This is a study on robust airline scheduling where flight block times are considered in two parts as cruise time and non-cruise time. Cruise times are controllable and non-cruise times are random variables. Cruise time controllability is used together with idle time insertion to handle uncertainty to guarantee passenger connection service levels while ensuring minimum costs. The nonlinearity of these cost functions are handled by representing them via second order conic inequalities. The uncertainty in non-cruise times are modeled through chance constraints on passenger connection service levels, which are expressed using second order conic inequalities using the closed form equations. Congestion levels of origin and destination airports are used to decide variability for each flight. Computational study shows exact solutions can be obtained by commercial solvers in seconds for a single hub schedule and in minutes for a 4-hub daily schedule of a major US carrie