In a peer-to-peer grid, tasks are distributed to grid nodes in a decentralized fashion. In this paper, we present an agent-based task handling mechanism in a peer-to-peer grid and then provide a macroscopic model to characterize the process of task handling. Our model consists of functional differential equations. Through case studies, (1) we show that our model is effective in characterizing the process of task handling; (2) we examine the effects of time delay, service time, etc. on the global performance of a peer-to-peer grid. Based on our model, we further simulate a complete process of task handling in a grid and show the main characteristics of a real grid.