Our purpose in this paper is to characterize buffer starvations for streaming services. The buffer is modeled as a FIFO queue with exponential service time and Poisson arrivals. When the buffer is empty, the service restarts after a certain amount of packets are prefetched. With this goal, we propose two approaches to obtain exact distribution of the number of buffer starvations, one of which is based on Ballot theorem, and the other uses recursive equations. The Ballot theorem approach gives an explicit result. We extend this approach to the scenario with a constant playback rate using Tàkacs Ballot theorem. The recursive approach, though not offering an explicit result, allows us to obtain the distribution of starvations with non-independent and identically distributed (i.i.d.) arrival process in which an ON/OFF bursty arrival process is considered. We further compute the starvation probability as a function of the amount of prefetched packets for a large number of files via a fluid analysis. Among many potential applications of starvation analysis, we show how to apply it to optimize objective quality of experience (QoE) of media streaming, by exploiting the tradeoff between startup/rebuffering delay and starvations.