At the entropy measures and their generalization path, in the direction of statistics and information science, recently, Sunoj & Linu (2012) proposed the cumulative residual Renyi's entropy of order α and its dynamic version and studied its main properties. In this paper, we introduce an alternative measure of cumulative residual Renyi's entropy (CRRE) of order α which, unlike the mentioned one, is positive for all distributions and all values of α. We also consider its dynamic version and study their main properties in the context of reliability theory and stochastic orders. We give an estimator of the proposed CRRE and investigate its exact and asymptotic distribution. Numerous examples illustrating the theory are also given.