This paper proposes a new model that generalizes the consecutive k-out-of-r-from-n:F system to the multi-state case. In this model (linear multi-state sliding window system), the system consists of n linearly ordered multi-state elements. Each element can have various states: from complete-failure up to perfect-functioning. A performance rate is associated with each state. The sliding window system fails if the sum of the performance rates of any r consecutive multi-state elements is lower than a minimum allowable level. An algorithm for system reliability evaluation is based on an extended universal moment-generating function. Examples of system reliability evaluation are presented.