Due to their ability to overcome many shortcomings associated with the contemporary client-server paradigm, Peer-to-Peer (P2P) networks have attracted phenomenal interests from researchers in both academia and industry. Interactive and multimedia streaming applications using P2P networks are, however, often prone to long startup delays, which disrupt the smooth playback and undermine users' perceived quality of service. In addition, P2P networks must be able to support a potential number of users while ensuring that the resources are efficiently utilized. In this paper, by addressing these shortcomings in the traditional P2P framework, we envision a novel scheme to effectively provide a Video-on-Demand (VoD) using P2P-based mesh overlay networks. The proposed scheme covers two main phases, namely requesting and scheduling modes. The former aims at dynamically selecting the required contents from the available peers. On the other hand, in the scheduling mode, the incoming requests are scheduled in a priority-based manner for minimizing the startup latency and sustaining the playback rate to an acceptable level. Computer simulations have been conducted to verify the effectiveness of the proposed scheme. The obtained results demonstrate the scalability of our envisioned scheme in addition to its capability to reduce the startup delay and provide a sustainable playback rate.