Large-scale distributed synchronization systems, using a cancel-on-completion redundancy mechanism
Alexander Stolyar
TL;DR
This work develops a unified mean-field framework for large-scale distributed synchronization with cancel-on-completion redundancy, allowing regulation on the left, right, both sides, or neither. It establishes the existence and uniqueness of mean-field fixed points (ML-FPs) and analyzes traveling-wave (ML-FP) solutions, including rigorous conditions for steady-state asymptotic independence (SSAI) as $n\to\infty$. The results characterize velocity ranges $[v_{min},v_{max}]$ for the free system and describe how SSAI and ML-FPs behave under left/right regulation, with explicit connections between regulated and unregulated models. Methods hinge on monotone couplings, reduced systems, and generator bounds, yielding sharp conclusions about load, stability, and asymptotic behavior across a broad class of c.o.c. redundancy systems. The findings unify and extend prior left-regulated results, offer insights into the speed of advance, and provide a framework applicable to other non-work-conserving mean-field systems with regulation boundaries.
Abstract
We consider a class of multi-agent distributed synchronization systems, which are modeled as $n$ particles moving on the real line. This class generalizes the model of a multi-server queueing system, considered in [15], employing so-called cancel-on-completion (c.o.c.) redundancy mechanism, but is motivated by other applications as well. The model in [15] is a particle system, regulated at the left boundary point. The more general model of this paper is such that we allow regulation boundaries on either side, or both sides, or no regulation at all. We consider the mean-field asymptotic regime, when the number of particles $n$ and the job arrival rates go to infinity, while the job arrival rates per particle remain constant. The system state for a given $n$ is the empirical distribution of the particles' locations. The results include: the existence/uniqueness of fixed points of mean-field limits (ML), which describe the limiting dynamics of the system; conditions for the steady-state asymptotic independence (concentration of the stationary distribution on a single ML fixed point); the limits of the average velocity at which unregulated (free) particle system advances. In particular, our results for the left-regulated system unify and generalize the corresponding results in [15]. Our technical approach is such that the systems with different types of regulation are analyzed within a unified framework.