Queueing theory of search processes with stochastic resetting
File(s)
Published version
Author(s)
Bressloff, Paul C
Type
Journal Article
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 xr. 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 Tr(xr ) and Tr+γ (xr ), 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.
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 xr. 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 Tr(xr ) and Tr+γ (xr ), 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.
Date Issued
2020-09
Date Acceptance
2020-08-21
Citation
Physical Review E, 2020, 102 (3)
ISSN
2470-0045
Publisher
American Physical Society (APS)
Journal / Book Title
Physical Review E
Volume
102
Issue
3
Copyright Statement
©2020 American Physical Society
Identifier
http://dx.doi.org/10.1103/physreve.102.032109
Publication Status
Published
Article Number
032109
Date Publish Online
2020-09-08