以特征函数为传递参数的CF-GERT及其矩阵法求解

本文构建了以特征函数和传递概率为传递函数的CF-GERT(characteristic function based GERT)模型,利用特征函数的性质,证明了CF-GERT网络串联结构、并联结构、自环结构的传递关系与信号流图等价传递参数计算完全相同,故可借鉴信号流图理论求解CF-GERT模型.考虑以梅森公式为基础的经典GERT解析算法需要分析复杂的网络拓扑结构,提出了CF-GERT网络的矩阵式表征方法,进而设计了CF-GERT的矩阵式求解算法,推导了期望、方差、等价特征函数的计算公式.若等价特征函数绝对可积,则利用傅里叶逆变换推导概率密度函数;否则,运用Fang等提出的COS方法推导.最后用两个案例说明了所提方法的有效性.

CF-GERT model conveying characteristic function and its matrix solution

This paper establishes a new GERT model named CF-GERT model, whose w-functions are products of the corresponding probability and characteristic function. The transform properties in the series structure, parallel structure and loop structure of CF-GERT are the same as the signal flow graph, flow graph is hence used to solve CF-GERT. Furthermore considering that it is tremendously difficult to analyze the topological characteristics of the CF-GERT network using traditional analytical algorithm for GERT which is based on Mason Formula, this paper proposes the matrix representation of CF-GERT network and then constructs the matrix solution algorithm. Then the equivalent characteristic function, mean and variance can be derived according to the properties of characteristic function. Finally, the probability density function can be derived by inversion Fourier transform if the equivalent characteristic function is absolutely integral function, otherwise COS method proposed by Fang will be used to obtain the probability density function. Two illustrative cases demonstrate the availability of the proposed method.