To determine how soon a TWC will have useful sea-level data after an earthquake, we calculated tsunami travel times (TTTs) from likely tsunami sources, i.e., thrust faults at subduction zones, to both coastal and deep-ocean sea level sensors monitored by the Pacific Tsunami Warning Center (PTWC). For this purpose we geographically sampled the axes of deep-sea trenches with a geodesic grid to ensure equal spacing of simulated sources. We then calculated global TTT grids for each of these sources and sampled each grid at the locations of coastal and deep-ocean sea-level gauges currently monitored by the Pacific Tsunami Warning Center (PTWC). We then added each gauge's transmission interval to the TTT values sampled at the gauges to determine the maximum time required for one, two, and three gauges to detect and transmit information about a tsunami generated by these sources. We also analyzed the network available to PTWC in 2005, showing how detection times have improved since then, and a hypothetical network in which all coastlines and oceans are saturated with sensors, showing that further improvements are still possible. We also used these data to quantify how the network can be compromised by sensor outages, identify those tsunami sources most in need of additional sea-level sensors for tsunami detection, and determine where future sensors should be located. As installing one deep-ocean sensor costs about 10 times as much as installing a coastal gauge, we also determine which sensor is more cost-effective for filling these network gaps. These analyses show that for global tsunami hazard mitigation the installation of about 100 additional carefully-selected coastal sea-level gauges could greatly improve the speed of tsunami detection and characterization.