Uncertain renewal process with general rewards

The renewal reward process describes the cumulative reward related to the renewals of a system. Usually, the rewards at the renewal times are assumed to be positive. Within the framework of uncertainty theory, this assumption is removed in this article such that the rewards can take both positive and negative values. The uncertainty distribution is derived, and the renewal reward theorems are proved for the general uncertain renewal reward process. In addition, the first hitting time that such a process reaches a pre-set level is investigated.