Uncertain Random Renewal Reward Process With Application to Block Replacement Policy

Abstract

As a mixture of uncertain variable and random variable, uncertain random variable is an important tool to describe indeterminacy phenomena. In order to model the evolution of uncertain random phenomena, a concept of uncertain random process has been proposed, and an uncertain random renewal process has been designed as an example. This paper aims to propose a new type of uncertain random process, called uncertain random renewal reward process, in which the interarrival times and the rewards are assumed to be random variables and uncertain variables, respectively. The chance distribution of the renewal reward process is obtained, and the reward rate is derived. A renewal reward theorem is verified, which shows that the reward rate converges in distribution to an uncertain variable derived from the random interarrival times and the uncertain rewards. As an application, this paper also proposes an uncertain random block replacement problem and formulates an unconstrained optimization model by using the uncertain random renewal reward process.