In this paper, we consider a weighted- $r$ -within-consecutive- $ k$ -out-of- $n:F$ system. The weighted system has in general $n$ components, each one having a positive integer weight ${w_i}, i= 1,2, \ldots,n$ . The weighted- $r$ -within-consecutive- $ k$ -out-of- $n:F$ system fails if and only if the total weight of failed components among $k$ consecutive components is at least $r$ . We introduce a binomial-type weighted scan statistic and study the reliability, the Birnbaum, improvement potential importance and Bayesian reliability importance of the system taken into consideration. We develop an explicit closed-form formula for the evaluation of reliability and reliability importance measures of a weighted- $r$ -within-consecutive- $ k$ -out-of- $n:F$ system and demonstrate the results numerically. We present a study showing the effectiveness of the method in terms of CPU time.