Adaptive video streaming improves users' quality of experience (QoE), while using the network efficiently. In the last few years, adaptive video streaming has seen widespread adoption and has attracted significant research effort. We study a dynamic system of random arrivals and departures for different classes of users using the adaptive streaming industry standard DASH (Dynamic Adaptive Streaming over HTTP). Using a Markov chain based analysis, we compute the user QoE metrics: probability of starvation, prefetching delay, average video quality and switching rate. We validate our model by simulations, which show a very close match. Our study of the playout buffer is based on client adaptation scheme, which makes efficient use of the network while improving users' QoE. We prove that for buffer-based variants, the average video bit-rate matches the average channel rate. Hence, we would see quality switches whenever the average channel rate does not match the available video bit rates. We give a sufficient condition for setting the playout buffer threshold to ensure that quality switches only between adjacent quality levels.