This paper presents mathematical formulations and methods to predict the effect of forced-flight variance reduction on Monte Carlo tally variance and calculation time. This includes deducing biasing operators that are then used to construct a history-score probability density function (HSPDF), which represents all possible Monte Carlo random walks and gives the probability of a Monte Carlo history scoring in a tally from a particular phase-space position. The history-score moment equations (HSMEs), the statistical moments of the HSPDF, are then derived to calculate the statistical behavior of the Monte Carlo tally when forced-flight variance reduction is applied. In addition, the future-time equation (FTE) is derived to predict the Monte Carlo computational time as a result of applying forced-flight variance reduction. The solutions of the HSMEs and FTE can be used to predict Monte Carlo computational cost. This work also describes a discrete ordinates method to solve the forced-flight HSMEs and FTE. Several 1-D and 2-D test problems verify that the derivations are performed and implemented correctly.

本文介绍了数学公式和方法来预测强制飞行方差减少对蒙特卡洛计数差异和计算时间的影响。这包括推导偏置算子，然后将其用于构建历史分数概率密度函数（HSPDF），该函数表示所有可能的蒙特卡洛随机游动，并给出从特定相空间位置对记数进行蒙特卡洛历史评分的概率。然后，推导出历史分数矩方程（HSMEs），HSPDF的统计矩，以计算应用强制飞行方差减少时的蒙特卡洛计数法的统计行为。另外，由于应用了强制飞行方差减少，因此得出了未来时间方程（FTE）以预测蒙特卡洛的计算时间。HSME和FTE的解决方案可用于预测蒙特卡洛的计算成本。这项工作还描述了一种离散的纵坐标方法来解决强制飞行的HSME和FTE。几个一维和二维测试问题验证了推导是否正确执行和实现。