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We propose a mathematical model to minimize the expected download time of cloud assisted Peer-to-Peer video on demand services. First, we define a simple fluid model that quantifies the evolution of peers, which are grouped into different classes regarding the number of concurrent video downloads. Then, analytical expressions for the expected download time are obtained under steady state, via Little's law. The goal is to minimize the expected download time with limited storage capacity in cache nodes of the network, called super-peers. The nature of this combinatorial problem is similar to the Multi-Knapsack Problem (MKP): the number of copies must be chosen for each video stream, with storage capacity constraints. We resolve the problem with a greedy randomized technique. The performance of this co-operative system is compared with a traditional content delivery network. Finally, the new caching policy is tested in a real scenario. The results confirm that the swarm assisted peer-to-peer service is both more economical and suitable to address massive scenarios, whereas the performance of both systems is similar in small scale instances.