Abstract
Restart -- interrupting a stochastic process followed by a new start -- is known to improve the mean time to its completion, and the general conditions under which such an improvement is achieved are now well understood. Here, we explore how restart affects other important metrics of first-passage phenomena, namely the median and the mode of the first-passage time distribution. Our analysis provides a general criterion for when restart lowers the median time, and demonstrates that restarting is always helpful in reducing the mode. Additionally, we show that simple non-uniform restart strategies allow to optimize the mean and the median first-passage times, regardless of the characteristic time scales of the underlying process. These findings are illustrated with the canonical example of a diffusive search with resetting.
Highlights
The mean first-passage time is widely used to quantify performance in diverse applications ranging from randomized search algorithms to kinetics of chemical reactions [1]
In an enzymatic reaction with an excess of substrate molecules [see Fig. 1(b)], the time taken for a significant change in concentration of the substrate is long compared with the expected catalysis time, and the latter is a more natural measure of how fast the reaction proceeds
The introduction of restart, which occasionally returns the particle to its initial position, reduces the median completion time T1/2 of diffusive search
Summary
The mean first-passage time is widely used to quantify performance in diverse applications ranging from randomized search algorithms to kinetics of chemical reactions [1]. This metrics can be significantly improved by implementation of restart, i.e., by interrupting the first-passage process just to start it anew. The development of a general renewal approach has provided a unified and model-independent treatment of first passage under restart [4,5,6] It has furnished a simple criterion for when restart helps to lower the expected completion time of first-passage processes and revealed universality in the behavior of the optimally restarted processes [5].
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