  We propose a method for analyzing the distributed random coordinate descent
algorithm for solving separable resource allocation problems in the context of
an open multiagent system, where agents can be replaced during the process. In
particular, we characterize the evolution of the distance to the minimizer in
expectation by following a time-varying optimization approach which builds on
two components. First, we establish the linear convergence of the algorithm in
closed systems, in terms of the estimate towards the minimizer, for general
graphs and appropriate step-size. Second, we estimate the change of the optimal
solution after a replacement, in order to evaluate its effect on the distance
between the current estimate and the minimizer. From these two elements, we
derive stability conditions in open systems and establish the linear
convergence of the algorithm towards a steady-state expected error. Our results
enable to characterize the trade-off between speed of convergence and
robustness to agent replacements, under the assumptions that local functions
are smooth, strongly convex, and have their minimizers located in a given ball.
The approach proposed in this paper can moreover be extended to other
algorithms guaranteeing linear convergence in closed system.
