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We consider the online scheduling problem on a single machine with the assumption that all jobs have their processing times in [p,(1+α)p], where p>0 and α=(5-1)/2. All jobs arrive over time, and each job and its processing time become known at its arrival time. The jobs should be first processed on a single machine and then delivered by a vehicle to some customer. When the capacity of the vehicle is infinite, we provide an online algorithm with the best competitive ratio of (5+1)/2. When the capacity of the vehicle is finite, that is, the vehicle can deliver at most c jobs at a time, we provide another best possible online algorithm with the competitive ratio of (5+1)/2.