First round hitting probability and firing delay time for tank with shooting gate

The modern tank is equipped with shooting gate to improve the first round hitting probability (FRHP) at a cost of the shooting delay time which is called the firing delay time (FDT). The relation between FRHP and FDT is studied for discrete-time and continuous-state system. The 2-dimension sample space is partitioned into many small rectangle griddings. Among these griddings, the faster transition probability matrix (FTPM) and slower transition probability matrix (STPM) are introduced to estimate transition probability matrix which is used to compute the approximative distribution of FDT and the expression of FRHP. The approximation error can be reduced by resizing the gridding. Then FDT is proved to be Phase-Type (PH) distribution, so are its upper and lower bound which arbitrarily trend to accord. At the same time, FRHP is expressed in terms of mean square variance and correlation coefficient of shooting error besides the size of shooting gate, and FDT also lies on the same parameters. At last, some numerical examples demonstrate their relation.