We propose the notion of weak simulation quasimetric as the quantitative counterpart of weak simulation for probabilistic processes. This is an asymmetric variant of the weak bisimulation metric of Desharnais et al. which maintains most of the properties of the original definition. However, our asymmetric version is particularly suitable to reason on protocols where the systems under consideration are not approximately equivalent. As a main application, we adopt our simulation theory in a simple probabilistic timed process calculus to derive an algebraic theory to evaluate the performances of gossip protocols.