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This paper considers two single-machine scheduling problems with a new type of aging effect, which is dominated by the processing speed of the machine. During the whole scheduling horizon, the machine is subject to an optional maintenance, and the duration of the maintenance depends on the length of the uptime before it. The objective is to schedule all jobs and find the location of the maintenance so as to minimize the makespan or the total completion times. The two problems are proved to be NP-complete, and two dynamic programming algorithms are proposed to solve the problems. We analyze the computation complexity of the algorithms, and show that the problems under study are solvable in polynomial time if the processing loads of all jobs are uniformly bounded. [ABSTRACT FROM AUTHOR]