THIS PAPER IS ELIGIBLE FOR THE STUDENT PAPER AWARD. The scheduling problem is a key class of optimization problems and has various kinds of applications both in practical and theoretical scenarios. In the scheduling problem, probabilistic analysis is a basic tool for investigating performance of scheduling algorithms, and therefore has been carried out by plenty amount of prior works. However, probabilistic analysis has several potential problems. For example, current research interest in the scheduling problem is limited to i.i.d. scenarios, due to its simplicity for analysis. This paper provides a new framework for probabilistic analysis in the scheduling problem and aims to deal with such problems. As a consequence, we obtain several theorems including a theoretical limit of scheduling problems which can be applied to general, non-i.i.d. probability distributions. Several information theoretic techniques, such as ¥emph{information-spectrum method}, turned out to be useful to prove our results. Since the scheduling problem has relations to many other research fields, our framework hopefully yields other interesting applications in the future.