  Asynchronous optimization algorithms are at the core of modern machine
learning and resource allocation systems. However, most convergence results
consider bounded information delays and several important algorithms lack
guarantees when they operate under total asynchrony. In this paper, we derive
explicit convergence rates for the proximal incremental aggregated gradient
(PIAG) and the asynchronous block-coordinate descent (Async-BCD) methods under
a specific model of total asynchrony, and show that the derived rates are
order-optimal. The convergence bounds provide an insightful understanding of
how the growth rate of the delays deteriorates the convergence times of the
algorithms. Our theoretical findings are demonstrated by a numerical example.
