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In a peer-to-peer grid, tasks are distributed to grid nodes in a decentralized fashion. 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.