本论文主要研究的是一类特殊的最优停止模型——马氏链收益模型的Gittins 指数.我们首先回顾了I. Sonin(2008) 关于经典Gittins 指数(GI)推广为广义Gittins 指数(GGI) 的处理过程, 以及“KV 指数”“Whittle 指数”“广义Gittins 指数”的等价关系.然后通过一个具体的算例,论证了在贴现率恒为常数的特殊条件下, 广义的GGI 并非是经典GI 的推广, 且无论涉及的是具体的击中时, 还是一般的停时情形, 这种差异都是存在的. 其次, 我们推广构造新旧过程的停时关系,从理论上证明了GGI 和GI 的关系. 最后, 依据他们的关系重新修正了GGI 的定义, 使之确实成为GI 的推广,并证明在修正的GGI 定义下, 上述三个指数等价关系依然保持.

This paper is mainly on the Gittins Index of a pecificOSP model which is called Markov Reward Model.We reach back to the method of the generalization from GI to GI and the equivalence among the "KV index", "Whittle index" and "GGI" formulated by I. Sonin(2008). An Example has been proposed to clarify a fact that the generalization cannot return to the classical case under the condition $\beta(x)$ is a constant. The gap between GI and GGI always exists, and it have no concern with the form of the stopping time. Then we construct the relationship between the stopping times of thenew and old models. And we get the relationship between GI and GGI. Finally, we try our best to modify the definition of GGI to make the generalization reasonable and prove that the equality of the three indices GGI, GW, and GKV is still preserved.