This paper considers a two-stage stochastic programming problem for airport runway scheduling under the uncertainty of arrival time on a single runway. The goal of airport runway scheduling is to schedule a set of aircrafts in a given time horizon and minimize a corresponding objective while satisfying separation requirements as well as other practical constraints. In order to boost runway elasticity and throughout, a mess of unpredictable factors, such as weather, pilot behavior and airport surface traffic, should be take into consideration by airport regulator. The arrival scheduling problem at airport can be decomposed into sequential decision problem, where the first stage determines the sequence of aircraft weight class, while the individual flight is assigned to positions in the weight class sequence in the second stage. The main mission of this work is to identify an optimal schedule involving the arrival time of flight is stochastic under different scenarios. A stochastic aircraft landing problem (SALP) formulation based on time-dependent traveling salesman problem (TDTSP) is proposed. Then a sample average approximation (SAA) algorithm is developed to solve this stochastic programming and the efficacy is verified by experimental result.