Optimization of Random Processes Completion by Restart
Belan S.

Received: June 25, 2026

DOI: 10.31857/S0044451026080088

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Restart - an interruption of a stochastic process before its completion, followed by the start of a new random realization of the same process - is a potential tool for control over the completion time statistical properties of stochastic dynamics (e.g., restart of randomized computational procedures). One of the key reasons underlying research interest in theoretical modelling of restart-induced effects on stochastic processes roots in observation that sometimes restart can accelerate process progress as evidenced by reduction of the mean completion time. After this observation was made, many subsequently resolved research questions arose, such as: the optimal schedule of restarting, the limits of the effectiveness of the restart method, the conditions under which this method works and the prospects for using this method to optimize performance measures other than the expected completion time. The most straightforward and general way of extracting theoretical prediction regarding completion time statistics of random processes in the presence of restart is provided by the renewal approach which is applicable to majority of particular stochastic models and restart protocols considered in the relevant literature. Supplemented by analysis of measures from descriptive statistics, linear response analysis, optimization analysis, and analysis based on probabilistic inequalities, the renewal approach allowed to pose and resolve many issues regarding completion statistics of generic random processes subject to various types of restart protocols as discussed in this review article. Keywords: stochastic processes, random walk processes, kinetic models, restart, renewal equation method