Restart Could Optimize the Probability of Success in a Bernoulli Trial

The recently noticed ability of restart to reduce the expected completion time of first-passage processes allows appealing opportunities for performance improvement in a variety of settings. However, complex stochastic processes often exhibit several possible scenarios of completion which are not equally desirable in terms of efficiency. Here we show that restart may have profound consequences on the splitting probabilities of a Bernoulli-like first-passage process, i.e., of a process which can end with one of two outcomes. Particularly intriguing, in this respect, is the class of problems where a carefully adjusted restart mechanism maximizes the probability that the process will complete in a desired way. We reveal the universal aspects of this kind of optimal behavior by applying the general approach recently proposed for the problem of first-passage under restart.

Figure 1

(Left) The probability of target detection $p_r$ versus the rate $r$ of Poisson restart for different values of the searcher’s decay constant $\alpha$. (Right) Optimal restart rate as a function of the decay constant.

Figure 2

(a) Bernoulli-like first-passage process under restart. (b) Pseudocode representation of Eq. (2).

Figure 3

The probability of success $p_r$ (gray circles) and the MFPTs, $⟨T_r⟩$ (blue circles) and $⟨T_r^s⟩$ (green circles), versus the rate $r$ of a Poisson restart obtained from the numerical simulations of a Bernoulli experiment in various settings: mortal Brownian searcher (a), mortal run-and-tumbling searcher (b), Brownian searcher and two competing targets (c), and two Brownian searchers competing for a target (d). As we see, the relation $(d\ln p_r/dr)=⟨T_r⟩-⟨T_r^s⟩$ [a simple rearrangement of Eq. (7)] is generally valid. The details of simulations can be found in [53].