This general conjecture is consistent with our results for the exponentially distributed jumps and is confirmed by simulations.

We consider a system consisting of n particles, moving forward on the real line. The particles move in jumps. The system state at a given time is the current empirical distribution of particle locations. Each particle gets "urges to jump" as an independent Poisson process of constant rate. However, a particle getting a jump urge actually jumps with the probability given by a decreasing function of the particle's location quantile within the current state (i.e., empirical distribution); hence this a mean-field type of particles' interaction with each other. When a particle does jump, the jump size is independent, distributed as a random variable Z > 0. We are interested in the system behavior when n is large. This model was introduced in [5,6] as an idealized model of distributed parallel simulation. In this case n particles represent n processors("sites") simulating different part of some large system, and a particle location is the current "local simulation time" of the corresponding processor. The following types of questions are of interest, as n becomes large: how the local times of the processors progress over time; do local times "stay closely together;" does the evolution of the empirical distribution of local times becomes that of a traveling 1