  We study the problem of distributed estimation over adaptive networks where
communication delays exist between nodes. In particular, we investigate the
diffusion Least-Mean- Square (LMS) strategy where delayed intermediate
estimates (due to the communication channels) are employed during the
combination step. One important question is: Do the delays affect the stability
condition and performance? To answer this question, we conduct a detailed
performance analysis in the mean and in the mean-square-error sense of the
diffusion LMS with delayed estimates. Stability conditions, transient and
steady-state mean-square-deviation (MSD) expressions are provided. One of the
main findings is that diffusion LMS with delays can still converge under the
same step-sizes condition of the diffusion LMS without delays. Finally,
simulation results illustrate the theoretical findings.
