Queueing theory of search processes with stochastic resetting.

Abstract

We use queueing theory to develop a general framework for analyzing search processes with stochastic resetting, under the additional assumption that following absorption by a target, the particle (searcher) delivers a packet of resources to the target and the search process restarts at the reset point x_{r}. This leads to a sequence of search-and-capture events, whereby resources accumulate in the target under the combined effects of resource supply and degradation. Combining the theory of G/M/∞ queues with a renewal method for analyzing resetting processes, we derive general expressions for the mean and variance of the number of resource packets within the target at steady state. These expressions apply to both exponential and nonexponential resetting protocols and take into account delays arising from various factors such as finite return times, refractory periods, and delays due to the loading or unloading of resources. In the case of exponential resetting, we show how the resource statistics can be expressed in terms of the MFPTs T_{r}(x_{r}) and T_{r+γ}(x_{r}), where r is the resetting rate and γ is the degradation rate. This allows us to derive various general results concerning the dependence of the mean and variance on the parameters r,γ. Our results are illustrated using several specific examples. Finally, we show how fluctuations can be reduced either by allowing the delivery of multiple packets that degrade independently or by having multiple independent searchers.