© 2018, Springer Nature Switzerland AG. In this paper we study an information flow security property for systems specified as terms of a quantitative process algebra, namely Performance Evaluation Process Algebra (PEPA). Intuitively, we propose a quantitative extension of the Non-Interference property used to secure systems from the functional point view by assuming that the observers are able to measure also the timing properties of the system, e.g., the response time or the throughput. We introduce the notion of Persistent Stochastic Non-Interference (PSNI) and provide two characterizations of it: one based on a bisimulation-like equivalence relation inducing a lumping on the underlying Markov chain, and another one based on unwinding conditions which demand properties of individual actions. These two different characterizations naturally lead to efficient methods for the verification and construction of secure systems. A decision algorithm for PSNI is presented and an application of PSNI to a queueing system is discussed.