Swarm robotic engineering aims to design robust, expandable and flexible swarm robotic systems to fulfill well-defined global goals.The collective behaviors ought to be analyzable and predicable so that the designer can determine whether the goal is achieved or not.In this paper we propose a novel macroscopic mathematical model for multi-robot task allocation.Our model can predict the global behavior emerging from local rules to check whether the swarm does precisely what it is designed to do or not.Besides it also provides the maneuver to gain insight of the properties of the system and find relationship between swarm size and the model parameter which makes an impact on the system performance.Especially, our novel model has no limitation on the number of task types.Quasi-birth-and-death process model is employed to model the dynamics of the state of the whole swarm directly.We introduce matrix analytic method to swarm robotics for calculating the closed-form expression in steady state, then we find the statistic law that task allocation descriptor obeys.We conduct simulations with hundred of robots, and the experi-ments verify the correctness and reliability of the proposed model.%以预测由个体局部规则涌现的全局行为为目的，提出一种新的群集机器人任务分配宏观模型，提供了解系统性能与作为模型参数的群集规模之间关系的分析手段，尤其是该模型对于任务类型和群集规模都没有限制。通过拟生灭过程模型建立系统任务分配动态性的演化方程，首次引入矩阵分析法到群集机器人领域，求解了宏观模型的闭式稳态解，发现了系统任务分配在稳态服从的统计规律。进行了包含上百个机器人的仿真，其结果说明了模型预测与分析的正确性和可靠性。
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