  Dynamic task allocation is an essential requirement for multi-robot systems
operating in unknown dynamic environments. It allows robots to change their
behavior in response to environmental changes or actions of other robots in
order to improve overall system performance. Emergent coordination algorithms
for task allocation that use only local sensing and no direct communication
between robots are attractive because they are robust and scalable. However, a
lack of formal analysis tools makes emergent coordination algorithms difficult
to design. In this paper we present a mathematical model of a general dynamic
task allocation mechanism. Robots using this mechanism have to choose between
two types of task, and the goal is to achieve a desired task division in the
absence of explicit communication and global knowledge. Robots estimate the
state of the environment from repeated local observations and decide which task
to choose based on these observations. We model the robots and observations as
stochastic processes and study the dynamics of the collective behavior.
Specifically, we analyze the effect that the number of observations and the
choice of the decision function have on the performance of the system. The
mathematical models are validated in a multi-robot multi-foraging scenario. The
model's predictions agree very closely with experimental results from
sensor-based simulations.
