Timing channels remain a difficult and important problem for information security. Recent work introduced predictive mitigation, a new way to mitigating leakage through timing channels; this mechanism works by predicting timing from past behavior, and then enforcing the predictions. This paper generalizes predictive mitigation to a larger and important class of systems: systems that receive input requests from multiple clients and deliver responses. The new insight is that timing predictions may be a function of any public information, rather than being a function simply of output events. Based on this insight, a more general mechanism and theory of predictive mitigation becomes possible. The result is that bounds on timing leakage can be tightened, achieving asymptotically logarithmic leakage under reasonable assumptions. By applying it to web applications, the generalized predictive mitigation mechanism is shown to be effective in practice.