Parallel search of long circular strands: modeling, analysis, and optimization

We introduce and explore a model of an ensemble of agents searching, in parallel, a long circular strand for a target site. The agents performing the search combine local-scanning (conducted by a one-dimensional motion along the strand) and random relocations on the strand. The agent-ensemble search-durations are analyzed, their limiting probability distributions are obtained in closed-form, and the optimal relocation strategies are derived. The results encompass the cases of parallel and massively parallel searches, taking place in the presence of either finite-mean or heavy-tailed relocation durations. The results are applicable to a wide spectrum of local-scans, including linear motions, Brownian motions, subdiffusive motions, fractional Brownian motions, and fractional Lévy motions.