Wafer quality issues are becoming essential concerns in semiconductor manufacturing industry. It is becoming increasingly important for fab managers to raise the wafer quality level. Quality variation across wafers and wafer lots is also recognized as of vital importance. Wafer waiting times, which occur between consecutive wafer processing steps, are critical for the quality and quality variation of wafers. To resolve these quality issues, we consider waiting time constraints and variation in a flowshop. To follow the actual operating features of the fab, we define a two-machine flowshop with jobs that can skip the first process step and are ready to enter the second step from the beginning of scheduling. This research thus examines a machine scheduling problem that minimizes the variation in job waiting times in a generalized two-machine flowshop with skipping jobs and waiting time constraints. The mathematical properties of the problem such as the dominance properties and feasibility conditions are vigorously analyzed. These analyses provide profound insights into reduction of the search space in the solution procedure. We also observe that the derived properties are intuitively consistent with the well-known principles of queueing theory. From these, we develop efficient approximation algorithms and present their computational performance.