Cloud-based systems often face the problem of dispatching a stream of jobs to run on cloud servers in an online manner. Each job has a size that defines the resource demand for running the job. Each job is assigned to run on a cloud server upon its arrival and the job departs after it completes. The departure time of a job, however, is not known at the time of its arrival. Each cloud server has a fixed resource capacity and the total resource demand of all the jobs running on a server cannot exceed its capacity at all times. The objective of job dispatching is to minimize the total cost of the servers used, where the cost of renting each cloud server is proportional to its running hours by "pay-as-you-go" billing. The above job dispatching problem can be modeled as a variant of the Dynamic Bin Packing (DBP) problem known as MinUsageTime DBP. In this paper, we develop new approaches to the competitive analysis of the commonly used First Fit packing algorithm for the MinUsageTime DBP problem, and establish a new upper bound of μ+4 on the competitive ratio of First Fit packing, where μ is the ratio of the maximum job duration to the minimum job duration. Our result significantly reduces the gap between the upper and lower bounds for the MinUsageTime DBP problem to a constant value independent of μ, and shows that First Fit packing is near optimal for MinUsageTime DBP.