The service capacities of a source peer at different times in a peer-to-peer (P2P) network exhibit temporal correlation. Unfortunately, there is no analytical result which clearly characterizes the expected download time from a source peer with stochastic and time-varying service capacity. The main contribution of this paper is to analyze the expected file download time in P2P networks with stochastic and time-varying service capacities. The service capacity of a source peer is treated as a stochastic process. Analytical results which characterize the expected download time from a source peer with stochastic and time-varying service capacity are derived for the autoregressive model of order 1. Simulation results are presented to validate our analytical results. Numerical data are given to show the impact of the degree of correlation and the strength of noise on the expected file download time. For any chunk allocation method, an analytical result of the expected parallel download time from a source peer with stochastic and time-varying service capacity is derived. It is shown that the algorithm which chooses chunk sizes proportional to the expected service capacities has better performance than the algorithm which chooses equal chunk sizes. It is also shown that multiple source peers do reduce the parallel download time significantly. Analyzing the file download time with stochastic and time-varying service capacities.Treating the service capacity of a source peer as a stochastic process.Validating analytical results by simulations.