We consider a peer-to-peer (P2P) network with two phases: the transient period and the stationary period. In the transient period, bursting demand causes a rapid increase in the number of servers, and users (customers) experience significant delays due to the lack of peers that can share the file. So, companies that would utilize P2P networks should seriously consider the transient period as the quality of service can be significantly degraded during that period. To understand the transient period and extract more explicit intuition on it in a P2P scheme, we build a simplified model and derive the closed form expressions and numerical results for performance measures. We also discuss the waiting time of users and the transient period when a file comprises multiple chunks.